# Calcolo Differenziale e Topologia in R^n > Una lezione universitaria completa sul calcolo in più variabili. Si parte dalla struttura euclidea di R^n: prodotto scalare, norma, Cauchy-Schwarz e disuguaglianza triangolare. Seguono la topologia (palle, punti interni, esterni e di frontiera, aperti, chiusi, chiusura, accumulazione) e le successioni, con completezza, Bolzano-Weierstrass e Heine-Borel. Poi limiti e continuità, retroimmagini di aperti, valori intermedi, Weierstrass e Heine-Cantor. Infine derivate parziali e direzionali, gradiente, differenziale e piano tangente, Jacobiana, Hessiana e Schwarz, regola della catena, Taylor al secondo ordine e classificazione dei punti critici con autovalori e criterio di Sylvester. - Canonical watch page: [Calcolo Differenziale e Topologia in R^n](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Mathematics - Published: 2026-10-29T22:02:25.532Z - Updated: 2026-10-29T22:02:25.532Z - Duration: PT881S (14 minutes 41 seconds) - Chapters: 5 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M64SNV1GG159WKTCK2FEM6WG/0/dark/master.m3u8) - Embed: [Player](https://academa.ai/embed/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M64SNV1GG159WKTCK2FEM6WG/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M64SNV1GG159WKTCK2FEM6WG/0/dark/poster.jpg) ## Description Lezione universitaria su R^n: norma, topologia, compattezza, continuità, gradiente, differenziale, Jacobiana, Hessiana e classificazione dei punti critici. ## Chapters - [00:00–03:0.974 · Lo spazio R^n e la sua topologia](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=0) - [03:0.974–05:36.063 · Successioni e compattezza](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=180.974125) - [05:36.063–07:50.439 · Continuità e teoremi fondamentali](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.0633333333334) - [07:50.439–10:54.287 · Derivate, differenziale e Jacobiana](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=470.43908333333343) - [10:54.287–14:41 · Hessiana e punti critici](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=654.2866666666667) ## Transcript ### [00:00 · Lo spazio R^n e la sua topologia](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=0) Benvenuti. In questa lezione costruiamo il calcolo differenziale per funzioni di più variabili, e prima ancora la topologia dello spazio in cui vivono. Ecco dove vogliamo arrivare. Questa è la matrice Hessiana, costruita con le derivate seconde. Alla fine sapremo leggere nei suoi autovalori se un punto critico è un minimo, un massimo o un punto di sella. Per arrivarci servono distanze, intorni, limiti e derivate. Un punto di erre enne è una enne-upla ordinata di numeri reali: le sue coordinate. Nel piano, con enne uguale a due, lo disegniamo come una freccia che parte dall'origine. Il prodotto scalare moltiplica le coordinate corrispondenti e somma i risultati. Da esso nasce la norma euclidea, la radice del prodotto scalare di x con sé stesso: per Pitagora, è proprio la lunghezza della freccia. La disuguaglianza di Cauchy-Schwarz afferma che il prodotto scalare, in valore assoluto, non supera mai il prodotto delle norme. L'uguaglianza vale solo per vettori paralleli. Da qui segue la disuguaglianza triangolare. Sviluppiamo il quadrato della norma della somma: compare il doppio prodotto scalare, che Cauchy-Schwarz maggiora con il doppio prodotto delle norme. Dunque la norma della somma non supera la somma delle norme. Geometricamente: mettiamo y in coda a x, e il terzo lato del triangolo non è mai più lungo degli altri due insieme. Per parlare di vicinanza servono gli intorni. La palla aperta di centro x zero e raggio erre contiene i punti che distano da x zero meno di erre. Nella palla chiusa ammettiamo anche la distanza uguale a erre. Ora fissiamo un insieme S, il disco azzurro. Un punto è interno a S se attorno a lui esiste una palla tutta contenuta in S, come quella verde. È esterno se esiste una palla tutta fuori da S, cioè contenuta nel complementare, come quella rossa. Infine è di frontiera se ogni palla centrata in lui, per quanto piccola, contiene sia punti di S sia punti del complementare. Il bordo del disco è fatto proprio di questi punti. Un insieme è aperto se tutti i suoi punti sono interni, come il disco senza bordo. È chiuso se il suo complementare è aperto, o equivalentemente se contiene tutta la sua frontiera. Aperto e chiuso non sono contrari. Aggiungiamo al disco questo punto rosso, lontano da tutto: l'insieme ottenuto non è né aperto né chiuso. La chiusura di S è S unito alla sua frontiera: il più piccolo chiuso che lo contiene. Qui significa aggiungere la circonferenza. Un punto è di accumulazione se ogni sua palla contiene punti di S diversi da lui: questo punto del bordo lo è, pur non appartenendo al disco aperto. Il punto rosso invece è isolato: una piccola palla lo separa dal resto. ### [03:0.974 · Successioni e compattezza](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=180.974125) Una successione in erre enne è una sequenza di punti, indicizzata dai numeri naturali. Diciamo che converge al punto x se la distanza fra x k e x tende a zero. In altre parole: fissato un raggio epsilon, da un certo indice in poi tutti i punti cadono nella palla di raggio epsilon attorno al limite. Se stringiamo epsilon l'indice cresce, ma esiste sempre. Poiché la norma controlla ogni coordinata, la convergenza in erre enne equivale alla convergenza di ciascuna delle enne successioni delle coordinate. Una successione è di Cauchy se i suoi termini diventano arbitrariamente vicini fra loro, senza nominare alcun limite. Ogni successione convergente è di Cauchy; il viceversa si chiama completezza. Erre enne è completo: ogni successione di Cauchy converge. Si dimostra coordinata per coordinata, perché ogni componente è di Cauchy in erre, e la retta reale è completa. Un insieme è limitato se sta dentro una palla abbastanza grande. Prendiamo ora una successione limitata qualsiasi: i suoi punti possono saltare avanti e indietro senza convergere. Il teorema di Bolzano-Weierstrass garantisce però che possiamo estrarre una sottosuccessione convergente. Qui basta tenere i termini di indice pari, quelli gialli, che si accumulano attorno a un punto. L'idea della prova: si dimezza ripetutamente il quadrato che contiene la successione, scegliendo ogni volta una parte con infiniti termini. I quadrati annidati si stringono su un unico punto, che esiste perché erre enne è completo. Questo suggerisce una definizione. Un insieme K è compatto se da ogni successione in K si estrae una sottosuccessione che converge, e converge a un punto di K stesso. In erre enne la compattezza ha una caratterizzazione concreta, il teorema di Heine-Borel: compatto significa chiuso e limitato. Limitato serve per Bolzano-Weierstrass; chiuso serve perché il limite non sfugga dall'insieme. Vediamo tre casi. Il disco chiuso è chiuso e limitato, quindi compatto: ogni successione al suo interno ha una sottosuccessione che converge dentro il disco. Nel disco aperto, questa successione si avvicina al bordo: converge, ma a un punto che non appartiene all'insieme. Manca la chiusura. Nella striscia illimitata, invece, i punti scappano verso destra e nessuna sottosuccessione può convergere. Manca la limitatezza. ### [05:36.063 · Continuità e teoremi fondamentali](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.0633333333334) Passiamo alle funzioni da erre enne a erre emme. La definizione di limite è quella del caso reale, con la norma al posto del valore assoluto. Per ogni palla di raggio epsilon attorno al limite elle esiste una palla di raggio delta attorno a x zero la cui immagine, escluso al più x zero, cade dentro la prima. Se stringiamo epsilon dobbiamo stringere anche delta, ma una scelta adatta esiste sempre. La funzione è continua in x zero se il limite coincide con il valore in x zero. Attenzione: in più variabili ci si avvicina a x zero lungo infinite rette e curve, e il limite deve essere lo stesso per tutte. Esiste un modo di dire la continuità senza epsilon e delta. Una funzione è continua se e solo se la retroimmagine di ogni aperto è aperta, o equivalentemente se la retroimmagine di ogni chiuso è chiusa. Questo è uno strumento pratico. Prendiamo f uguale a x quadro più y quadro, che è continua. La corona dove f sta strettamente fra uno e quattro è la retroimmagine di un intervallo aperto: dunque è aperta, senza alcun conto. Allo stesso modo il disco dove f è minore o uguale a uno è la retroimmagine dell'intervallo chiuso da zero a uno, ed è quindi chiuso. Tre teoremi reggono tutta la teoria. Il primo è il teorema dei valori intermedi: su un insieme connesso per archi, due punti qualsiasi a e b sono uniti da un cammino continuo gamma, tutto dentro l'insieme. Lungo il cammino, f composta con gamma è una funzione continua di una sola variabile, e il teorema classico in erre fornisce un istante in cui assume il valore c. Il secondo è il teorema di Weierstrass: una funzione continua su un compatto ha massimo e minimo assoluti. Da una successione di valori che tende all'estremo superiore, la compattezza estrae una sottosuccessione convergente, e la continuità fa il resto. Il terzo è il teorema di Heine-Cantor. La continuità uniforme chiede che lo stesso delta funzioni in tutti i punti insieme; su un compatto questo è automatico. ### [07:50.439 · Derivate, differenziale e Jacobiana](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=470.43908333333343) Veniamo alle derivate. Prendiamo il paraboloide zeta uguale a x quadro più y quadro, e un punto P sul suo grafico. La derivata parziale rispetto a x i congela tutte le altre variabili: è la derivata ordinaria lungo un asse. Geometricamente è la pendenza della curva ottenuta tagliando il grafico con un piano verticale. Più in generale, scelto un vettore unitario v, la derivata direzionale misura come varia f muovendosi lungo v. Le derivate parziali sono il caso dei vettori della base canonica. Il vettore delle derivate parziali si chiama gradiente, nabla f. I punti in cui il gradiente si annulla si dicono stazionari, o critici: sono i candidati a massimi e minimi. Diciamo che f è differenziabile in x zero se il suo incremento è uguale a una parte lineare in h, più un errore che va a zero più velocemente della norma di h. La parte lineare definisce il piano tangente: il piano che meglio approssima il grafico vicino a P, e contiene le due tangenti di prima. Quando enne supera due si parla di iperpiano tangente. Se f è differenziabile, prendiamo h uguale a t per v e dividiamo per t. Il limite dà la formula fondamentale: la derivata direzionale è il prodotto scalare fra gradiente e v. Per Cauchy-Schwarz questo prodotto è massimo quando v è parallelo al gradiente. Il gradiente indica quindi la direzione di massima crescita, e la sua norma è la pendenza massima. Infine, la differenziabilità implica la continuità: l'incremento di f tende a zero insieme a h. Attenzione: in più variabili, l'esistenza di tutte le derivate parziali non garantisce nemmeno la continuità. Guardiamo questa funzione: sugli assi vale zero, quindi le due derivate parziali nell'origine esistono e sono nulle. Ma lungo la retta y uguale a emme x vale costantemente emme fratto uno più emme quadro: un mezzo sulla bisettrice, meno un mezzo sull'altra. Il limite nell'origine non esiste, e f non è continua. Il teorema del differenziale totale fornisce una condizione sufficiente: se le derivate parziali esistono vicino a x zero e sono continue in x zero, allora f è differenziabile. Ogni funzione di classe C uno è dunque differenziabile e continua. Per un campo vettoriale da erre enne a erre emme, ogni componente ha il suo gradiente. Disposti come righe, formano la matrice Jacobiana, con emme righe ed enne colonne. Per esempio, per questa f del piano in sé, la Jacobiana nel punto uno, uno è la matrice due, meno uno, uno, uno. Il suo significato è questo: vicino al punto, f si comporta come l'applicazione lineare data dalla Jacobiana. Guardate come questa matrice deforma il quadrato verde: è ciò che f fa, in prima approssimazione, a un piccolo intorno. ### [10:54.287 · Hessiana e punti critici](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=654.2866666666667) Derivando di nuovo le derivate parziali otteniamo le derivate seconde. Raccolte in una matrice enne per enne, formano la Hessiana. Proviamo con x al cubo per y, più y quadro. Le derivate prime sono tre x quadro y, e x al cubo più due y. Ora deriviamo f x rispetto a y, e f y rispetto a x: in entrambi i casi otteniamo tre x quadro. Non è una coincidenza. È il teorema di Schwarz: se le derivate miste sono continue, l'ordine di derivazione non conta. La Hessiana di una funzione di classe C due è quindi simmetrica, e avrà autovalori reali. Come si deriva una composizione? Seguiamo una curva gi di t nel piano, mentre f ha queste curve di livello. La derivata di f lungo la curva è il gradiente di f nel punto, scalare la velocità della curva. Il caso generale è altrettanto pulito: la Jacobiana della composizione è il prodotto delle Jacobiane, calcolate nei punti giusti. Scritto per componenti, è una somma su tutte le variabili intermedie. Con la Hessiana possiamo scrivere lo sviluppo di Taylor al secondo ordine. Il valore, più il termine lineare del gradiente, più metà della forma quadratica della Hessiana, più un resto di Peano, piccolo rispetto al quadrato della norma di h. Se f è di classe C due vale anche la forma di Lagrange: niente resto, ma la Hessiana è calcolata in un punto xi intermedio sul segmento fra x zero e x zero più h. Ora il punto chiave. In un punto stazionario il gradiente è nullo, e la differenza fra f di x zero più h e f di x zero è governata dalla forma quadratica q. Poiché la Hessiana è simmetrica, il teorema spettrale la diagonalizza in una base ortonormale di autovettori. In quelle coordinate la forma quadratica diventa una somma di autovalori per quadrati. Se tutti gli autovalori sono positivi, la forma è definita positiva e il punto è di minimo locale: il grafico è una coppa. Se sono tutti negativi, la forma è definita negativa: massimo locale, il grafico è una cupola. Se ci sono autovalori di segno opposto, la forma è indefinita e il punto è una sella: si sale lungo una direzione e si scende lungo l'altra. Se invece qualche autovalore è nullo, il secondo ordine non basta a decidere. Calcolare gli autovalori può essere faticoso. Il criterio di Sylvester usa invece i minori principali di testa: i determinanti dei blocchi quadrati in alto a sinistra della Hessiana. Se sono tutti positivi, la Hessiana è definita positiva: minimo locale. Se hanno segni alternati, a partire da un primo minore negativo, è definita negativa: massimo locale. In due variabili, un determinante negativo indica una sella. Chiudiamo con un esempio completo: f uguale a x al cubo meno tre x, più y quadro. Il grafico ha una valle e un passo di montagna. Il gradiente si annulla quando x quadro vale uno e y vale zero: due punti stazionari, uno zero e meno uno zero. La Hessiana è diagonale, con sei x e due. In uno zero i minori valgono sei e dodici, entrambi positivi: è un minimo locale, il fondo della valle. In meno uno zero il determinante vale meno dodici: è una sella. Lungo x la funzione ha un massimo, lungo y un minimo. Abbiamo percorso la strada intera: dalla norma e dagli intorni ai limiti, dalle derivate al piano tangente, fino a leggere nella Hessiana la forma di una funzione vicino ai suoi punti critici. