# Fondamenti di Statica dei Fluidi e Tensore degli Sforzi

> Una lezione universitaria sui fondamenti della statica dei fluidi. Si parte dalla definizione di fluido e dall'ipotesi del mezzo continuo, con il volume rappresentativo e i campi di densità, pressione e velocità. Si mostra perché un fluido in quiete non sostiene sforzi di taglio, si riduce il tensore degli sforzi alla forma idrostatica sigma\_ij = -P delta\_ij e si dimostra con il prisma infinitesimo che la pressione è isotropa. Infine si ricava l'equazione fondamentale grad P = rho f e, per forze conservative, si prova che isobare, equipotenziali e isopicne coincidono e sono ortogonali alle forze.

- Canonical watch page: [Fondamenti di Statica dei Fluidi e Tensore degli Sforzi](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi)
- Publisher: [Academa, Inc.](https://academa.ai)
- Subject: Physics
- Published: 2026-10-11T11:04:23.192Z
- Updated: 2026-10-11T11:04:23.192Z
- Duration: PT384S (6 minutes 24 seconds)
- Chapters: 4
- Views: 2
- Language: en-US
- Access: Free
- Video stream: [HLS content](https://academa.ai/media/l/01M4N99AMEY4ZTYVJYXKSJ7578/0/dark/master.m3u8)
- Embed: [Player](https://academa.ai/embed/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi)
- Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M4N99AMEY4ZTYVJYXKSJ7578/0/semantic.json)
- Thumbnail: [Image](https://academa.ai/media/l/01M4N99AMEY4ZTYVJYXKSJ7578/0/dark/poster.jpg)

## Description

Fondamenti di statica dei fluidi: mezzo continuo, tensore degli sforzi a riposo, pressione isotropa, legge ∇P = ρf e superfici isobare.

## Chapters

- [00:00–01:9.441 · Il fluido come mezzo continuo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0)
- [01:9.441–02:24.654 · Niente taglio a riposo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999)
- [02:24.654–04:19.021 · La pressione è uno scalare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=144.65354166666668)
- [04:19.021–06:24 · L'equazione fondamentale e le superfici isobare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=259.021125)

## Transcript

### [00:00 · Il fluido come mezzo continuo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0)

Un fluido, liquido o gas, è una sostanza che si deforma con continuità sotto uno sforzo di taglio, per quanto piccolo. Non possiede una forma propria: un liquido conserva il suo volume, mentre un gas occupa tutto il recipiente. Al microscopio, però, la materia è discreta: molecole separate da spazio vuoto. Nessuno vuole risolvere le equazioni del moto per dieci alla ventitré particelle. Prendiamo allora un piccolo volume, delta V, e misuriamo la massa che contiene. Se lo restringiamo troppo contiene pochissime molecole, e il rapporto tra massa e volume oscilla in modo casuale. Ci fermiamo quindi a una scala delta V star: piccolissima rispetto al sistema, ma abbastanza grande da contenere moltissime molecole. La densità è il limite del rapporto tra massa e volume verso questa scala. Con lo stesso limite, ogni punto riceve i suoi campi: la densità in chilogrammi al metro cubo, la pressione in pascal, e la velocità in metri al secondo. Da qui in avanti il fluido è un mezzo continuo: campi regolari, che possiamo derivare come qualsiasi funzione.

### [01:9.441 · Niente taglio a riposo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999)

Spingiamo parallelamente alla faccia superiore di un blocco solido. Il blocco si inclina di un certo angolo e poi si ferma: una forza tangenziale interna equilibra la spinta. Con un fluido la stessa spinta non trova alcuna nuova posizione di equilibrio: finché agisce, il fluido continua a scorrere. Quindi, in un fluido in quiete, gli sforzi tangenziali sono nulli: tau uguale a zero. Le forze tra parti adiacenti, o contro le pareti, sono soltanto perpendicolari alle superfici. Le forze interne di un mezzo continuo si descrivono con il tensore degli sforzi. Su un elemento di superficie d A, con normale n, agisce la forza sigma per n, d A. In tre dimensioni sigma è una matrice tre per tre. Sulla diagonale stanno gli sforzi normali, perpendicolari alle facce; fuori diagonale gli sforzi di taglio, paralleli alle facce. A riposo il taglio è nullo, quindi tutti i termini fuori diagonale si annullano. Restano soltanto i tre sforzi normali. Sono uguali fra loro? Per ora sappiamo soltanto che sono compressioni. Per deciderlo ci serve un piccolo prisma di fluido.

### [02:24.654 · La pressione è uno scalare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=144.65354166666668)

La pressione in un punto è il rapporto tra la forza normale e la superficie su cui agisce. La forza è un vettore; la pressione, invece, non dipende da come orientiamo la superficie. Dimostriamolo. Prendiamo un prisma triangolare di fluido, con cateti d x e d y e ipotenusa d s. La profondità d z è comune a tutte le facce, e la semplifichiamo fin da subito. Su ogni faccia la pressione spinge verso il fluido. Chiamiamo theta l'angolo alla base: allora d s seno theta vale d y, e d s coseno theta vale d x. Lungo x: P x per d y, meno P n per d s seno theta, fa zero. Con la geometria i due prodotti hanno lo stesso fattore d y, e resta P x uguale a P n. Lungo y compare anche il peso del prisma, proporzionale alla sua area. Dividendo per d x resta P y meno P n meno un mezzo rho g d y, uguale a zero. Ora restringiamo il prisma verso un punto. Il peso è di ordine superiore e svanisce con d y, quindi anche P y è uguale a P n. Dunque P x, P y e P n coincidono, qualunque sia l'angolo theta: la pressione è la stessa in tutte le direzioni. È uno scalare. Torniamo al tensore, che a riposo era diagonale. Ogni sforzo normale è una compressione di modulo P, quindi tutti e tre valgono meno P. Raccogliendo meno P resta la matrice identità, che in notazione indiciale è la delta di Kronecker, delta i j. In una sola riga, sigma i j uguale a meno P delta i j dice due cose: se i è diverso da j lo sforzo è nullo, niente taglio; se i è uguale a j vale meno P, la stessa in ogni direzione.

### [04:19.021 · L'equazione fondamentale e le superfici isobare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=259.021125)

Ora mettiamo in equilibrio un cubetto di fluido di volume d V. Su di esso agiscono forze di superficie, dovute alla pressione, e forze di volume, proporzionali alla massa. Lungo x la faccia sinistra riceve P di x per d y d z, quella destra P di x più d x per la stessa area. La differenza è meno la derivata parziale di P rispetto a x, per d V. Lo stesso accade lungo y e lungo z: la forza netta di pressione è meno il gradiente di P per d V. La forza di volume è rho f d V, dove f è la forza per unità di massa. In equilibrio la somma è nulla. Dividendo per d V si ottiene l'equazione fondamentale della statica dei fluidi: il gradiente di pressione è uguale a rho f. Supponiamo ora che la forza sia conservativa: f è meno il gradiente di un potenziale U. Qui U cresce con la distanza dal centro, le sue superfici di livello sono questi cerchi, e le frecce di forza puntano verso il centro. Sostituendo, il gradiente di P vale meno rho per il gradiente di U. Ora applichiamo il rotore: il rotore di un gradiente è nullo, quindi anche il rotore di rho per il gradiente di U deve annullarsi. Sviluppando il prodotto, il termine con il rotore del gradiente di U sparisce, e resta: gradiente di rho, vettore gradiente di U, uguale a zero. Un prodotto vettoriale è nullo solo se i due gradienti sono paralleli. In questo punto il gradiente di U punta in fuori, quello di P verso il centro, sulla stessa retta. Un gradiente di densità inclinato, come questo, darebbe un prodotto diverso da zero: non è ammesso. La densità può variare solo lungo la stessa direzione, ovunque. Quindi pressione, potenziale e densità sono costanti sulle stesse superfici: isobare, equipotenziali e isopicne coincidono, e sono ortogonali alle linee di forza. In un campo di gravità uniforme queste superfici diventano piani orizzontali, come il pelo libero di un lago in quiete.