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M64SNV1GG159WKTCK2FEM6WG/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Lo spazio R^n e la sua topologia](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=0) Span: 00:00–03:0.974 (0s–180.974125s). #### Objects - b\_fr: a Circle \[yellow\] drawn in blob (center=(3.8, 2.0), radius=0.45) - b\_in: a Circle \[green\] drawn in blob (center=(2.0, 2.3), radius=0.45) - b\_out: a Circle \[red\] drawn in blob (center=(5.0, 3.0), radius=0.45) - ball: a Math \[text\] that says "$B\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \< r\}$" - blob: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)) - card: a Title that says "Analisi Matematica 2 — Calcolo Differenziale e Topologia in R^n" - cball: a Math \[text\] that says "$overline(B)\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \<= r\}$" - cs: a Math \[text\] that says "$abs(x dot y) \<= norm(x) thin norm(y)$" - disk: a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True) - dot\_def: a Math \[text\] that says "$x dot y = sum\_(i=1)^n x\_i y\_i$" - goal: a Math \[text\] that says "$H\_f (x\_0) = mat(f\_(x x), f\_(x y); f\_(y x), f\_(y y))$" - goal\_note: a Text \[text\] that says "Il segno della forma quadratica associata distingue minimi, massimi e punti di sella." - head\_balls: a Heading that says "Palle e tipi di punto" - head\_goal: a Heading that says "Dove vogliamo arrivare" - head\_rn: a Heading that says "La struttura euclidea di $RR^n$" - head\_sets: a Heading that says "Aperti, chiusi, chiusura" - iso: a Point \[red\] labelled "x\_4" drawn in blob (location=(5.2, 0.8)) - iso\_ball: a Circle \[red\] drawn in blob (center=(5.2, 0.8), radius=0.4) - norm\_def: a Math \[text\] that says "$norm(x) = sqrt(x dot x)$" - origin: a Point \[gray\] labelled "0" drawn in plane - p\_fr: a Point \[yellow\] labelled "x\_3" drawn in blob (location=(3.8, 2.0)) - p\_in: a Point \[green\] labelled "x\_1" drawn in blob (location=(2.0, 2.3)) - p\_out: a Point \[red\] labelled "x\_2" drawn in blob (location=(5.0, 3.0)) - plane: a Figure (x\_range=(-0.5, 4.5), y\_range=(-0.5, 3.5)) - props: a Table \[text\] that says "nozione definizione aperto ogni punto di $S$ è interno chiuso $S^c$ è aperto chiusura $overline(S) = S union partial S$ accumulazione punti di $S$ diversi da $x\_0$ in ogni $B\_r (x\_0)$ isolato $B\_r (x\_0) inter S = \{x\_0\}$ per qualche $r$" (rows=(('nozione', 'definizione'), ('aperto', 'ogni punto di $S$ è in…, header=True) - rim: a Circle \[yellow\] drawn in blob (center=(2.4, 2.0), radius=1.4) - sq: a Math \[text\] that says "$norm(x + y)^2 = norm(x)^2 + 2 x dot y + norm(y)^2$" - tri: a Math \[text\] that says "$norm(x + y) \<= norm(x) + norm(y)$" - tuple\_def: a Math \[text\] that says "$x = (x\_1, dots, x\_n) in RR^n$" - types: a Table \[text\] that says "punto condizione interno esiste $B\_r (x\_0) subset S$ esterno esiste $B\_r (x\_0) subset S^c$ di frontiera ogni $B\_r (x\_0)$ incontra $S$ e $S^c$" (rows=(('punto', 'condizione'), ('interno', 'esiste $B\_r (x\_0) subset…, header=True) - vs: a Vector \[red\] labelled "x + y" drawn in plane (end=(4.0, 3.0)) - vx: a Vector \[blue\] labelled "x" drawn in plane (end=(3.0, 1.0)) - vy: a Vector \[green\] labelled "y" drawn in plane (start=(3.0, 1.0), end=(4.0, 3.0)) #### Beats ##### [00:00](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=0) Narration: Benvenuti. In questa lezione costruiamo il calcolo differenziale per funzioni di più variabili, e prima ancora la topologia dello spazio in cui vivono. Board: Empty. Actions: - [00:00](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=0): card is shown on the screen, written out. - [00:1.842](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=1.8418165304268843): card: enter:write-left-to-right. - [00:9.41](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=9.41): card is hidden from the screen — left the board. ##### [00:10.01](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=10.01) Narration: Ecco dove vogliamo arrivare. Questa è la matrice Hessiana, costruita con le derivate seconde. Alla fine sapremo leggere nei suoi autovalori se un punto critico è un minimo, un massimo o un punto di sella. Per arrivarci servono distanze, intorni, limiti e derivate. Board: Unchanged from the preceding beat in this scene. Actions: - [00:10.01](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=10.01): head\_goal is shown on the screen, written out. - [00:12.68](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=12.68): goal is shown on the screen, written out. - [00:17.92](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=17.92): goal\_note is shown on the screen, written out. - [00:27.11](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=27.11): goal is hidden from the screen — left the board. - [00:27.11](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=27.11): goal\_note is hidden from the screen — left the board. - [00:27.11](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=27.11): head\_goal is hidden from the screen — left the board. ##### [00:28.31](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=28.310000000000002) Narration: Un punto di erre enne è una enne-upla ordinata di numeri reali: le sue coordinate. Nel piano, con enne uguale a due, lo disegniamo come una freccia che parte dall'origine. Board: Unchanged from the preceding beat in this scene. Actions: - [00:28.31](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=28.310000000000002): head\_rn is shown on the screen, written out. - [00:31.2](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=31.2): tuple\_def is shown on the screen, written out. - [00:34.467](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=34.467): plane is shown on the screen, written out. - [00:34.467](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=34.467): origin is shown on the screen, written out. - [00:37.24](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=37.239999999999995): vx is shown on the screen, written out. ##### [00:39.568](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=39.5675) Narration: Il prodotto scalare moltiplica le coordinate corrispondenti e somma i risultati. Da esso nasce la norma euclidea, la radice del prodotto scalare di x con sé stesso: per Pitagora, è proprio la lunghezza della freccia. Board: tuple\_def — a Math \[text\] that says "$x = (x\_1, dots, x\_n) in RR^n$"; plane — a Figure (x\_range=(-0.5, 4.5), y\_range=(-0.5, 3.5)); head\_rn — a Heading that says "La struttura euclidea di $RR^n$"; origin — a Point \[gray\] labelled "0" drawn in plane; vx — a Vector \[blue\] labelled "x" drawn in plane (end=(3.0, 1.0)) Actions: - [00:40](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=40.00000000000001): dot\_def is shown on the screen, written out. - [00:45.84](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=45.84000000000001): norm\_def is shown on the screen, written out. - [00:51.76](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=51.76000000000002): vx is indicated — a transient flash. ##### [00:53.845](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=53.845) Narration: La disuguaglianza di Cauchy-Schwarz afferma che il prodotto scalare, in valore assoluto, non supera mai il prodotto delle norme. L'uguaglianza vale solo per vettori paralleli. Board: tuple\_def — a Math \[text\] that says "$x = (x\_1, dots, x\_n) in RR^n$"; dot\_def — a Math \[text\] that says "$x dot y = sum\_(i=1)^n x\_i y\_i$"; norm\_def — a Math \[text\] that says "$norm(x) = sqrt(x dot x)$"; plane — a Figure (x\_range=(-0.5, 4.5), y\_range=(-0.5, 3.5)); head\_rn — a Heading that says "La struttura euclidea di $RR^n$"; origin — a Point \[gray\] labelled "0" drawn in plane; vx — a Vector \[blue\] labelled "x" drawn in plane (end=(3.0, 1.0)) Actions: - [00:56.04](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=56.04000000000001): cs is shown on the screen, written out. ##### [01:5.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=65.0225) Narration: Da qui segue la disuguaglianza triangolare. Sviluppiamo il quadrato della norma della somma: compare il doppio prodotto scalare, che Cauchy-Schwarz maggiora con il doppio prodotto delle norme. Board: tuple\_def — a Math \[text\] that says "$x = (x\_1, dots, x\_n) in RR^n$"; dot\_def — a Math \[text\] that says "$x dot y = sum\_(i=1)^n x\_i y\_i$"; norm\_def — a Math \[text\] that says "$norm(x) = sqrt(x dot x)$"; cs — a Math \[text\] that says "$abs(x dot y) \<= norm(x) thin norm(y)$"; plane — a Figure (x\_range=(-0.5, 4.5), y\_range=(-0.5, 3.5)); head\_rn — a Heading that says "La struttura euclidea di $RR^n$"; origin — a Point \[gray\] labelled "0" drawn in plane; vx — a Vector \[blue\] labelled "x" drawn in plane (end=(3.0, 1.0)) Actions: - [01:8.32](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=68.32000000000001): sq is shown on the screen, written out. - [01:11.52](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=71.52000000000001): sq (the "2 x dot y" part) is emphasized. - [01:16.297](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=76.2975): sq (the "2 x dot y" part) is no longer emphasized. ##### [01:16.897](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=76.8975) Narration: Dunque la norma della somma non supera la somma delle norme. Geometricamente: mettiamo y in coda a x, e il terzo lato del triangolo non è mai più lungo degli altri due insieme. Board: tuple\_def — a Math \[text\] that says "$x = (x\_1, dots, x\_n) in RR^n$"; dot\_def — a Math \[text\] that says "$x dot y = sum\_(i=1)^n x\_i y\_i$"; norm\_def — a Math \[text\] that says "$norm(x) = sqrt(x dot x)$"; cs — a Math \[text\] that says "$abs(x dot y) \<= norm(x) thin norm(y)$"; sq — a Math \[text\] that says "$norm(x + y)^2 = norm(x)^2 + 2 x dot y + norm(y)^2$"; plane — a Figure (x\_range=(-0.5, 4.5), y\_range=(-0.5, 3.5)); head\_rn — a Heading that says "La struttura euclidea di $RR^n$"; origin — a Point \[gray\] labelled "0" drawn in plane; vx — a Vector \[blue\] labelled "x" drawn in plane (end=(3.0, 1.0)) Actions: - [01:17.16](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=77.16000000000001): tri is shown on the screen, written out. - [01:23.28](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=83.28000000000002): vy is shown on the screen, written out. - [01:24.52](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=84.52000000000001): vs is shown on the screen, written out. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): cs is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): dot\_def is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): head\_rn is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): norm\_def is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): plane is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): origin is hidden from the screen — plane left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): vx is hidden from the screen — plane left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): vy is hidden from the screen — plane left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): vs is hidden from the screen — plane left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): sq is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): tri is hidden from the screen — left the board. - [01:28.515](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=88.51499999999999): tuple\_def is hidden from the screen — left the board. ##### [01:29.715](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=89.71499999999999) Narration: Per parlare di vicinanza servono gli intorni. La palla aperta di centro x zero e raggio erre contiene i punti che distano da x zero meno di erre. Nella palla chiusa ammettiamo anche la distanza uguale a erre. Board: Empty. Actions: - [01:29.715](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=89.71499999999999): head\_balls is shown on the screen, written out. - [01:32.08](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=92.08000000000003): blob is shown on the screen, written out. - [01:33.4](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=93.40000000000002): ball is shown on the screen, written out. - [01:33.4](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=93.40000000000002): b\_in is shown on the screen, written out. - [01:34.12](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=94.12000000000002): p\_in is shown on the screen, written out. - [01:40.24](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=100.24000000000002): cball is shown on the screen, written out. ##### [01:44.017](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=104.01749999999998) Narration: Ora fissiamo un insieme S, il disco azzurro. Un punto è interno a S se attorno a lui esiste una palla tutta contenuta in S, come quella verde. Board: ball — a Math \[text\] that says "$B\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \< r\}$"; cball — a Math \[text\] that says "$overline(B)\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \<= r\}$"; blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); head\_balls — a Heading that says "Palle e tipi di punto"; p\_in — a Point \[green\] labelled "x\_1" drawn in blob (location=(2.0, 2.3)); b\_in — a Circle \[green\] drawn in blob (center=(2.0, 2.3), radius=0.45) Actions: - [01:44.8](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=104.80000000000003): disk is shown on the screen, written out. - [01:45.4](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=105.40000000000003): types is shown on the screen, written out. - [01:48.28](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=108.28000000000003): types is shown on the screen, written out. - [01:53.16](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=113.16000000000004): b\_in is indicated — a transient flash. ##### [01:54.275](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=114.27499999999998) Narration: È esterno se esiste una palla tutta fuori da S, cioè contenuta nel complementare, come quella rossa. Board: ball — a Math \[text\] that says "$B\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \< r\}$"; cball — a Math \[text\] that says "$overline(B)\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \<= r\}$"; blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); head\_balls — a Heading that says "Palle e tipi di punto"; p\_in — a Point \[green\] labelled "x\_1" drawn in blob (location=(2.0, 2.3)); b\_in — a Circle \[green\] drawn in blob (center=(2.0, 2.3), radius=0.45); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True) Actions: - [01:54.68](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=114.68000000000004): p\_out is shown on the screen, written out. - [01:54.68](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=114.68000000000004): types is shown on the screen, written out. - [01:56.84](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=116.84000000000003): b\_out is shown on the screen, written out. ##### [02:1.35](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=121.34999999999998) Narration: Infine è di frontiera se ogni palla centrata in lui, per quanto piccola, contiene sia punti di S sia punti del complementare. Il bordo del disco è fatto proprio di questi punti. Board: ball — a Math \[text\] that says "$B\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \< r\}$"; cball — a Math \[text\] that says "$overline(B)\_r (x\_0) = \{x in RR^n : norm(x - x\_0) \<= r\}$"; blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); head\_balls — a Heading that says "Palle e tipi di punto"; p\_in — a Point \[green\] labelled "x\_1" drawn in blob (location=(2.0, 2.3)); b\_in — a Circle \[green\] drawn in blob (center=(2.0, 2.3), radius=0.45); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True); p\_out — a Point \[red\] labelled "x\_2" drawn in blob (location=(5.0, 3.0)); b\_out — a Circle \[red\] drawn in blob (center=(5.0, 3.0), radius=0.45) Actions: - [02:2.64](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=122.64000000000003): p\_fr is shown on the screen, written out. - [02:2.64](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=122.64000000000003): types is shown on the screen, written out. - [02:5.44](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=125.44000000000003): b\_fr is shown on the screen, written out. - [02:12.862](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=132.86249999999998): ball is hidden from the screen — left the board. - [02:12.862](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=132.86249999999998): cball is hidden from the screen — left the board. - [02:12.862](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=132.86249999999998): head\_balls is hidden from the screen — left the board. - [02:12.862](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=132.86249999999998): types is hidden from the screen — left the board. ##### [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997) Narration: Un insieme è aperto se tutti i suoi punti sono interni, come il disco senza bordo. È chiuso se il suo complementare è aperto, o equivalentemente se contiene tutta la sua frontiera. Board: blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); p\_in — a Point \[green\] labelled "x\_1" drawn in blob (location=(2.0, 2.3)); b\_in — a Circle \[green\] drawn in blob (center=(2.0, 2.3), radius=0.45); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True); p\_out — a Point \[red\] labelled "x\_2" drawn in blob (location=(5.0, 3.0)); b\_out — a Circle \[red\] drawn in blob (center=(5.0, 3.0), radius=0.45); p\_fr — a Point \[yellow\] labelled "x\_3" drawn in blob (location=(3.8, 2.0)); b\_fr — a Circle \[yellow\] drawn in blob (center=(3.8, 2.0), radius=0.45) Actions: - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): head\_sets is shown on the screen, written out. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): p\_in is hidden from the screen. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): b\_in is hidden from the screen. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): p\_out is hidden from the screen. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): b\_out is hidden from the screen. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): b\_fr is hidden from the screen. - [02:14.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=134.06249999999997): props is shown on the screen, written out. - [02:15.36](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=135.36000000000004): props is shown on the screen, written out. - [02:19.92](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=139.92000000000002): props is shown on the screen, written out. ##### [02:26.18](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=146.17999999999998) Narration: Aperto e chiuso non sono contrari. Aggiungiamo al disco questo punto rosso, lontano da tutto: l'insieme ottenuto non è né aperto né chiuso. Board: blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True); p\_fr — a Point \[yellow\] labelled "x\_3" drawn in blob (location=(3.8, 2.0)); head\_sets — a Heading that says "Aperti, chiusi, chiusura" Actions: - [02:30.72](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=150.72000000000003): iso is shown on the screen, written out. ##### [02:36.32](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=156.31999999999996) Narration: La chiusura di S è S unito alla sua frontiera: il più piccolo chiuso che lo contiene. Qui significa aggiungere la circonferenza. Board: blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True); p\_fr — a Point \[yellow\] labelled "x\_3" drawn in blob (location=(3.8, 2.0)); head\_sets — a Heading that says "Aperti, chiusi, chiusura"; iso — a Point \[red\] labelled "x\_4" drawn in blob (location=(5.2, 0.8)) Actions: - [02:36.96](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=156.96000000000004): props is shown on the screen, written out. - [02:44.08](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=164.08): rim is shown on the screen, written out. ##### [02:45.57](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=165.56999999999996) Narration: Un punto è di accumulazione se ogni sua palla contiene punti di S diversi da lui: questo punto del bordo lo è, pur non appartenendo al disco aperto. Il punto rosso invece è isolato: una piccola palla lo separa dal resto. Board: blob — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); disk — a Circle \[blue\] labelled "S" drawn in blob (center=(2.4, 2.0), radius=1.4, filled=True); p\_fr — a Point \[yellow\] labelled "x\_3" drawn in blob (location=(3.8, 2.0)); head\_sets — a Heading that says "Aperti, chiusi, chiusura"; iso — a Point \[red\] labelled "x\_4" drawn in blob (location=(5.2, 0.8)); rim — a Circle \[yellow\] drawn in blob (center=(2.4, 2.0), radius=1.4) Actions: - [02:46.72](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=166.72): props is shown on the screen, written out. - [02:51.96](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=171.96): p\_fr is indicated — a transient flash. - [02:55.48](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=175.48000000000002): iso\_ball is shown on the screen, written out. - [02:56.52](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=176.52): props is shown on the screen, written out. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): blob is hidden from the screen — left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): disk is hidden from the screen — blob left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): p\_fr is hidden from the screen — blob left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): iso is hidden from the screen — blob left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): rim is hidden from the screen — blob left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): iso\_ball is hidden from the screen — blob left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): head\_sets is hidden from the screen — left the board. - [02:59.932](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=179.9324583333333): props is hidden from the screen — left the board. ### Scene 2: [Successioni e compattezza](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=180.974125) Span: 03:0.974–05:36.063 (180.974125s–336.0633333333334s). #### Objects - bound: a Circle \[gray\] drawn in bw\_fig (radius=2.0) - bounded: a Math \[text\] that says "$S thin upright("limitato") \<=\> exists R : S subset B\_R (0)$" - bw\_fig: a Figure (x\_range=(-2.5, 2.5), y\_range=(-2.5, 2.5), aspect=(1.0, 1.0)) - bw\_thm: a Panel that says "Ogni successione limitata in $RR^n$ ammette una sottosuccessione convergente." - c1f: a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)) - c2f: a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)) - c2lim: a Point \[red\] labelled "x" drawn in c2f (location=(1.0, 0.0)) - c2pts: a Point \[yellow\] drawn in c2f (location=(0.4, 0.0), marker\_radius=0.05) - c2pts\_2: a Point \[yellow\] drawn in c2f (location=(0.7, 0.0), marker\_radius=0.05) - c2pts\_3: a Point \[yellow\] drawn in c2f (location=(0.8, 0.0), marker\_radius=0.05) - c2pts\_4: a Point \[yellow\] drawn in c2f (location=(0.85, 0.0), marker\_radius=0.05) - c2pts\_5: a Point \[yellow\] drawn in c2f (location=(0.88, 0.0), marker\_radius=0.05) - c2pts\_6: a Point \[yellow\] drawn in c2f (location=(0.9, 0.0), marker\_radius=0.05) - c2pts\_7: a Point \[yellow\] drawn in c2f (location=(0.9142857142857143, 0.0), marker\_radius=0.05) - c3f: a Figure (x\_range=(-0.3, 4.7), y\_range=(-1.5, 1.5), aspect=(5.0, 3.0)) - c3pts: a Point \[yellow\] drawn in c3f (location=(0.3, 0.0), marker\_radius=0.05) - c3pts\_2: a Point \[yellow\] drawn in c3f (location=(0.8999999999999999, 0.0), marker\_radius=0.05) - c3pts\_3: a Point \[yellow\] drawn in c3f (location=(1.5, 0.0), marker\_radius=0.05) - c3pts\_4: a Point \[yellow\] drawn in c3f (location=(2.0999999999999996, 0.0), marker\_radius=0.05) - c3pts\_5: a Point \[yellow\] drawn in c3f (location=(2.6999999999999997, 0.0), marker\_radius=0.05) - c3pts\_6: a Point \[yellow\] drawn in c3f (location=(3.3, 0.0), marker\_radius=0.05) - c3pts\_7: a Point \[yellow\] drawn in c3f (location=(3.8999999999999995, 0.0), marker\_radius=0.05) - cap1: a Math \[text\] that says "$overline(B\_1 (0)) : upright("compatto")$" - cap2: a Math \[text\] that says "$B\_1 (0) : upright("non chiuso")$" - cap3: a Math \[text\] that says "$S : upright("non limitato")$" - comp\_def: a Panel that says "$K subset RR^n$ è compatto se ogni successione in $K$ ammette una sottosuccessione convergente a un punto di $K$." - d1: a Circle \[green\] drawn in c1f (filled=True) - d2: a Circle \[blue\] drawn in c2f (filled=True) - ell: a Point \[red\] labelled "x^\*" drawn in bw\_fig (location=(0.9, 0.7)) - eps: a VariableNumber (initial\_value=1.2) - eps\_ball: a Circle \[red\] drawn in seq\_fig (center=(2.0, 2.0), radius=\) - hb: a Panel that says "$K subset RR^n$ è compatto se e solo se è chiuso e limitato." - head\_bw: a Heading that says "Limitatezza e sottosuccessioni" - head\_cases: a Heading that says "Heine-Borel in tre casi" - head\_comp: a Heading that says "Insiemi compatti" - head\_seq: a Heading that says "Successioni in $RR^n$" - lim\_pt: a Point \[yellow\] labelled "x" drawn in seq\_fig (location=(2.0, 2.0)) - point: a Point \[blue\] drawn in seq\_fig (location=(3.6, 2.0), marker\_radius=0.05) - point\_10: a Point \[blue\] drawn in seq\_fig (location=(1.9260211487288768, 1.9619339669126623), marker\_radius=0.05) - point\_11: a Point \[yellow\] drawn in bw\_fig (location=(1.5, 0.7), marker\_radius=0.05) - point\_12: a Point \[gray\] drawn in bw\_fig (location=(-1.1248440509641426, -0.5272107719522956), marker\_radius=0.05) - point\_13: a Point \[yellow\] drawn in bw\_fig (location=(0.7692712758272776, 0.5486395009384143), marker\_radius=0.05) - point\_14: a Point \[gray\] drawn in bw\_fig (location=(-0.8559744570024451, -0.8419123247298389), marker\_radius=0.05) - point\_15: a Point \[yellow\] drawn in bw\_fig (location=(0.8825399959429664, 0.8187229895948058), marker\_radius=0.05) - point\_16: a Point \[gray\] drawn in bw\_fig (location=(-1.0839071529076452, -0.854402111088937), marker\_radius=0.05) - point\_17: a Point \[yellow\] drawn in bw\_fig (location=(0.972330339319928, 0.6540080355999627), marker\_radius=0.05) - point\_18: a Point \[gray\] drawn in bw\_fig (location=(-0.9897447086344124, -0.7257044483228847), marker\_radius=0.05) - point\_19: a Point \[yellow\] drawn in bw\_fig (location=(0.8361560346451077, 0.6808064455556623), marker\_radius=0.05) - point\_2: a Point \[blue\] drawn in seq\_fig (location=(2.522542731882265, 3.0266708787907737), marker\_radius=0.05) - point\_20: a Point \[gray\] drawn in bw\_fig (location=(-0.9603809975053552, -0.8450592348063006), marker\_radius=0.05) - point\_21: a Point \[yellow\] drawn in bw\_fig (location=(0.9222590215534577, 0.7497970136760523), marker\_radius=0.05) - point\_22: a Point \[gray\] drawn in bw\_fig (location=(-1.0499980413197318, -0.8004425654645202), marker\_radius=0.05) - point\_3: a Point \[blue\] drawn in seq\_fig (location=(1.511873633303726, 2.670599257184121), marker\_radius=0.05) - point\_4: a Point \[blue\] drawn in seq\_fig (location=(1.4102802413456896, 1.9057947762438732), marker\_radius=0.05) - point\_5: a Point \[blue\] drawn in seq\_fig (location=(1.8678524913301318, 1.5908285263518687), marker\_radius=0.05) - point\_6: a Point \[blue\] drawn in seq\_fig (location=(2.219394822646659, 1.7815740134346925), marker\_radius=0.05) - point\_7: a Point \[blue\] drawn in seq\_fig (location=(2.2118092309777224, 2.069443352271798), marker\_radius=0.05) - point\_8: a Point \[blue\] drawn in seq\_fig (location=(2.024614941681198, 2.158590930194415), marker\_radius=0.05) - point\_9: a Point \[blue\] drawn in seq\_fig (location=(1.9062760432501684, 2.067588738576724), marker\_radius=0.05) - q1: a Polygon \[green\] drawn in bw\_fig (vertices=((-2, -2), (2, -2), (2, 2), (-2, 2)), filled=False, dashed=True) - q2: a Polygon \[green\] drawn in bw\_fig (vertices=((0, 0), (2, 0), (2, 2), (0, 2)), filled=False, dashed=True) - q3: a Polygon \[green\] drawn in bw\_fig (vertices=((0, 0), (1, 0), (1, 1), (0, 1)), filled=False, dashed=True) - q4: a Polygon \[green\] drawn in bw\_fig (vertices=((0.5, 0.5), (1, 0.5), (1, 1), (0.5, 1)), filled=False, dashed=True) - s1: a Math \[text\] that says "$x\_k -\> x \<=\> norm(x\_k - x) -\> 0$" - s2: a Math \[text\] that says "$x\_k -\> x \<=\> x\_k^((i)) -\> x^((i)) quad forall i$" - s3: a Math \[text\] that says "$norm(x\_k - x\_h) \< epsilon quad forall k, h \>= N\_epsilon$" - s4: a Math \[text\] that says "$upright("Cauchy") \<=\> upright("convergente")$" - seq\_fig: a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)) - strip: a Polygon \[blue\] drawn in c3f (vertices=((0, -0.6), (4.6, -0.6), (4.6, 0.6), (0, 0.6))) #### Beats ##### [03:0.974](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=180.974125) Narration: Una successione in erre enne è una sequenza di punti, indicizzata dai numeri naturali. Diciamo che converge al punto x se la distanza fra x k e x tende a zero. Board: Empty. Actions: - [03:0.974](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=180.974125): head\_seq is shown on the screen, written out. - [03:3.174](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=183.17412499999998): seq\_fig is shown on the screen, written out. - [03:3.801](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=183.80112499999998): point is shown on the screen, written out. - [03:4.057](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.0573803925798): point\_2 is shown on the screen, written out. - [03:4.314](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.31363578515962): point\_3 is shown on the screen, written out. - [03:4.53](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.52983969657748): point\_4 is shown on the screen, written out. - [03:4.651](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.65063348432557): point\_5 is shown on the screen, written out. - [03:4.771](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.77142727207362): point\_6 is shown on the screen, written out. - [03:4.892](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=184.89222105982168): point\_7 is shown on the screen, written out. - [03:5.013](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=185.01301484756974): point\_8 is shown on the screen, written out. - [03:5.157](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=185.15657538826574): point\_9 is shown on the screen, written out. - [03:5.304](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=185.30427167817084): point\_10 is shown on the screen, written out. - [03:7.214](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=187.214125): lim\_pt is shown on the screen, written out. - [03:9.054](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=189.054125): seq\_fig moves to a new place on the board. - [03:9.054](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=189.054125): s1 is shown on the screen, written out. ##### [03:12.762](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=192.76162499999998) Narration: In altre parole: fissato un raggio epsilon, da un certo indice in poi tutti i punti cadono nella palla di raggio epsilon attorno al limite. Se stringiamo epsilon l'indice cresce, ma esiste sempre. Board: s1 — a Math \[text\] that says "$x\_k -\> x \<=\> norm(x\_k - x) -\> 0$"; seq\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); head\_seq — a Heading that says "Successioni in $RR^n$"; point — a Point \[blue\] drawn in seq\_fig (location=(3.6, 2.0), marker\_radius=0.05); point\_2 — a Point \[blue\] drawn in seq\_fig (location=(2.522542731882265, 3.0266708787907737), marker\_radius=0.05); point\_3 — a Point \[blue\] drawn in seq\_fig (location=(1.511873633303726, 2.670599257184121), marker\_radius=0.05); point\_4 — a Point \[blue\] drawn in seq\_fig (location=(1.4102802413456896, 1.9057947762438732), marker\_radius=0.05); point\_5 — a Point \[blue\] drawn in seq\_fig (location=(1.8678524913301318, 1.5908285263518687), marker\_radius=0.05); point\_6 — a Point \[blue\] drawn in seq\_fig (location=(2.219394822646659, 1.7815740134346925), marker\_radius=0.05); point\_7 — a Point \[blue\] drawn in seq\_fig (location=(2.2118092309777224, 2.069443352271798), marker\_radius=0.05); point\_8 — a Point \[blue\] drawn in seq\_fig (location=(2.024614941681198, 2.158590930194415), marker\_radius=0.05); point\_9 — a Point \[blue\] drawn in seq\_fig (location=(1.9062760432501684, 2.067588738576724), marker\_radius=0.05); point\_10 — a Point \[blue\] drawn in seq\_fig (location=(1.9260211487288768, 1.9619339669126623), marker\_radius=0.05); lim\_pt — a Point \[yellow\] labelled "x" drawn in seq\_fig (location=(2.0, 2.0)) Actions: - [03:14.134](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=194.13412499999998): eps\_ball is shown on the screen, written out. - [03:21.414](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=201.41412499999998): eps\_ball is redrawn as the numbers it depends on change. - [03:21.414](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=201.41412499999998): eps ticks to 0.4. ##### [03:25.547](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=205.54662499999998) Narration: Poiché la norma controlla ogni coordinata, la convergenza in erre enne equivale alla convergenza di ciascuna delle enne successioni delle coordinate. Board: s1 — a Math \[text\] that says "$x\_k -\> x \<=\> norm(x\_k - x) -\> 0$"; seq\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); head\_seq — a Heading that says "Successioni in $RR^n$"; point — a Point \[blue\] drawn in seq\_fig (location=(3.6, 2.0), marker\_radius=0.05); point\_2 — a Point \[blue\] drawn in seq\_fig (location=(2.522542731882265, 3.0266708787907737), marker\_radius=0.05); point\_3 — a Point \[blue\] drawn in seq\_fig (location=(1.511873633303726, 2.670599257184121), marker\_radius=0.05); point\_4 — a Point \[blue\] drawn in seq\_fig (location=(1.4102802413456896, 1.9057947762438732), marker\_radius=0.05); point\_5 — a Point \[blue\] drawn in seq\_fig (location=(1.8678524913301318, 1.5908285263518687), marker\_radius=0.05); point\_6 — a Point \[blue\] drawn in seq\_fig (location=(2.219394822646659, 1.7815740134346925), marker\_radius=0.05); point\_7 — a Point \[blue\] drawn in seq\_fig (location=(2.2118092309777224, 2.069443352271798), marker\_radius=0.05); point\_8 — a Point \[blue\] drawn in seq\_fig (location=(2.024614941681198, 2.158590930194415), marker\_radius=0.05); point\_9 — a Point \[blue\] drawn in seq\_fig (location=(1.9062760432501684, 2.067588738576724), marker\_radius=0.05); point\_10 — a Point \[blue\] drawn in seq\_fig (location=(1.9260211487288768, 1.9619339669126623), marker\_radius=0.05); lim\_pt — a Point \[yellow\] labelled "x" drawn in seq\_fig (location=(2.0, 2.0)); eps\_ball — a Circle \[red\] drawn in seq\_fig (center=(2.0, 2.0), radius=\) Actions: - [03:29.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=209.85412499999998): s2 is shown on the screen, written out. ##### [03:34.927](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=214.926625) Narration: Una successione è di Cauchy se i suoi termini diventano arbitrariamente vicini fra loro, senza nominare alcun limite. Ogni successione convergente è di Cauchy; il viceversa si chiama completezza. Board: s1 — a Math \[text\] that says "$x\_k -\> x \<=\> norm(x\_k - x) -\> 0$"; s2 — a Math \[text\] that says "$x\_k -\> x \<=\> x\_k^((i)) -\> x^((i)) quad forall i$"; seq\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); head\_seq — a Heading that says "Successioni in $RR^n$"; point — a Point \[blue\] drawn in seq\_fig (location=(3.6, 2.0), marker\_radius=0.05); point\_2 — a Point \[blue\] drawn in seq\_fig (location=(2.522542731882265, 3.0266708787907737), marker\_radius=0.05); point\_3 — a Point \[blue\] drawn in seq\_fig (location=(1.511873633303726, 2.670599257184121), marker\_radius=0.05); point\_4 — a Point \[blue\] drawn in seq\_fig (location=(1.4102802413456896, 1.9057947762438732), marker\_radius=0.05); point\_5 — a Point \[blue\] drawn in seq\_fig (location=(1.8678524913301318, 1.5908285263518687), marker\_radius=0.05); point\_6 — a Point \[blue\] drawn in seq\_fig (location=(2.219394822646659, 1.7815740134346925), marker\_radius=0.05); point\_7 — a Point \[blue\] drawn in seq\_fig (location=(2.2118092309777224, 2.069443352271798), marker\_radius=0.05); point\_8 — a Point \[blue\] drawn in seq\_fig (location=(2.024614941681198, 2.158590930194415), marker\_radius=0.05); point\_9 — a Point \[blue\] drawn in seq\_fig (location=(1.9062760432501684, 2.067588738576724), marker\_radius=0.05); point\_10 — a Point \[blue\] drawn in seq\_fig (location=(1.9260211487288768, 1.9619339669126623), marker\_radius=0.05); lim\_pt — a Point \[yellow\] labelled "x" drawn in seq\_fig (location=(2.0, 2.0)); eps\_ball — a Circle \[red\] drawn in seq\_fig (center=(2.0, 2.0), radius=\) Actions: - [03:39.