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## Complete audiovisual record

Immutable source: [semantic.json](https://academa.ai/media/l/01M4N99AMEY4ZTYVJYXKSJ7578/0/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: [Il fluido come mezzo continuo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0)

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#### Beats

##### [00:00](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0)

Narration: Un fluido, liquido o gas, è una sostanza che si deforma con continuità sotto uno sforzo di taglio, per quanto piccolo. Non possiede una forma propria: un liquido conserva il suo volume, mentre un gas occupa tutto il recipiente.

Board: Empty.

Actions:
- [00:00](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0): heading is shown on the screen, written out.
- [00:00](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=0): definition is shown on the screen, written out.
- [00:6.08](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=6.08): definition (the "sforzo di taglio" part) is emphasized.
- [00:8.32](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=8.32): definition (the "sforzo di taglio" part) is no longer emphasized.
- [00:14.623](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=14.6225): definition is hidden from the screen — left the board.

##### [00:15.223](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.2225)

Narration: Al microscopio, però, la materia è discreta: molecole separate da spazio vuoto. Nessuno vuole risolvere le equazioni del moto per dieci alla ventitré particelle.

Board: heading — a Heading that says "Il fluido come mezzo continuo"

Actions:
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- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_7 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_8 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_9 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_10 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_11 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_12 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_13 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_14 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_15 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_16 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_17 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_18 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_19 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_20 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_21 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_22 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_23 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_24 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_25 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_26 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_27 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_28 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_29 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_30 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_31 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_32 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_33 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_34 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_35 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_36 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_37 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_38 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_39 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_40 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_41 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_42 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_43 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_44 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_45 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_46 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_47 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_48 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_49 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_50 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_51 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_52 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_53 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_54 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_55 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_56 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_57 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_58 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_59 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_60 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_61 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_62 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_63 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_64 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_65 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_66 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_67 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_68 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_69 is shown on the screen, written out.
- [00:15.76](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=15.76): molecules\_70 is shown on the screen, written out.

##### [00:25.718](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=25.7175)

Narration: Prendiamo allora un piccolo volume, delta V, e misuriamo la massa che contiene. Se lo restringiamo troppo contiene pochissime molecole, e il rapporto tra massa e volume oscilla in modo casuale.