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=219.09412499999996): s3 is shown on the screen, written out. ##### [03:47.799](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=227.799125) Narration: Erre enne è completo: ogni successione di Cauchy converge. Si dimostra coordinata per coordinata, perché ogni componente è di Cauchy in erre, e la retta reale è completa. Board: s1 — a Math \[text\] that says "$x\_k -\> x \<=\> norm(x\_k - x) -\> 0$"; s2 — a Math \[text\] that says "$x\_k -\> x \<=\> x\_k^((i)) -\> x^((i)) quad forall i$"; s3 — a Math \[text\] that says "$norm(x\_k - x\_h) \< epsilon quad forall k, h \>= N\_epsilon$"; seq\_fig — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 4.0), aspect=(1.0, 1.0)); head\_seq — a Heading that says "Successioni in $RR^n$"; point — a Point \[blue\] drawn in seq\_fig (location=(3.6, 2.0), marker\_radius=0.05); point\_2 — a Point \[blue\] drawn in seq\_fig (location=(2.522542731882265, 3.0266708787907737), marker\_radius=0.05); point\_3 — a Point \[blue\] drawn in seq\_fig (location=(1.511873633303726, 2.670599257184121), marker\_radius=0.05); point\_4 — a Point \[blue\] drawn in seq\_fig (location=(1.4102802413456896, 1.9057947762438732), marker\_radius=0.05); point\_5 — a Point \[blue\] drawn in seq\_fig (location=(1.8678524913301318, 1.5908285263518687), marker\_radius=0.05); point\_6 — a Point \[blue\] drawn in seq\_fig (location=(2.219394822646659, 1.7815740134346925), marker\_radius=0.05); point\_7 — a Point \[blue\] drawn in seq\_fig (location=(2.2118092309777224, 2.069443352271798), marker\_radius=0.05); point\_8 — a Point \[blue\] drawn in seq\_fig (location=(2.024614941681198, 2.158590930194415), marker\_radius=0.05); point\_9 — a Point \[blue\] drawn in seq\_fig (location=(1.9062760432501684, 2.067588738576724), marker\_radius=0.05); point\_10 — a Point \[blue\] drawn in seq\_fig (location=(1.9260211487288768, 1.9619339669126623), marker\_radius=0.05); lim\_pt — a Point \[yellow\] labelled "x" drawn in seq\_fig (location=(2.0, 2.0)); eps\_ball — a Circle \[red\] drawn in seq\_fig (center=(2.0, 2.0), radius=\) Actions: - [03:49.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=229.094125): s4 is shown on the screen, written out. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): head\_seq is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): s1 is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): s2 is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): s3 is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): s4 is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): seq\_fig is hidden from the screen — left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_2 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_3 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_4 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_5 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_6 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_7 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_8 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_9 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): point\_10 is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): lim\_pt is hidden from the screen — seq\_fig left the board. - [03:59.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=239.14412499999997): eps\_ball is hidden from the screen — seq\_fig left the board. ##### [04:0.344](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=240.344125) Narration: Un insieme è limitato se sta dentro una palla abbastanza grande. Prendiamo ora una successione limitata qualsiasi: i suoi punti possono saltare avanti e indietro senza convergere. Board: Empty. Actions: - [04:0.344](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=240.344125): head\_bw is shown on the screen, written out. - [04:1.534](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=241.534125): bounded is shown on the screen, written out. - [04:2.894](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=242.894125): bw\_fig is shown on the screen, written out. - [04:3.774](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=243.774125): bound is shown on the screen, written out. - [04:8.414](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.41412499999998): point\_11 is shown on the screen, written out. - [04:8.518](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.51765260736195): point\_12 is shown on the screen, written out. - [04:8.621](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.6211802147239): point\_13 is shown on the screen, written out. - [04:8.725](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.72470782208586): point\_14 is shown on the screen, written out. - [04:8.829](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.82863858457492): point\_15 is shown on the screen, written out. - [04:8.935](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=248.9351241235758): point\_16 is shown on the screen, written out. - [04:9.042](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.04160966257666): point\_17 is shown on the screen, written out. - [04:9.148](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.14809520157752): point\_18 is shown on the screen, written out. - [04:9.268](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.2679900306748): point\_19 is shown on the screen, written out. - [04:9.381](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.38132336400815): point\_20 is shown on the screen, written out. - [04:9.492](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.49175281186092): point\_21 is shown on the screen, written out. - [04:9.602](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=249.6021822597137): point\_22 is shown on the screen, written out. ##### [04:11.674](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=251.674125) Narration: Il teorema di Bolzano-Weierstrass garantisce però che possiamo estrarre una sottosuccessione convergente. Qui basta tenere i termini di indice pari, quelli gialli, che si accumulano attorno a un punto. Board: bounded — a Math \[text\] that says "$S thin upright("limitato") \<=\> exists R : S subset B\_R (0)$"; bw\_fig — a Figure (x\_range=(-2.5, 2.5), y\_range=(-2.5, 2.5), aspect=(1.0, 1.0)); head\_bw — a Heading that says "Limitatezza e sottosuccessioni"; bound — a Circle \[gray\] drawn in bw\_fig (radius=2.0); point\_11 — a Point \[yellow\] drawn in bw\_fig (location=(1.5, 0.7), marker\_radius=0.05); point\_12 — a Point \[gray\] drawn in bw\_fig (location=(-1.1248440509641426, -0.5272107719522956), marker\_radius=0.05); point\_13 — a Point \[yellow\] drawn in bw\_fig (location=(0.7692712758272776, 0.5486395009384143), marker\_radius=0.05); point\_14 — a Point \[gray\] drawn in bw\_fig (location=(-0.8559744570024451, -0.8419123247298389), marker\_radius=0.05); point\_15 — a Point \[yellow\] drawn in bw\_fig (location=(0.8825399959429664, 0.8187229895948058), marker\_radius=0.05); point\_16 — a Point \[gray\] drawn in bw\_fig (location=(-1.0839071529076452, -0.854402111088937), marker\_radius=0.05); point\_17 — a Point \[yellow\] drawn in bw\_fig (location=(0.972330339319928, 0.6540080355999627), marker\_radius=0.05); point\_18 — a Point \[gray\] drawn in bw\_fig (location=(-0.9897447086344124, -0.7257044483228847), marker\_radius=0.05); point\_19 — a Point \[yellow\] drawn in bw\_fig (location=(0.8361560346451077, 0.6808064455556623), marker\_radius=0.05); point\_20 — a Point \[gray\] drawn in bw\_fig (location=(-0.9603809975053552, -0.8450592348063006), marker\_radius=0.05); point\_21 — a Point \[yellow\] drawn in bw\_fig (location=(0.9222590215534577, 0.7497970136760523), marker\_radius=0.05); point\_22 — a Point \[gray\] drawn in bw\_fig (location=(-1.0499980413197318, -0.8004425654645202), marker\_radius=0.05) Actions: - [04:12.134](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=252.13412499999998): bw\_thm is shown on the screen, written out. - [04:21.814](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=261.814125): ell is shown on the screen, written out. ##### [04:24.244](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=264.244125) Narration: L'idea della prova: si dimezza ripetutamente il quadrato che contiene la successione, scegliendo ogni volta una parte con infiniti termini. I quadrati annidati si stringono su un unico punto, che esiste perché erre enne è completo. Board: bounded — a Math \[text\] that says "$S thin upright("limitato") \<=\> exists R : S subset B\_R (0)$"; bw\_thm — a Panel that says "Ogni successione limitata in $RR^n$ ammette una sottosuccessione convergente."; bw\_fig — a Figure (x\_range=(-2.5, 2.5), y\_range=(-2.5, 2.5), aspect=(1.0, 1.0)); head\_bw — a Heading that says "Limitatezza e sottosuccessioni"; bound — a Circle \[gray\] drawn in bw\_fig (radius=2.0); point\_11 — a Point \[yellow\] drawn in bw\_fig (location=(1.5, 0.7), marker\_radius=0.05); point\_12 — a Point \[gray\] drawn in bw\_fig (location=(-1.1248440509641426, -0.5272107719522956), marker\_radius=0.05); point\_13 — a Point \[yellow\] drawn in bw\_fig (location=(0.7692712758272776, 0.5486395009384143), marker\_radius=0.05); point\_14 — a Point \[gray\] drawn in bw\_fig (location=(-0.8559744570024451, -0.8419123247298389), marker\_radius=0.05); point\_15 — a Point \[yellow\] drawn in bw\_fig (location=(0.8825399959429664, 0.8187229895948058), marker\_radius=0.05); point\_16 — a Point \[gray\] drawn in bw\_fig (location=(-1.0839071529076452, -0.854402111088937), marker\_radius=0.05); point\_17 — a Point \[yellow\] drawn in bw\_fig (location=(0.972330339319928, 0.6540080355999627), marker\_radius=0.05); point\_18 — a Point \[gray\] drawn in bw\_fig (location=(-0.9897447086344124, -0.7257044483228847), marker\_radius=0.05); point\_19 — a Point \[yellow\] drawn in bw\_fig (location=(0.8361560346451077, 0.6808064455556623), marker\_radius=0.05); point\_20 — a Point \[gray\] drawn in bw\_fig (location=(-0.9603809975053552, -0.8450592348063006), marker\_radius=0.05); point\_21 — a Point \[yellow\] drawn in bw\_fig (location=(0.9222590215534577, 0.7497970136760523), marker\_radius=0.05); point\_22 — a Point \[gray\] drawn in bw\_fig (location=(-1.0499980413197318, -0.8004425654645202), marker\_radius=0.05); ell — a Point \[red\] labelled "x^\*" drawn in bw\_fig (location=(0.9, 0.7)) Actions: - [04:27.454](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=267.454125): q1 is shown on the screen, written out. - [04:29.494](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=269.494125): q2 is shown on the screen, written out. - [04:33.454](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=273.45412500000003): q3 is shown on the screen, written out. - [04:34.174](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=274.174125): q4 is shown on the screen, written out. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): bounded is hidden from the screen — left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): bw\_fig is hidden from the screen — left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): bound is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_11 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_12 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_13 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_14 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_15 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_16 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_17 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_18 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_19 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_20 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_21 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): point\_22 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): ell is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): q1 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): q2 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): q3 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): q4 is hidden from the screen — bw\_fig left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): bw\_thm is hidden from the screen — left the board. - [04:38.619](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=278.619125): head\_bw is hidden from the screen — left the board. ##### [04:39.819](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=279.819125) Narration: Questo suggerisce una definizione. Un insieme K è compatto se da ogni successione in K si estrae una sottosuccessione che converge, e converge a un punto di K stesso. Board: Empty. Actions: - [04:39.819](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=279.819125): head\_comp is shown on the screen, written out. - [04:41.294](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=281.294125): comp\_def is shown on the screen, written out. ##### [04:52.129](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=292.129125) Narration: In erre enne la compattezza ha una caratterizzazione concreta, il teorema di Heine-Borel: compatto significa chiuso e limitato. Limitato serve per Bolzano-Weierstrass; chiuso serve perché il limite non sfugga dall'insieme. Board: comp\_def — a Panel that says "$K subset RR^n$ è compatto se ogni successione in $K$ ammette una sottosuccessione convergente a un punto di $K$."; head\_comp — a Heading that says "Insiemi compatti" Actions: - [04:54.334](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=294.334125): hb is shown on the screen, written out. - [05:6.744](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=306.744125): comp\_def is hidden from the screen — left the board. - [05:6.744](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=306.744125): hb is hidden from the screen — left the board. - [05:6.744](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=306.744125): head\_comp is hidden from the screen — left the board. ##### [05:7.344](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=307.344125) Narration: Vediamo tre casi. Il disco chiuso è chiuso e limitato, quindi compatto: ogni successione al suo interno ha una sottosuccessione che converge dentro il disco. Board: Empty. Actions: - [05:7.344](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=307.344125): head\_cases is shown on the screen, written out. - [05:9.214](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=309.21412499999997): c1f is shown on the screen, written out. - [05:9.214](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=309.21412499999997): d1 is shown on the screen, written out. - [05:11.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=311.854125): c1f moves to a new place on the board. - [05:11.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=311.854125): cap1 is shown on the screen, written out. ##### [05:18.069](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=318.069125) Narration: Nel disco aperto, questa successione si avvicina al bordo: converge, ma a un punto che non appartiene all'insieme. Manca la chiusura. Board: c1f — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); cap1 — a Math \[text\] that says "$overline(B\_1 (0)) : upright("compatto")$"; head\_cases — a Heading that says "Heine-Borel in tre casi"; d1 — a Circle \[green\] drawn in c1f (filled=True) Actions: - [05:18.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=318.854125): c2f is shown on the screen, written out. - [05:18.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=318.854125): d2 is shown on the screen, written out. - [05:19.734](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=319.734125): c2pts is shown on the screen, written out. - [05:19.838](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=319.83767986862443): c2pts\_2 is shown on the screen, written out. - [05:19.941](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=319.94123473724886): c2pts\_3 is shown on the screen, written out. - [05:20.045](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=320.04478960587323): c2pts\_4 is shown on the screen, written out. - [05:20.148](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=320.14834447449766): c2pts\_5 is shown on the screen, written out. - [05:20.252](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=320.2518993431221): c2pts\_6 is shown on the screen, written out. - [05:20.351](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=320.3511883693972): c2pts\_7 is shown on the screen, written out. - [05:21.014](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=321.01412500000004): c2lim is shown on the screen, written out. - [05:25.014](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=325.01412500000004): cap2 is shown on the screen, written out. ##### [05:26.764](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=326.76412500000004) Narration: Nella striscia illimitata, invece, i punti scappano verso destra e nessuna sottosuccessione può convergere. Manca la limitatezza. Board: c1f — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); cap1 — a Math \[text\] that says "$overline(B\_1 (0)) : upright("compatto")$"; c2f — a Figure (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), aspect=(1.0, 1.0)); cap2 — a Math \[text\] that says "$B\_1 (0) : upright("non chiuso")$"; head\_cases — a Heading that says "Heine-Borel in tre casi"; d1 — a Circle \[green\] drawn in c1f (filled=True); d2 — a Circle \[blue\] drawn in c2f (filled=True); c2pts — a Point \[yellow\] drawn in c2f (location=(0.4, 0.0), marker\_radius=0.05); c2pts\_2 — a Point \[yellow\] drawn in c2f (location=(0.7, 0.0), marker\_radius=0.05); c2pts\_3 — a Point \[yellow\] drawn in c2f (location=(0.8, 0.0), marker\_radius=0.05); c2pts\_4 — a Point \[yellow\] drawn in c2f (location=(0.85, 0.0), marker\_radius=0.05); c2pts\_5 — a Point \[yellow\] drawn in c2f (location=(0.88, 0.0), marker\_radius=0.05); c2pts\_6 — a Point \[yellow\] drawn in c2f (location=(0.9, 0.0), marker\_radius=0.05); c2pts\_7 — a Point \[yellow\] drawn in c2f (location=(0.9142857142857143, 0.0), marker\_radius=0.05); c2lim — a Point \[red\] labelled "x" drawn in c2f (location=(1.0, 0.0)) Actions: - [05:27.294](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=327.294125): c3f is shown on the screen, written out. - [05:27.294](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=327.294125): strip is shown on the screen, written out. - [05:29.454](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=329.454125): c3pts is shown on the screen, written out. - [05:29.583](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=329.58299736679055): c3pts\_2 is shown on the screen, written out. - [05:29.712](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=329.7118697335812): c3pts\_3 is shown on the screen, written out. - [05:29.841](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=329.84074210037176): c3pts\_4 is shown on the screen, written out. - [05:29.97](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=329.9696144671623): c3pts\_5 is shown on the screen, written out. - [05:30.068](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=330.06835760754115): c3pts\_6 is shown on the screen, written out. - [05:30.151](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=330.15120412904935): c3pts\_7 is shown on the screen, written out. - [05:33.454](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=333.454125): cap3 is shown on the screen, written out. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c1f is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): d1 is hidden from the screen — c1f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2f is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): d2 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_2 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_3 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_4 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_5 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_6 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2pts\_7 is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c2lim is hidden from the screen — c2f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3f is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): strip is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_2 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_3 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_4 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_5 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_6 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): c3pts\_7 is hidden from the screen — c3f left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): cap1 is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): cap2 is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): cap3 is hidden from the screen — left the board. - [05:35.022](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=335.02166666666665): head\_cases is hidden from the screen — left the board. ### Scene 3: [Continuità e teoremi fondamentali](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.0633333333334) Span: 05:36.063–07:50.439 (336.0633333333334s–470.43908333333343s). #### Objects - a\_pt: a Point \[green\] labelled "a" drawn in path\_fig (location=(1.2, 1.6)) - b\_pt: a Point \[green\] labelled "b" drawn in path\_fig (location=(4.8, 2.6)) - cod: a Circle \[gray\] labelled "RR^m" drawn in lim\_fig (center=(6.0, 2.0), radius=1.6) - cont\_def: a Math \[text\] that says "$f thin upright("continua in") thin x\_0 \<=\> lim\_(x -\> x\_0) f(x) = f(x\_0)$" - dball: a Circle \[green\] drawn in lim\_fig (center=(2.0, 2.0), radius=(0.6 \* eps)) - disk\_closed: a Circle \[blue\] drawn in rax (filled=True) - dom: a Circle \[gray\] labelled "RR^n" drawn in lim\_fig (center=(2.0, 2.0), radius=1.6) - eball: a Circle \[red\] drawn in lim\_fig (center=(6.0, 2.0), radius=\) - eps: a VariableNumber (initial\_value=1.0) - ex0: a Math \[text\] that says "$f(x, y) = x^2 + y^2$" - ex1: a Math \[text\] that says "$\{1 \< x^2 + y^2 \< 4\} = f^(-1)((1, 4))$" - ex2: a Math \[text\] that says "$\{x^2 + y^2 \<= 1\} = f^(-1)(\[0, 1\])$" - fmap: a CurvedArrow \[text\] labelled "f" drawn in lim\_fig (start=(2.9, 3.4), end=(5.1, 3.4), bend=0.4) - gamma: a ParametricCurve \[yellow\] drawn in path\_fig (function=\) - head\_lim: a Heading that says "Limiti e continuità" - head\_thm: a Heading that says "Tre teoremi fondamentali" - head\_top: a Heading that says "Continuità e insiemi aperti" - img: a Circle \[yellow\] drawn in lim\_fig (center=(6.1, 2.05), radius=(0.45 \* eps), filled=True) - lim\_def: a Panel that says "Sia $f: A subset RR^n -\> RR^m$ e $x\_0$ di accumulazione per $A$. Allora $lim\_(x -\> x\_0) f(x) = L$ se per ogni $epsilon \> 0$ esiste $delta \> 0$ tale che $0 \< norm(x - x\_0) \< delta$, con $x in A$, implica $norm(f(x) - L) \< epsilon$." - lim\_fig: a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 4.0), aspect=(2.0, 1.0)) - lp: a Point \[red\] labelled "L" drawn in lim\_fig (location=(6.0, 2.0)) - path\_fig: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)) - pc: a ParametricCurve \[yellow\] drawn in lim\_fig (function=\, t\_range=(-1.0, 1.0)) - pl1: a Line \[gray\] drawn in lim\_fig (start=(0.7, 2.0), end=(3.3, 2.0), dashed=True) - pl2: a Line \[gray\] drawn in lim\_fig (start=(1.0, 1.0), end=(3.0, 3.0), dashed=True) - rax: an Axes (x\_range=(-2.5, 2.5), y\_range=(-2.5, 2.5), aspect=(1.0, 1.0)) - region: a Polygon \[blue\] labelled "S" drawn in path\_fig (vertices=((0.5, 1.0), (2.0, 0.4), (4.0, 0.8), (5.5, 2.0), (4.5, 3.5), (2…) - rider: a Point \[red\] labelled "gamma(t)" drawn in path\_fig (location=((1.2 + (3.6 \* tt)), ((1.6 + tt) + (0.5 \* sin((3.14159265358979…) - ring: a Region \[green\] drawn in rax (predicates=(\.inside at 0x2af6cb7d0510\>,), x\_range=(-2.000019038558531, 2.000019038558531), y\_range=(-2.000019038558531, 2.000019038558531)) - t1: a Text \[text\] that says "Valori intermedi: se $S$ è connesso per archi, $f: S -\> RR$ è continua e $f(a) \< c \< f(b)$, esiste $x in S$ con $f(x) = c$." - t2: a Text \[text\] that says "Weierstrass: se $K$ è compatto e $f: K -\> RR$ è continua, $f$ ha massimo e minimo assoluti su $K$." - t3: a Text \[text\] that says "Heine-Cantor: se $K$ è compatto, ogni $f$ continua su $K$ è uniformemente continua: un solo $delta$ vale per tutti i punti." - top\_thm: a Panel that says "$f: RR^n -\> RR^m$ è continua se e solo se $f^(-1)(A)$ è aperto per ogni aperto $A subset RR^m$, ovvero se e solo se la retroimmagine di ogni chiuso è chiusa." - tt: a VariableNumber - x0: a Point \[green\] labelled "x\_0" drawn in lim\_fig (location=(2.0, 2.0)) #### Beats ##### [05:36.063](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.0633333333334) Narration: Passiamo alle funzioni da erre enne a erre emme. La definizione di limite è quella del caso reale, con la norma al posto del valore assoluto. Board: Empty. Actions: - [05:36.063](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.0633333333334): head\_lim is shown on the screen, written out. - [05:36.703](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.7033333333334): lim\_fig is shown on the screen, written out. - [05:36.703](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.7033333333334): dom is shown on the screen, written out. - [05:36.703](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=336.7033333333334): cod is shown on the screen, written out. - [05:40.063](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=340.0633333333334): lim\_fig moves to a new place on the board. - [05:40.063](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=340.0633333333334): lim\_def is shown on the screen, written out. ##### [05:45.923](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=345.9233333333334) Narration: Per ogni palla di raggio epsilon attorno al limite elle esiste una palla di raggio delta attorno a x zero la cui immagine, escluso al più x zero, cade dentro la prima. Board: lim\_def — a Panel that says "Sia $f: A subset RR^n -\> RR^m$ e $x\_0$ di accumulazione per $A$. Allora $lim\_(x -\> x\_0) f(x) = L$ se per ogni $epsilon \> 0$ esiste $delta \> 0$ tale che $0 \< norm(x - x\_0) \< delta$, con $x in A$, implica $norm(f(x) - L) \< epsilon$."; lim\_fig — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 4.0), aspect=(2.0, 1.0)); head\_lim — a Heading that says "Limiti e continuità"; dom — a Circle \[gray\] labelled "RR^n" drawn in lim\_fig (center=(2.0, 2.0), radius=1.6); cod — a Circle \[gray\] labelled "RR^m" drawn in lim\_fig (center=(6.0, 2.0), radius=1.6) Actions: - [05:47.543](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=347.5433333333334): eball is shown on the screen, written out. - [05:48.743](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=348.7433333333334): lp is shown on the screen, written out. - [05:50.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=350.6633333333334): dball is shown on the screen, written out. - [05:50.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=350.6633333333334): x0 is shown on the screen, written out. - [05:52.783](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=352.7833333333334): fmap is shown on the screen, written out. - [05:55.223](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=355.2233333333334): img is shown on the screen, written out. ##### [05:57.206](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=357.2058333333334) Narration: Se stringiamo epsilon dobbiamo stringere anche delta, ma una scelta adatta esiste sempre. Board: lim\_def — a Panel that says "Sia $f: A subset RR^n -\> RR^m$ e $x\_0$ di accumulazione per $A$. Allora $lim\_(x -\> x\_0) f(x) = L$ se per ogni $epsilon \> 0$ esiste $delta \> 0$ tale che $0 \< norm(x - x\_0) \< delta$, con $x in A$, implica $norm(f(x) - L) \< epsilon$."; lim\_fig — a Figure (x\_range=(0.0, 8.0), y\_range=(0.0, 4.0), aspect=(2.0, 1.0)); head\_lim — a Heading that says "Limiti e continuità"; dom — a Circle \[gray\] labelled "RR^n" drawn in lim\_fig (center=(2.0, 2.0), radius=1.6); cod — a Circle \[gray\] labelled "RR^m" drawn in lim\_fig (center=(6.0, 2.0), radius=1.6); eball — a Circle \[red\] drawn in lim\_fig (center=(6.0, 2.0), radius=\); lp — a Point \[red\] labelled "L" drawn in lim\_fig (location=(6.0, 2.0)); dball — a Circle \[green\] drawn in lim\_fig (center=(2.0, 2.0), radius=(0.6 \* eps)); x0 — a Point \[green\] labelled "x\_0" drawn in lim\_fig (location=(2.0, 2.0)); fmap — a CurvedArrow \[text\] labelled "f" drawn in lim\_fig (start=(2.9, 3.4), end=(5.1, 3.4), bend=0.4); img — a Circle \[yellow\] drawn in lim\_fig (center=(6.1, 2.05), radius=(0.45 \* eps), filled=True) Actions: - [05:57.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=357.6633333333334): eball is redrawn as the numbers it depends on change. - [05:57.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=357.6633333333334): dball is redrawn as the numbers it depends on change. - [05:57.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=357.6633333333334): img is redrawn as the numbers it depends on change. - [05:57.663](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=357.6633333333334): eps ticks to 0.5. ##### [06:3.466](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=363.46583333333336) Narration: La funzione è continua in x zero se il limite coincide con il valore in x zero. Attenzione: in più variabili ci si avvicina a x zero lungo infinite rette e curve, e il limite deve essere lo stesso per tutte. Board: Unchanged from the preceding beat in this scene. Actions: - [06:4.583](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=364.58333333333337): cont\_def is shown on the screen, written out. - [06:13.543](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=373.5433333333334): pl1 is shown on the screen, written out. - [06:13.854](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=373.8538840964389): pl2 is shown on the screen, written out. - [06:14.023](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=374.02333333333337): pc is shown on the screen, written out. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): cont\_def is hidden from the screen — left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): head\_lim is hidden from the screen — left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): lim\_def is hidden from the screen — left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): lim\_fig is hidden from the screen — left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): dom is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): cod is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): eball is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): lp is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): dball is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): x0 is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): fmap is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): img is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): pl1 is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): pl2 is hidden from the screen — lim\_fig left the board. - [06:17.413](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=377.4133333333334): pc is hidden from the screen — lim\_fig left the board. ##### [06:18.613](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=378.6133333333334) Narration: Esiste un modo di dire la continuità senza epsilon e delta. Una funzione è continua se e solo se la retroimmagine di ogni aperto è aperta, o equivalentemente se la retroimmagine di ogni chiuso è chiusa. Board: Empty. Actions: - [06:18.613](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=378.6133333333334): head\_top is shown on the screen, written out. - [06:23.823](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=383.8233333333334): top\_thm is shown on the screen, written out. ##### [06:32.926](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=392.9258333333334) Narration: Questo è uno strumento pratico. Prendiamo f uguale a x quadro più y quadro, che è continua. La corona dove f sta strettamente fra uno e quattro è la retroimmagine di un intervallo aperto: dunque è aperta, senza alcun conto. Board: top\_thm — a Panel that says "$f: RR^n -\> RR^m$ è continua se e solo se $f^(-1)(A)$ è aperto per ogni aperto $A subset RR^m$, ovvero se e solo se la retroimmagine di ogni chiuso è chiusa."; head\_top — a Heading that says "Continuità e insiemi aperti" Actions: - [06:35.183](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=395.1833333333334): ex0 is shown on the screen, written out. - [06:38.823](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=398.8233333333334): rax is shown on the screen, written out. - [06:39.903](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=399.90333333333336): ring is shown on the screen, written out. - [06:42.943](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=402.9433333333334): ex1 is shown on the screen, written out. ##### [06:48.178](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=408.1783333333334) Narration: Allo stesso modo il disco dove f è minore o uguale a uno è la retroimmagine dell'intervallo chiuso da zero a uno, ed è quindi chiuso. Board: top\_thm — a Panel that says "$f: RR^n -\> RR^m$ è continua se e solo se $f^(-1)(A)$ è aperto per ogni aperto $A subset RR^m$, ovvero se e solo se la retroimmagine di ogni chiuso è chiusa."; ex0 — a Math \[text\] that says "$f(x, y) = x^2 + y^2$"; ex1 — a Math \[text\] that says "$\{1 \< x^2 + y^2 \< 4\} = f^(-1)((1, 4))$"; rax — an Axes (x\_range=(-2.5, 2.5), y\_range=(-2.5, 2.5), aspect=(1.0, 1.0)); head\_top — a Heading that says "Continuità e insiemi aperti"; ring — a Region \[green\] drawn in rax (predicates=(\.inside at 0x2af6cb7d0510\>,), x\_range=(-2.000019038558531, 2.000019038558531), y\_range=(-2.000019038558531, 2.000019038558531)) Actions: - [06:49.623](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=409.6233333333334): disk\_closed is shown on the screen, written out. - [06:52.263](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=412.2633333333334): ex2 is shown on the screen, written out. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): ex0 is hidden from the screen — left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): ex1 is hidden from the screen — left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): ex2 is hidden from the screen — left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): head\_top is hidden from the screen — left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): rax is hidden from the screen — left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): ring is hidden from the screen — rax left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): disk\_closed is hidden from the screen — rax left the board. - [06:56.393](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=416.3933333333334): top\_thm is hidden from the screen — left the board. ##### [06:57.593](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=417.59333333333336) Narration: Tre teoremi reggono tutta la teoria. Il primo è il teorema dei valori intermedi: su un insieme connesso per archi, due punti qualsiasi a e b sono uniti da un cammino continuo gamma, tutto dentro l'insieme. Board: Empty. Actions: - [06:57.593](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=417.59333333333336): head\_thm is shown on the screen, written out. - [07:0.823](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=420.8233333333334): t1 is shown on the screen, written out. - [07:3.623](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=423.6233333333334): path\_fig is shown on the screen, written out. - [07:3.623](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=423.6233333333334): region is shown on the screen, written out. - [07:5.783](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=425.7833333333334): a\_pt is shown on the screen, written out. - [07:5.783](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=425.7833333333334): b\_pt is shown on the screen, written out. - [07:8.263](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=428.26333333333343): gamma is shown on the screen, written out. - [07:9.343](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=429.3433333333334): rider is shown on the screen, written out. ##### [07:11.836](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=431.8358333333334) Narration: Lungo il cammino, f composta con gamma è una funzione continua di una sola variabile, e il teorema classico in erre fornisce un istante in cui assume il valore c. Board: t1 — a Text \[text\] that says "Valori intermedi: se $S$ è connesso per archi, $f: S -\> RR$ è continua e $f(a) \< c \< f(b)$, esiste $x in S$ con $f(x) = c$."; path\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); head\_thm — a Heading that says "Tre teoremi fondamentali"; region — a Polygon \[blue\] labelled "S" drawn in path\_fig (vertices=((0.5, 1.0), (2.0, 0.4), (4.0, 0.8), (5.5, 2.0), (4.5, 3.5), (2…); a\_pt — a Point \[green\] labelled "a" drawn in path\_fig (location=(1.2, 1.6)); b\_pt — a Point \[green\] labelled "b" drawn in path\_fig (location=(4.8, 2.6)); gamma — a ParametricCurve \[yellow\] drawn in path\_fig (function=\); rider — a Point \[red\] labelled "gamma(t)" drawn in path\_fig (location=((1.2 + (3.6 \* tt)), ((1.6 + tt) + (0.5 \* sin((3.14159265358979…) Actions: - [07:12.143](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=432.14333333333343): rider is redrawn as the numbers it depends on change. - [07:12.143](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=432.14333333333343): tt ticks to 1.0. - [07:19.863](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=439.8633333333334): rider is redrawn as the numbers it depends on change. - [07:19.863](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=439.8633333333334): tt ticks to 0.55. ##### [07:22.708](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=442.70833333333337) Narration: Il secondo è il teorema di Weierstrass: una funzione continua su un compatto ha massimo e minimo assoluti. Da una successione di valori che tende all'estremo superiore, la compattezza estrae una sottosuccessione convergente, e la continuità fa il resto. Board: Unchanged from the preceding beat in this scene. Actions: - [07:23.223](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=443.2233333333334): t2 is shown on the screen, written out. ##### [07:38.886](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=458.8858333333334) Narration: Il terzo è il teorema di Heine-Cantor. La continuità uniforme chiede che lo stesso delta funzioni in tutti i punti insieme; su un compatto questo è automatico. Board: t1 — a Text \[text\] that says "Valori intermedi: se $S$ è connesso per archi, $f: S -\> RR$ è continua e $f(a) \< c \< f(b)$, esiste $x in S$ con $f(x) = c$."; t2 — a Text \[text\] that says "Weierstrass: se $K$ è compatto e $f: K -\> RR$ è continua, $f$ ha massimo e minimo assoluti su $K$."; path\_fig — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); head\_thm — a Heading that says "Tre teoremi fondamentali"; region — a Polygon \[blue\] labelled "S" drawn in path\_fig (vertices=((0.5, 1.0), (2.0, 0.4), (4.0, 0.8), (5.5, 2.0), (4.5, 3.5), (2…); a\_pt — a Point \[green\] labelled "a" drawn in path\_fig (location=(1.2, 1.6)); b\_pt — a Point \[green\] labelled "b" drawn in path\_fig (location=(4.8, 2.6)); gamma — a ParametricCurve \[yellow\] drawn in path\_fig (function=\); rider — a Point \[red\] labelled "gamma(t)" drawn in path\_fig (location=((1.2 + (3.6 \* tt)), ((1.6 + tt) + (0.5 \* sin((3.14159265358979…) Actions: - [07:39.463](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=459.4633333333334): t3 is shown on the screen, written out. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): head\_thm is hidden from the screen — left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): path\_fig is hidden from the screen — left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): region is hidden from the screen — path\_fig left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): a\_pt is hidden from the screen — path\_fig left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): b\_pt is hidden from the screen — path\_fig left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): gamma is hidden from the screen — path\_fig left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): rider is hidden from the screen — path\_fig left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): t1 is hidden from the screen — left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): t2 is hidden from the screen — left the board. - [07:49.397](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=469.39741666666674): t3 is hidden from the screen — left the board. ### Scene 4: [Derivate, differenziale e Jacobiana](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=470.43908333333343) Span: 07:50.439–10:54.287 (470.43908333333343s–654.2866666666667s). #### Objects - axes3: an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)) - cax: an Axes (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), aspect=(1.0, 1.0)) - cex: a Math \[text\] that says "$f(x, y) = frac(x y, x^2 + y^2), quad f(0, 0) = 0$" - cex\_m: a Math \[text\] that says "$f(x, m x) = frac(m, 1 + m^2)$" - dd: a Math \[text\] that says "$D\_v f(x\_0) = lim\_(h -\> 0) frac(f(x\_0 + h v) - f(x\_0), h), quad norm(v) = 1$" - dif: a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + o(norm(h))$" - dvf: a Math \[text\] that says "$D\_v f(x\_0) = nabla f(x\_0) dot v$" - grad: a Math \[text\] that says "$nabla f(x\_0) = (frac(partial f, partial x\_1), dots, frac(partial f, partial x\_n))$" - grid: a Gridlines \[gray\] drawn in jax (x\_range=(-1.0, 1.0), y\_range=(-1.0, 1.0), step=0.5) - gvec: a Vector \[red\] labelled "nabla f" drawn in axes3 (start=(1.0, 0.5, 0.0), end=(1.8, 0.9, 0.0)) - head\_cex: a Heading that says "Derivate parziali senza differenziabilità" - head\_d: a Heading that says "Derivate parziali e direzionali" - head\_diff: a Heading that says "Differenziabilità e piano tangente" - head\_jac: a Heading that says "La matrice Jacobiana" - imp: a Math \[text\] that says "$upright("differenziabile") =\> upright("continua")$" - j0: a Math \[text\] that says "$J\_f (x\_0) = (frac(partial f\_i, partial x\_j)(x\_0))\_(i, j), quad m times n$" - j1: a Math \[text\] that says "$f(x, y) = (x^2 - y, thin x y)$" - j2: a Math \[text\] that says "$J\_f (x, y) = mat(2 x, -1; y, x), quad J\_f (1, 1) = mat(2, -1; 1, 1)$" - j3: a Math \[text\] that says "$f(x\_0 + h) approx f(x\_0) + J\_f (x\_0) h$" - jax: an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0)) - l0: a Line \[blue\] labelled "f = 0" drawn in cax (start=(-1.8, 0.0), end=(1.8, 0.0)) - l1: a Line \[green\] labelled "f = 1/2" drawn in cax (start=(-1.4, -1.4), end=(1.4, 1.4)) - l2: a Line \[red\] labelled "f = -1/2" drawn in cax (start=(-1.4, 1.4), end=(1.4, -1.4)) - pd: a Math \[text\] that says "$frac(partial f, partial x\_i)(x\_0) = lim\_(h -\> 0) frac(f(x\_0 + h e\_i) - f(x\_0), h)$" - pt: a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)) - square: a Polygon \[green\] drawn in jax (vertices=((0, 0), (1, 0), (1, 1), (0, 1))) - stat: a Math \[text\] that says "$nabla f(x\_0) = 0 : x\_0 thin upright("stazionario")$" - surf: a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)) - tan: a Math \[text\] that says "$z = f(x\_0) + nabla f(x\_0) dot (x - x\_0)$" - tdt: a Panel that says "Se le derivate parziali esistono in un intorno di $x\_0$ e sono continue in $x\_0$, allora $f$ è differenziabile in $x\_0$. Dunque $f in C^1$ implica $f$ differenziabile, quindi continua." - tp: a Plane \[red\] drawn in axes3 (point=(1.0, 0.5, 1.25), normal=(-0.8164965809390571, -0.4082482904581974, 0.4082482904468662), edge\_direction=(0.9999999999732445, 0.0, 2.0000000000575113)) - tu: a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)) - tv: a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)) #### Beats ##### [07:50.439](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=470.43908333333343) Narration: Veniamo alle derivate. Prendiamo il paraboloide zeta uguale a x quadro più y quadro, e un punto P sul suo grafico. Board: Empty. Actions: - [07:50.439](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=470.43908333333343): head\_d is shown on the screen, written out. - [07:52.679](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=472.67908333333344): axes3 is shown on the screen, written out. - [07:52.679](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=472.67908333333344): surf is shown on the screen, written out. - [07:56.599](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=476.59908333333345): axes3 turns to a new orientation in its own slot. - [07:56.599](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=476.59908333333345): pt is shown on the screen, written out. ##### [07:59.042](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=479.04158333333345) Narration: La derivata parziale rispetto a x i congela tutte le altre variabili: è la derivata ordinaria lungo un asse. Geometricamente è la pendenza della curva ottenuta tagliando il grafico con un piano verticale. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); head\_d — a Heading that says "Derivate parziali e direzionali"; surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)) Actions: - [07:59.839](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=479.8390833333334): axes3 moves to a new place on the board. - [07:59.839](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=479.8390833333334): pd is shown on the screen, written out. - [08:7.559](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=487.55908333333343): tu is shown on the screen, written out. - [08:9.119](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=489.11908333333344): tv is shown on the screen, written out. ##### [08:12.167](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=492.16658333333345) Narration: Più in generale, scelto un vettore unitario v, la derivata direzionale misura come varia f muovendosi lungo v. Le derivate parziali sono il caso dei vettori della base canonica. Board: pd — a Math \[text\] that says "$frac(partial f, partial x\_i)(x\_0) = lim\_(h -\> 0) frac(f(x\_0 + h e\_i) - f(x\_0), h)$"; axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); head\_d — a Heading that says "Derivate parziali e direzionali"; surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)) Actions: - [08:15.959](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=495.9590833333334): dd is shown on the screen, written out. ##### [08:24.192](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=504.1915833333334) Narration: Il vettore delle derivate parziali si chiama gradiente, nabla f. I punti in cui il gradiente si annulla si dicono stazionari, o critici: sono i candidati a massimi e minimi. Board: pd — a Math \[text\] that says "$frac(partial f, partial x\_i)(x\_0) = lim\_(h -\> 0) frac(f(x\_0 + h e\_i) - f(x\_0), h)$"; dd — a Math \[text\] that says "$D\_v f(x\_0) = lim\_(h -\> 0) frac(f(x\_0 + h v) - f(x\_0), h), quad norm(v) = 1$"; axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); head\_d — a Heading that says "Derivate parziali e direzionali"; surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)) Actions: - [08:26.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=506.7190833333334): grad is shown on the screen, written out. - [08:31.439](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=511.43908333333343): stat is shown on the screen, written out. - [08:35.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=515.7015833333335): dd is hidden from the screen — left the board. - [08:35.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=515.7015833333335): grad is hidden from the screen — left the board. - [08:35.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=515.7015833333335): head\_d is hidden from the screen — left the board. - [08:35.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=515.7015833333335): pd is hidden from the screen — left the board. - [08:35.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=515.7015833333335): stat is hidden from the screen — left the board. ##### [08:36.902](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=516.9015833333334) Narration: Diciamo che f è differenziabile in x zero se il suo incremento è uguale a una parte lineare in h, più un errore che va a zero più velocemente della norma di h. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)) Actions: - [08:36.902](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=516.9015833333334): head\_diff is shown on the screen, written out. - [08:38.279](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=518.2790833333335): dif is shown on the screen, written out. - [08:43.599](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=523.5990833333334): dif (the "o(norm(h))" part) is emphasized. - [08:46.837](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=526.8365833333335): dif (the "o(norm(h))" part) is no longer emphasized. ##### [08:47.437](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=527.4365833333334) Narration: La parte lineare definisce il piano tangente: il piano che meglio approssima il grafico vicino a P, e contiene le due tangenti di prima. Quando enne supera due si parla di iperpiano tangente. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)); dif — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + o(norm(h))$"; head\_diff — a Heading that says "Differenziabilità e piano tangente" Actions: - [08:48.679](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=528.6790833333334): tan is shown on the screen, written out. - [08:49.319](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=529.3190833333334): tp is shown on the screen, written out. ##### [09:0.239](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=540.2390833333334) Narration: Se f è differenziabile, prendiamo h uguale a t per v e dividiamo per t. Il limite dà la formula fondamentale: la derivata direzionale è il prodotto scalare fra gradiente e v. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)); dif — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + o(norm(h))$"; tan — a Math \[text\] that says "$z = f(x\_0) + nabla f(x\_0) dot (x - x\_0)$"; head\_diff — a Heading that says "Differenziabilità e piano tangente"; tp — a Plane \[red\] drawn in axes3 (point=(1.0, 0.5, 1.25), normal=(-0.8164965809390571, -0.4082482904581974, 0.4082482904468662), edge\_direction=(0.9999999999732445, 0.0, 2.0000000000575113)) Actions: - [09:6.799](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=546.7990833333334): dvf is shown on the screen, written out. ##### [09:12.869](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=552.8690833333334) Narration: Per Cauchy-Schwarz questo prodotto è massimo quando v è parallelo al gradiente. Il gradiente indica quindi la direzione di massima crescita, e la sua norma è la pendenza massima. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)); dif — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + o(norm(h))$"; tan — a Math \[text\] that says "$z = f(x\_0) + nabla f(x\_0) dot (x - x\_0)$"; dvf — a Math \[text\] that says "$D\_v f(x\_0) = nabla f(x\_0) dot v$"; head\_diff — a Heading that says "Differenziabilità e piano tangente"; tp — a Plane \[red\] drawn in axes3 (point=(1.0, 0.5, 1.25), normal=(-0.8164965809390571, -0.4082482904581974, 0.4082482904468662), edge\_direction=(0.9999999999732445, 0.0, 2.0000000000575113)) Actions: - [09:15.159](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=555.1590833333335): dvf (the "nabla f(x\_0) dot v" part) is emphasized. - [09:19.639](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=559.6390833333335): gvec is shown on the screen, written out. - [09:21.879](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=561.8790833333335): dvf (the "nabla f(x\_0) dot v" part) is no longer emphasized. ##### [09:24.354](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=564.3540833333334) Narration: Infine, la differenziabilità implica la continuità: l'incremento di f tende a zero insieme a h. Board: axes3 — an Axes3D (x\_range=(-1.5, 1.5), y\_range=(-1.5, 1.5), z\_range=(0.0, 4.5)); surf — a Surface \[blue\] drawn in axes3 (function=\, u\_range=(-1.5, 1.5), v\_range=(-1.5, 1.5)); pt — a Point \[yellow\] labelled "P" drawn in axes3 (location=(1.0, 0.5, 1.25)); tu — a Vector \[yellow\] labelled "partial\_x" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.4472135954800978, 0.5, 2.144427191009846)); tv — a Vector \[green\] labelled "partial\_y" drawn in axes3 (start=(1.0, 0.5, 1.25), end=(1.0, 1.2071067811767344, 1.9571067811963605)); dif — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + o(norm(h))$"; tan — a Math \[text\] that says "$z = f(x\_0) + nabla f(x\_0) dot (x - x\_0)$"; dvf — a Math \[text\] that says "$D\_v f(x\_0) = nabla f(x\_0) dot v$"; head\_diff — a Heading that says "Differenziabilità e piano tangente"; tp — a Plane \[red\] drawn in axes3 (point=(1.0, 0.5, 1.25), normal=(-0.8164965809390571, -0.4082482904581974, 0.4082482904468662), edge\_direction=(0.9999999999732445, 0.0, 2.0000000000575113)); gvec — a Vector \[red\] labelled "nabla f" drawn in axes3 (start=(1.0, 0.5, 0.0), end=(1.8, 0.9, 0.0)) Actions: - [09:26.599](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=566.5990833333335): imp is shown on the screen, written out. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): axes3 is hidden from the screen — left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): surf is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): pt is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): tu is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): tv is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): tp is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): gvec is hidden from the screen — axes3 left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): dif is hidden from the screen — left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): dvf is hidden from the screen — left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): head\_diff is hidden from the screen — left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): imp is hidden from the screen — left the board. - [09:31.342](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=571.3415833333335): tan is hidden from the screen — left the board. ##### [09:32.542](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=572.5415833333334) Narration: Attenzione: in più variabili, l'esistenza di tutte le derivate parziali non garantisce nemmeno la continuità. Guardiamo questa funzione: sugli assi vale zero, quindi le due derivate parziali nell'origine esistono e sono nulle. Board: Empty. Actions: - [09:32.542](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=572.5415833333334): head\_cex is shown on the screen, written out. - [09:39.439](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=579.4390833333334): cex is shown on the screen, written out. - [09:41.359](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=581.3590833333335): cax is shown on the screen, written out. - [09:42.079](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=582.0790833333335): l0 is shown on the screen, written out. ##### [09:47.237](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=587.2365833333334) Narration: Ma lungo la retta y uguale a emme x vale costantemente emme fratto uno più emme quadro: un mezzo sulla bisettrice, meno un mezzo sull'altra. Il limite nell'origine non esiste, e f non è continua. Board: cex — a Math \[text\] that says "$f(x, y) = frac(x y, x^2 + y^2), quad f(0, 0) = 0$"; cax — an Axes (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), aspect=(1.0, 1.0)); head\_cex — a Heading that says "Derivate parziali senza differenziabilità"; l0 — a Line \[blue\] labelled "f = 0" drawn in cax (start=(-1.8, 0.0), end=(1.8, 0.0)) Actions: - [09:47.959](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=587.9590833333334): cex\_m is shown on the screen, written out. - [09:54.399](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=594.3990833333335): l1 is shown on the screen, written out. - [09:55.239](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=595.2390833333335): l2 is shown on the screen, written out. ##### [10:1.322](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=601.3215833333334) Narration: Il teorema del differenziale totale fornisce una condizione sufficiente: se le derivate parziali esistono vicino a x zero e sono continue in x zero, allora f è differenziabile. Ogni funzione di classe C uno è dunque differenziabile e continua. Board: cex — a Math \[text\] that says "$f(x, y) = frac(x y, x^2 + y^2), quad f(0, 0) = 0$"; cex\_m — a Math \[text\] that says "$f(x, m x) = frac(m, 1 + m^2)$"; cax — an Axes (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), aspect=(1.0, 1.0)); head\_cex — a Heading that says "Derivate parziali senza differenziabilità"; l0 — a Line \[blue\] labelled "f = 0" drawn in cax (start=(-1.8, 0.0), end=(1.8, 0.0)); l1 — a Line \[green\] labelled "f = 1/2" drawn in cax (start=(-1.4, -1.4), end=(1.4, 1.4)); l2 — a Line \[red\] labelled "f = -1/2" drawn in cax (start=(-1.4, 1.4), end=(1.4, -1.4)) Actions: - [10:2.199](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=602.1990833333334): tdt is shown on the screen, written out. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): cax is hidden from the screen — left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): l0 is hidden from the screen — cax left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): l1 is hidden from the screen — cax left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): l2 is hidden from the screen — cax left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): cex is hidden from the screen — left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): cex\_m is hidden from the screen — left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): head\_cex is hidden from the screen — left the board. - [10:17.094](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=617.0940833333334): tdt is hidden from the screen — left the board. ##### [10:18.294](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=618.2940833333334) Narration: Per un campo vettoriale da erre enne a erre emme, ogni componente ha il suo gradiente. Disposti come righe, formano la matrice Jacobiana, con emme righe ed enne colonne. Board: Empty. Actions: - [10:18.294](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=618.2940833333334): head\_jac is shown on the screen, written out. - [10:26.239](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=626.2390833333335): j0 is shown on the screen, written out. ##### [10:29.924](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=629.9240833333334) Narration: Per esempio, per questa f del piano in sé, la Jacobiana nel punto uno, uno è la matrice due, meno uno, uno, uno. Board: j0 — a Math \[text\] that says "$J\_f (x\_0) = (frac(partial f\_i, partial x\_j)(x\_0))\_(i, j), quad m times n$"; head\_jac — a Heading that says "La matrice Jacobiana" Actions: - [10:30.279](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=630.2790833333335): j1 is shown on the screen, written out. - [10:33.159](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=633.1590833333335): j2 is shown on the screen, written out. ##### [10:38.484](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=638.4840833333334) Narration: Il suo significato è questo: vicino al punto, f si comporta come l'applicazione lineare data dalla Jacobiana. Guardate come questa matrice deforma il quadrato verde: è ciò che f fa, in prima approssimazione, a un piccolo intorno. Board: j0 — a Math \[text\] that says "$J\_f (x\_0) = (frac(partial f\_i, partial x\_j)(x\_0))\_(i, j), quad m times n$"; j1 — a Math \[text\] that says "$f(x, y) = (x^2 - y, thin x y)$"; j2 — a Math \[text\] that says "$J\_f (x, y) = mat(2 x, -1; y, x), quad J\_f (1, 1) = mat(2, -1; 1, 1)$"; head\_jac — a Heading that says "La matrice Jacobiana" Actions: - [10:38.999](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=638.9990833333335): jax is shown on the screen, written out. - [10:38.999](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=638.9990833333335): grid is shown on the screen, written out. - [10:38.999](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=638.9990833333335): square is shown on the screen, written out. - [10:40.639](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=640.6390833333335): j3 is shown on the screen, written out. - [10:47.559](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=647.5590833333334): jax: put this frame's data space back where it started (reset\_matrix). - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): head\_jac is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): j0 is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): j1 is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): j2 is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): j3 is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): jax is hidden from the screen — left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): grid is hidden from the screen — jax left the board. - [10:53.245](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=653.2450000000001): square is hidden from the screen — jax left the board. ### Scene 5: [Hessiana e punti critici](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=654.2866666666667) Span: 10:54.287–14:40.969 (654.2866666666667s–880.9693541666668s). #### Objects - c0: a Math \[text\] that says "$frac(dif, dif t) f(g(t)) = nabla f(g(t)) dot g'(t)$" - c1: a Math \[text\] that says "$h = f(g(x)) =\> J\_h (x) = J\_f (g(x)) thin J\_g (x)$" - c2: a Math \[text\] that says "$frac(partial h\_i, partial x\_j) = sum\_k frac(partial f\_i, partial y\_k) frac(partial g\_k, partial x\_j)$" - cap\_max: a Math \[text\] that says "$lambda\_i \< 0 : upright("massimo")$" - cap\_min: a Math \[text\] that says "$lambda\_i \> 0 : upright("minimo")$" - cap\_sad: a Math \[text\] that says "$lambda\_1 lambda\_2 \< 0 : upright("sella")$" - cax2: an Axes (x\_range=(-2.0, 2.0), y\_range=(0.0, 2.5)) - frame: an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)) - frame\_2: an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)) - frame\_3: an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)) - gradv: a Vector \[red\] labelled "nabla f" drawn in cax2 (start=(0.8, 1.008), end=(1.44, 1.814)) - h0: a Math \[text\] that says "$H\_f (x) = (frac(partial^2 f, partial x\_i partial x\_j)(x))\_(i, j = 1, dots, n)$" - h1: a Math \[text\] that says "$f(x, y) = x^3 y + y^2$" - h2: a Math \[text\] that says "$f\_x = 3 x^2 y, quad f\_y = x^3 + 2 y$" - h3: a Math \[text\] that says "$f\_(x y) = 3 x^2 = f\_(y x)$" - head\_chain: a Heading that says "Derivata della funzione composta" - head\_cls: a Heading that says "Il segno di $q(h) = lambda\_1 y\_1^2 + dots + lambda\_n y\_n^2$" - head\_ex: a Heading that says "Un esempio completo" - head\_h: a Heading that says "Derivate seconde e teorema di Schwarz" - head\_syl: a Heading that says "Criterio di Sylvester" - head\_tay: a Heading that says "Formula di Taylor al secondo ordine" - lev: a LevelCurves \[gray\] drawn in cax2 (function=\, values=(0.5, 1.0, 2.0, 3.0)) - mark: a Point \[yellow\] labelled "g(t\_0)" drawn in cax2 (location=(0.8, 1.008)) - path: a ParametricCurve \[blue\] labelled "g(t)" drawn in cax2 (function=\, t\_range=(-1.8, 1.8)) - pmin: a Point \[green\] labelled "(1, 0)" drawn in wax (location=(1.0, 0.0, -2.0)) - psad: a Point \[red\] labelled "(-1, 0)" drawn in wax (location=(-1.0, 0.0, 2.0)) - schwarz: a Panel that says "Se le derivate seconde miste sono continue, allora $f\_(x\_i x\_j) = f\_(x\_j x\_i)$: la matrice Hessiana di una funzione $C^2$ è simmetrica." - shape: a Surface \[green\] drawn in frame (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)) - shape\_2: a Surface \[red\] drawn in frame\_2 (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)) - shape\_3: a Surface \[yellow\] drawn in frame\_3 (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)) - sy0: a Math \[text\] that says "$D\_k = det (f\_(x\_i x\_j))\_(i, j \<= k), quad k = 1, dots, n$" - sy1: a Math \[text\] that says "$n = 2 : quad D\_1 = f\_(x x), quad D\_2 = f\_(x x) f\_(y y) - f\_(x y)^2$" - sytab: a Table \[text\] that says "condizione punto stazionario $D\_k \> 0$ per ogni $k$ minimo locale $(-1)^k D\_k \> 0$ per ogni $k$ massimo locale $n = 2$, $D\_2 \< 0$ sella" (rows=(('condizione', 'punto stazionario'), ('$D\_k \> 0$ per ogni $k$'…, header=True) - t0: a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + frac(1, 2) h^T H\_f (x\_0) h + o(norm(h)^2)$" - t1: a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + frac(1, 2) h^T H\_f (xi) h, quad xi in \[x\_0, x\_0 + h\]$" - t2: a Math \[text\] that says "$nabla f(x\_0) = 0 =\> f(x\_0 + h) - f(x\_0) = frac(1, 2) q(h) + o(norm(h)^2)$" - t3: a Math \[text\] that says "$q(h) = h^T H\_f (x\_0) h = sum\_(i, j) f\_(x\_i x\_j)(x\_0) h\_i h\_j$" - vel: a Vector \[green\] labelled "g'(t\_0)" drawn in cax2 (start=(0.8, 1.008), end=(1.6, 0.624)) - w0: a Math \[text\] that says "$f(x, y) = x^3 - 3 x + y^2$" - w1: a Math \[text\] that says "$nabla f = (3 x^2 - 3, thin 2 y) = 0 =\> (plus.minus 1, 0)$" - w2: a Math \[text\] that says "$H\_f = mat(6 x, 0; 0, 2)$" - w3: a Math \[text\] that says "$(1, 0) : D\_1 = 6, thin D\_2 = 12 =\> upright("minimo")$" - w4: a Math \[text\] that says "$(-1, 0) : D\_2 = -12 \< 0 =\> upright("sella")$" - wax: an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-1.5, 1.5), z\_range=(-2.5, 4.5)) - wsurf: a Surface \[blue\] drawn in wax (function=\, u\_range=(-2.0, 2.0), v\_range=(-1.5, 1.5)) #### Beats ##### [10:54.287](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=654.2866666666667) Narration: Derivando di nuovo le derivate parziali otteniamo le derivate seconde. Raccolte in una matrice enne per enne, formano la Hessiana. Board: Empty. Actions: - [10:54.287](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=654.2866666666667): head\_h is shown on the screen, written out. - [10:58.767](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=658.7666666666668): h0 is shown on the screen, written out. ##### [11:3.062](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=663.0616666666667) Narration: Proviamo con x al cubo per y, più y quadro. Le derivate prime sono tre x quadro y, e x al cubo più due y. Board: h0 — a Math \[text\] that says "$H\_f (x) = (frac(partial^2 f, partial x\_i partial x\_j)(x))\_(i, j = 1, dots, n)$"; head\_h — a Heading that says "Derivate seconde e teorema di Schwarz" Actions: - [11:3.207](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=663.2066666666667): h1 is shown on the screen, written out. - [11:7.447](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=667.4466666666667): h2 is shown on the screen, written out. ##### [11:12.674](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=672.6741666666668) Narration: Ora deriviamo f x rispetto a y, e f y rispetto a x: in entrambi i casi otteniamo tre x quadro. Non è una coincidenza. Board: h0 — a Math \[text\] that says "$H\_f (x) = (frac(partial^2 f, partial x\_i partial x\_j)(x))\_(i, j = 1, dots, n)$"; h1 — a Math \[text\] that says "$f(x, y) = x^3 y + y^2$"; h2 — a Math \[text\] that says "$f\_x = 3 x^2 y, quad f\_y = x^3 + 2 y$"; head\_h — a Heading that says "Derivate seconde e teorema di Schwarz" Actions: - [11:13.167](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=673.1666666666667): h3 is shown on the screen, written out. ##### [11:23.089](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=683.0891666666668) Narration: È il teorema di Schwarz: se le derivate miste sono continue, l'ordine di derivazione non conta. La Hessiana di una funzione di classe C due è quindi simmetrica, e avrà autovalori reali. Board: h0 — a Math \[text\] that says "$H\_f (x) = (frac(partial^2 f, partial x\_i partial x\_j)(x))\_(i, j = 1, dots, n)$"; h1 — a Math \[text\] that says "$f(x, y) = x^3 y + y^2$"; h2 — a Math \[text\] that says "$f\_x = 3 x^2 y, quad f\_y = x^3 + 2 y$"; h3 — a Math \[text\] that says "$f\_(x y) = 3 x^2 = f\_(y x)$"; head\_h — a Heading that says "Derivate seconde e teorema di Schwarz" Actions: - [11:23.967](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=683.9666666666667): schwarz is shown on the screen, written out. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): h0 is hidden from the screen — left the board. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): h1 is hidden from the screen — left the board. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): h2 is hidden from the screen — left the board. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): h3 is hidden from the screen — left the board. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): head\_h is hidden from the screen — left the board. - [11:34.719](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=694.7191666666668): schwarz is hidden from the screen — left the board. ##### [11:35.919](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=695.9191666666668) Narration: Come si deriva una composizione? Seguiamo una curva gi di t nel piano, mentre f ha queste curve di livello. La derivata di f lungo la curva è il gradiente di f nel punto, scalare la velocità della curva. Board: Empty. Actions: - [11:35.919](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=695.9191666666668): head\_chain is shown on the screen, written out. - [11:38.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=698.2466666666668): cax2 is shown on the screen, written out. - [11:38.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=698.2466666666668): path is shown on the screen, written out. - [11:42.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=702.2466666666668): lev is shown on the screen, written out. - [11:43.207](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=703.2066666666667): c0 is shown on the screen, written out. - [11:45.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=705.2466666666668): gradv is shown on the screen, written out. - [11:46.287](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=706.2866666666667): mark is shown on the screen, written out. - [11:46.927](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=706.9266666666667): vel is shown on the screen, written out. ##### [11:49.217](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=709.2166666666667) Narration: Il caso generale è altrettanto pulito: la Jacobiana della composizione è il prodotto delle Jacobiane, calcolate nei punti giusti. Scritto per componenti, è una somma su tutte le variabili intermedie. Board: c0 — a Math \[text\] that says "$frac(dif, dif t) f(g(t)) = nabla f(g(t)) dot g'(t)$"; cax2 — an Axes (x\_range=(-2.0, 2.0), y\_range=(0.0, 2.5)); head\_chain — a Heading that says "Derivata della funzione composta"; path — a ParametricCurve \[blue\] labelled "g(t)" drawn in cax2 (function=\, t\_range=(-1.8, 1.8)); lev — a LevelCurves \[gray\] drawn in cax2 (function=\, values=(0.5, 1.0, 2.0, 3.0)); mark — a Point \[yellow\] labelled "g(t\_0)" drawn in cax2 (location=(0.8, 1.008)); gradv — a Vector \[red\] labelled "nabla f" drawn in cax2 (start=(0.8, 1.008), end=(1.44, 1.814)); vel — a Vector \[green\] labelled "g'(t\_0)" drawn in cax2 (start=(0.8, 1.008), end=(1.6, 0.624)) Actions: - [11:49.807](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=709.8066666666667): c1 is shown on the screen, written out. - [11:57.767](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=717.7666666666668): c2 is shown on the screen, written out. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): c0 is hidden from the screen — left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): c1 is hidden from the screen — left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): c2 is hidden from the screen — left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): cax2 is hidden from the screen — left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): path is hidden from the screen — cax2 left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): lev is hidden from the screen — cax2 left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): mark is hidden from the screen — cax2 left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): gradv is hidden from the screen — cax2 left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): vel is hidden from the screen — cax2 left the board. - [12:0.877](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=720.8766666666668): head\_chain is hidden from the screen — left the board. ##### [12:2.077](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=722.0766666666667) Narration: Con la Hessiana possiamo scrivere lo sviluppo di Taylor al secondo ordine. Il valore, più il termine lineare del gradiente, più metà della forma quadratica della Hessiana, più un resto di Peano, piccolo rispetto al quadrato della norma di h. Board: Empty. Actions: - [12:2.077](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=722.0766666666667): head\_tay is shown on the screen, written out. - [12:4.087](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=724.0866666666667): t0 is shown on the screen, written out. - [12:8.047](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=728.0466666666667): t0 (the "nabla f(x\_0) dot h" part) is emphasized. - [12:10.527](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=730.5266666666668): t0 (the "frac(1, 2) h^T H\_f (x\_0) h" part) is emphasized. - [12:10.527](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=730.5266666666668): t0 (the "nabla f(x\_0) dot h" part) is no longer emphasized. - [12:13.007](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=733.0066666666668): t0 (the "frac(1, 2) h^T H\_f (x\_0) h" part) is no longer emphasized. - [12:13.007](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=733.0066666666668): t0 (the "o(norm(h)^2)" part) is emphasized. - [12:16.222](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=736.2216666666667): t0 (the "o(norm(h)^2)" part) is no longer emphasized. ##### [12:16.822](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=736.8216666666667) Narration: Se f è di classe C due vale anche la forma di Lagrange: niente resto, ma la Hessiana è calcolata in un punto xi intermedio sul segmento fra x zero e x zero più h. Board: t0 — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + frac(1, 2) h^T H\_f (x\_0) h + o(norm(h)^2)$"; head\_tay — a Heading that says "Formula di Taylor al secondo ordine" Actions: - [12:19.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=739.7266666666667): t1 is shown on the screen, written out. ##### [12:28.407](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=748.4066666666668) Narration: Ora il punto chiave. In un punto stazionario il gradiente è nullo, e la differenza fra f di x zero più h e f di x zero è governata dalla forma quadratica q. Board: t0 — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + frac(1, 2) h^T H\_f (x\_0) h + o(norm(h)^2)$"; t1 — a Math \[text\] that says "$f(x\_0 + h) = f(x\_0) + nabla f(x\_0) dot h + frac(1, 2) h^T H\_f (xi) h, quad xi in \[x\_0, x\_0 + h\]$"; head\_tay — a Heading that says "Formula di Taylor al secondo ordine" Actions: - [12:30.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=750.7266666666667): t2 is shown on the screen, written out. - [12:33.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=753.2466666666668): t3 is shown on the screen, written out. - [12:39.227](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=759.2266666666667): head\_tay is hidden from the screen — left the board. - [12:39.227](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=759.2266666666667): t0 is hidden from the screen — left the board. - [12:39.227](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=759.2266666666667): t1 is hidden from the screen — left the board. - [12:39.227](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=759.2266666666667): t2 is hidden from the screen — left the board. - [12:39.227](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=759.2266666666667): t3 is hidden from the screen — left the board. ##### [12:40.427](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=760.4266666666667) Narration: Poiché la Hessiana è simmetrica, il teorema spettrale la diagonalizza in una base ortonormale di autovettori. In quelle coordinate la forma quadratica diventa una somma di autovalori per quadrati. Board: Empty. Actions: - [12:40.427](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=760.4266666666667): head\_cls is shown on the screen, written out. ##### [12:52.124](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=772.1241666666667) Narration: Se tutti gli autovalori sono positivi, la forma è definita positiva e il punto è di minimo locale: il grafico è una coppa. Board: head\_cls — a Heading that says "Il segno di $q(h) = lambda\_1 y\_1^2 + dots + lambda\_n y\_n^2$" Actions: - [12:53.687](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=773.6866666666667): frame is shown on the screen, written out. - [12:53.687](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=773.6866666666667): shape is shown on the screen, written out. - [12:57.127](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=777.1266666666668): frame moves to a new place on the board. - [12:57.127](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=777.1266666666668): cap\_min is shown on the screen, written out. ##### [13:0.299](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=780.2991666666667) Narration: Se sono tutti negativi, la forma è definita negativa: massimo locale, il grafico è una cupola. Board: frame — an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)); cap\_min — a Math \[text\] that says "$lambda\_i \> 0 : upright("minimo")$"; head\_cls — a Heading that says "Il segno di $q(h) = lambda\_1 y\_1^2 + dots + lambda\_n y\_n^2$"; shape — a Surface \[green\] drawn in frame (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)) Actions: - [13:1.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=781.2466666666668): frame\_2 is shown on the screen, written out. - [13:1.247](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=781.2466666666668): shape\_2 is shown on the screen, written out. - [13:3.847](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=783.8466666666668): cap\_max is shown on the screen, written out. ##### [13:7.024](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=787.0241666666667) Narration: Se ci sono autovalori di segno opposto, la forma è indefinita e il punto è una sella: si sale lungo una direzione e si scende lungo l'altra. Se invece qualche autovalore è nullo, il secondo ordine non basta a decidere. Board: frame — an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)); cap\_min — a Math \[text\] that says "$lambda\_i \> 0 : upright("minimo")$"; frame\_2 — an Axes3D (x\_range=(-1.2, 1.2), y\_range=(-1.2, 1.2), z\_range=(-2.2, 2.2)); cap\_max — a Math \[text\] that says "$lambda\_i \< 0 : upright("massimo")$"; head\_cls — a Heading that says "Il segno di $q(h) = lambda\_1 y\_1^2 + dots + lambda\_n y\_n^2$"; shape — a Surface \[green\] drawn in frame (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)); shape\_2 — a Surface \[red\] drawn in frame\_2 (function=\, u\_range=(-1.0, 1.0), v\_range=(-1.0, 1.0)) Actions: - [13:8.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=788.7266666666668): frame\_3 is shown on the screen, written out. - [13:8.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=788.7266666666668): shape\_3 is shown on the screen, written out. - [13:11.767](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=791.7666666666668): cap\_sad is shown on the screen, written out. - [13:12.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=792.7266666666668): frame\_3 turns to a new orientation in its own slot. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): cap\_max is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): cap\_min is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): cap\_sad is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): frame is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): shape is hidden from the screen — frame left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): frame\_2 is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): shape\_2 is hidden from the screen — frame\_2 left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): frame\_3 is hidden from the screen — left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): shape\_3 is hidden from the screen — frame\_3 left the board. - [13:20.139](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=800.1391666666667): head\_cls is hidden from the screen — left the board. ##### [13:21.339](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=801.3391666666668) Narration: Calcolare gli autovalori può essere faticoso. Il criterio di Sylvester usa invece i minori principali di testa: i determinanti dei blocchi quadrati in alto a sinistra della Hessiana. Board: Empty. Actions: - [13:21.339](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=801.3391666666668): head\_syl is shown on the screen, written out. - [13:26.167](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=806.1666666666667): sy0 is shown on the screen, written out. ##### [13:32.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=812.1441666666667) Narration: Se sono tutti positivi, la Hessiana è definita positiva: minimo locale. Se hanno segni alternati, a partire da un primo minore negativo, è definita negativa: massimo locale. In due variabili, un determinante negativo indica una sella. Board: sy0 — a Math \[text\] that says "$D\_k = det (f\_(x\_i x\_j))\_(i, j \<= k), quad k = 1, dots, n$"; head\_syl — a Heading that says "Criterio di Sylvester" Actions: - [13:32.144](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=812.1441666666667): sytab is shown on the screen, written out. - [13:33.087](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=813.0866666666667): sytab is shown on the screen, written out. - [13:37.807](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=817.8066666666667): sytab is shown on the screen, written out. - [13:44.287](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=824.2866666666667): sy1 is shown on the screen, written out. - [13:47.087](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=827.0866666666667): sytab is shown on the screen, written out. - [13:47.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=827.7016666666667): head\_syl is hidden from the screen — left the board. - [13:47.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=827.7016666666667): sy0 is hidden from the screen — left the board. - [13:47.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=827.7016666666667): sy1 is hidden from the screen — left the board. - [13:47.702](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=827.7016666666667): sytab is hidden from the screen — left the board. ##### [13:48.902](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=828.9016666666668) Narration: Chiudiamo con un esempio completo: f uguale a x al cubo meno tre x, più y quadro. Il grafico ha una valle e un passo di montagna. Board: Empty. Actions: - [13:48.902](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=828.9016666666668): head\_ex is shown on the screen, written out. - [13:49.967](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=829.9666666666667): w0 is shown on the screen, written out. - [13:52.807](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=832.8066666666667): wax is shown on the screen, written out. - [13:52.807](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=832.8066666666667): wsurf is shown on the screen, written out. - [13:55.727](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=835.7266666666667): wax turns to a new orientation in its own slot. ##### [13:59.069](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=839.0691666666667) Narration: Il gradiente si annulla quando x quadro vale uno e y vale zero: due punti stazionari, uno zero e meno uno zero. La Hessiana è diagonale, con sei x e due. Board: w0 — a Math \[text\] that says "$f(x, y) = x^3 - 3 x + y^2$"; wax — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-1.5, 1.5), z\_range=(-2.5, 4.5)); head\_ex — a Heading that says "Un esempio completo"; wsurf — a Surface \[blue\] drawn in wax (function=\, u\_range=(-2.0, 2.0), v\_range=(-1.5, 1.5)) Actions: - [13:59.447](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=839.4466666666667): w1 is shown on the screen, written out. - [14:7.767](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=847.7666666666667): w2 is shown on the screen, written out. ##### [14:11.554](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=851.5541666666668) Narration: In uno zero i minori valgono sei e dodici, entrambi positivi: è un minimo locale, il fondo della valle. Board: w0 — a Math \[text\] that says "$f(x, y) = x^3 - 3 x + y^2$"; w1 — a Math \[text\] that says "$nabla f = (3 x^2 - 3, thin 2 y) = 0 =\> (plus.minus 1, 0)$"; w2 — a Math \[text\] that says "$H\_f = mat(6 x, 0; 0, 2)$"; wax — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-1.5, 1.5), z\_range=(-2.5, 4.5)); head\_ex — a Heading that says "Un esempio completo"; wsurf — a Surface \[blue\] drawn in wax (function=\, u\_range=(-2.0, 2.0), v\_range=(-1.5, 1.5)) Actions: - [14:12.767](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=852.7666666666667): w3 is shown on the screen, written out. - [14:16.447](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=856.4466666666667): pmin is shown on the screen, written out. ##### [14:19.399](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=859.3991666666667) Narration: In meno uno zero il determinante vale meno dodici: è una sella. Lungo x la funzione ha un massimo, lungo y un minimo. Board: w0 — a Math \[text\] that says "$f(x, y) = x^3 - 3 x + y^2$"; w1 — a Math \[text\] that says "$nabla f = (3 x^2 - 3, thin 2 y) = 0 =\> (plus.minus 1, 0)$"; w2 — a Math \[text\] that says "$H\_f = mat(6 x, 0; 0, 2)$"; w3 — a Math \[text\] that says "$(1, 0) : D\_1 = 6, thin D\_2 = 12 =\> upright("minimo")$"; wax — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-1.5, 1.5), z\_range=(-2.5, 4.5)); head\_ex — a Heading that says "Un esempio completo"; wsurf — a Surface \[blue\] drawn in wax (function=\, u\_range=(-2.0, 2.0), v\_range=(-1.5, 1.5)); pmin — a Point \[green\] labelled "(1, 0)" drawn in wax (location=(1.0, 0.0, -2.0)) Actions: - [14:20.967](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=860.9666666666667): w4 is shown on the screen, written out. - [14:23.207](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=863.2066666666667): psad is shown on the screen, written out. ##### [14:28.589](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=868.5891666666668) Narration: Abbiamo percorso la strada intera: dalla norma e dagli intorni ai limiti, dalle derivate al piano tangente, fino a leggere nella Hessiana la forma di una funzione vicino ai suoi punti critici. Board: w0 — a Math \[text\] that says "$f(x, y) = x^3 - 3 x + y^2$"; w1 — a Math \[text\] that says "$nabla f = (3 x^2 - 3, thin 2 y) = 0 =\> (plus.minus 1, 0)$"; w2 — a Math \[text\] that says "$H\_f = mat(6 x, 0; 0, 2)$"; w3 — a Math \[text\] that says "$(1, 0) : D\_1 = 6, thin D\_2 = 12 =\> upright("minimo")$"; w4 — a Math \[text\] that says "$(-1, 0) : D\_2 = -12 \< 0 =\> upright("sella")$"; wax — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-1.5, 1.5), z\_range=(-2.5, 4.5)); head\_ex — a Heading that says "Un esempio completo"; wsurf — a Surface \[blue\] drawn in wax (function=\, u\_range=(-2.0, 2.0), v\_range=(-1.5, 1.5)); pmin — a Point \[green\] labelled "(1, 0)" drawn in wax (location=(1.0, 0.0, -2.0)); psad — a Point \[red\] labelled "(-1, 0)" drawn in wax (location=(-1.0, 0.0, 2.0)) Actions: - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): head\_ex is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): w0 is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): w1 is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): w2 is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): w3 is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): w4 is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): wax is hidden from the screen — left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): wsurf is hidden from the screen — wax left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): pmin is hidden from the screen — wax left the board. - [14:39.928](https://academa.ai/@gawsli/lectures/calcolo-differenziale-e-topologia-in-r-n?t=879.9276875): psad is hidden from the screen — wax left the board.