Board: heading — a Heading that says "Il fluido come mezzo continuo"; cloud — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); molecules — a Point \[blue\] drawn in cloud (location=(3.543513910472393, 1.2773533949628473), marker\_radius=0.05); molecules\_2 — a Point \[blue\] drawn in cloud (location=(2.0573781721293924, 2.691104256175459), marker\_radius=0.05); molecules\_3 — a Point \[blue\] drawn in cloud (location=(5.759593617916107, 0.569746738858521), marker\_radius=0.05); molecules\_4 — a Point \[blue\] drawn in cloud (location=(5.674672777205705, 3.6926537020131946), marker\_radius=0.05); molecules\_5 — a Point \[blue\] drawn in cloud (location=(1.5646392419934272, 1.3989436024799944), marker\_radius=0.05); molecules\_6 — a Point \[blue\] drawn in cloud (location=(2.116310992091894, 2.9864718889817596), marker\_radius=0.05); molecules\_7 — a Point \[blue\] drawn in cloud (location=(5.263862040638924, 3.042646126635373), marker\_radius=0.05); molecules\_8 — a Point \[blue\] drawn in cloud (location=(0.328107712417841, 0.23548282068222762), marker\_radius=0.05); molecules\_9 — a Point \[blue\] drawn in cloud (location=(2.6242713674902918, 2.798792658187449), marker\_radius=0.05); molecules\_10 — a Point \[blue\] drawn in cloud (location=(3.472802882641554, 1.0195996364578606), marker\_radius=0.05); molecules\_11 — a Point \[blue\] drawn in cloud (location=(0.24313170313835145, 2.228868296928704), marker\_radius=0.05); molecules\_12 — a Point \[blue\] drawn in cloud (location=(1.0680497772991657, 3.4583320332691074), marker\_radius=0.05); molecules\_13 — a Point \[blue\] drawn in cloud (location=(3.2867321863770487, 1.806684192456305), marker\_radius=0.05); molecules\_14 — a Point \[blue\] drawn in cloud (location=(5.607215782254934, 2.207612437568605), marker\_radius=0.05); molecules\_15 — a Point \[blue\] drawn in cloud (location=(4.724468645453453, 1.8601255910471082), marker\_radius=0.05); molecules\_16 — a Point \[blue\] drawn in cloud (location=(2.5879956759512424, 1.6183409271761775), marker\_radius=0.05); molecules\_17 — a Point \[blue\] drawn in cloud (location=(4.78323812931776, 0.6933686325326562), marker\_radius=0.05); molecules\_18 — a Point \[blue\] drawn in cloud (location=(4.873837410658598, 2.844148903153837), marker\_radius=0.05); molecules\_19 — a Point \[blue\] drawn in cloud (location=(3.747145938873291, 0.5770658971741796), marker\_radius=0.05); molecules\_20 — a Point \[blue\] drawn in cloud (location=(1.9131803937256335, 3.485881511680782), marker\_radius=0.05); molecules\_21 — a Point \[blue\] drawn in cloud (location=(3.62752515822649, 1.6940007025375963), marker\_radius=0.05); molecules\_22 — a Point \[blue\] drawn in cloud (location=(0.3415062688291073, 3.3602988159283997), marker\_radius=0.05); molecules\_23 — a Point \[blue\] drawn in cloud (location=(1.0282686859369277, 1.9277101503685117), marker\_radius=0.05); molecules\_24 — a Point \[blue\] drawn in cloud (location=(2.4912021972239016, 0.9804957179352642), marker\_radius=0.05); molecules\_25 — a Point \[blue\] drawn in cloud (location=(1.1314737662672996, 0.5815769234672189), marker\_radius=0.05); molecules\_26 — a Point \[blue\] drawn in cloud (location=(1.7342366389930246, 3.1355406304821374), marker\_radius=0.05); molecules\_27 — a Point \[blue\] drawn in cloud (location=(1.5283990204334257, 2.3158898325636987), marker\_radius=0.05); molecules\_28 — a Point \[blue\] drawn in cloud (location=(3.5826476342976092, 3.3568328989669682), marker\_radius=0.05); molecules\_29 — a Point \[blue\] drawn in cloud (location=(3.820733977854252, 1.3903731131926178), marker\_radius=0.05); molecules\_30 — a Point \[blue\] drawn in cloud (location=(0.27184358462691305, 2.0586789732798936), marker\_radius=0.05); molecules\_31 — a Point \[blue\] drawn in cloud (location=(1.7171811074018477, 3.6996844725683333), marker\_radius=0.05); molecules\_32 — a Point \[blue\] drawn in cloud (location=(2.9517165340483187, 2.5405873296782375), marker\_radius=0.05); molecules\_33 — a Point \[blue\] drawn in cloud (location=(3.550350348651409, 1.889381886459887), marker\_radius=0.05); molecules\_34 — a Point \[blue\] drawn in cloud (location=(3.2105679504573343, 1.638782142661512), marker\_radius=0.05); molecules\_35 — a Point \[blue\] drawn in cloud (location=(4.338966143131256, 0.63985914606601), marker\_radius=0.05); molecules\_36 — a Point \[blue\] drawn in cloud (location=(1.6568470068275927, 2.814435706846416), marker\_radius=0.05); molecules\_37 — a Point \[blue\] drawn in cloud (location=(4.5203869655728335, 0.20575015489012005), marker\_radius=0.05); molecules\_38 — a Point \[blue\] drawn in cloud (location=(3.2837013624608518, 2.0661196088418365), marker\_radius=0.05); molecules\_39 — a Point \[blue\] drawn in cloud (location=(1.678437355160713, 2.8983019756153228), marker\_radius=0.05); molecules\_40 — a Point \[blue\] drawn in cloud (location=(5.341340444236994, 3.253974649123848), marker\_radius=0.05); molecules\_41 — a Point \[blue\] drawn in cloud (location=(1.9915524467825887, 0.9982166269794106), marker\_radius=0.05); molecules\_42 — a Point \[blue\] drawn in cloud (location=(0.6422112278640271, 2.423099513910711), marker\_radius=0.05); molecules\_43 — a Point \[blue\] drawn in cloud (location=(5.726235002279282, 3.586461630649865), marker\_radius=0.05); molecules\_44 — a Point \[blue\] drawn in cloud (location=(5.223986519128084, 1.3136581970378756), marker\_radius=0.05); molecules\_45 — a Point \[blue\] drawn in cloud (location=(2.3008249416947364, 3.1805493080988527), marker\_radius=0.05); molecules\_46 — a Point \[blue\] drawn in cloud (location=(4.495365735143423, 0.4452061718329787), marker\_radius=0.05); molecules\_47 — a Point \[blue\] drawn in cloud (location=(2.044581374526024, 0.86378482747823), marker\_radius=0.05); molecules\_48 — a Point \[blue\] drawn in cloud (location=(0.3896881856024265, 3.1627353129908444), marker\_radius=0.05); molecules\_49 — a Point \[blue\] drawn in cloud (location=(5.777077130973339, 1.5949830004945398), marker\_radius=0.05); molecules\_50 — a Point \[blue\] drawn in cloud (location=(3.2513588540256024, 2.631937568448484), marker\_radius=0.05); molecules\_51 — a Point \[blue\] drawn in cloud (location=(0.9329057931900024, 1.1861538292840124), marker\_radius=0.05); molecules\_52 — a Point \[blue\] drawn in cloud (location=(5.380086679011583, 1.171187381260097), marker\_radius=0.05); molecules\_53 — a Point \[blue\] drawn in cloud (location=(2.9567012265324593, 2.635656803287566), marker\_radius=0.05); molecules\_54 — a Point \[blue\] drawn in cloud (location=(3.9579353354871274, 1.5077428614720703), marker\_radius=0.05); molecules\_55 — a Point \[blue\] drawn in cloud (location=(1.0934696108102797, 0.5043239461258053), marker\_radius=0.05); molecules\_56 — a Point \[blue\] drawn in cloud (location=(3.884186515957117, 1.3456657079979777), marker\_radius=0.05); molecules\_57 — a Point \[blue\] drawn in cloud (location=(0.8992339715361595, 3.4220516124740246), marker\_radius=0.05); molecules\_58 — a Point \[blue\] drawn in cloud (location=(1.0430157117545604, 2.8329598804935814), marker\_radius=0.05); molecules\_59 — a Point \[blue\] drawn in cloud (location=(5.796991211175919, 1.012361034937203), marker\_radius=0.05); molecules\_60 — a Point \[blue\] drawn in cloud (location=(5.32898049429059, 3.51542902123183), marker\_radius=0.05); molecules\_61 — a Point \[blue\] drawn in cloud (location=(1.2029816403985023, 2.9224276209250095), marker\_radius=0.05); molecules\_62 — a Point \[blue\] drawn in cloud (location=(3.1553212963044643, 1.964752626977861), marker\_radius=0.05); molecules\_63 — a Point \[blue\] drawn in cloud (location=(1.948271009325981, 2.246078999526799), marker\_radius=0.05); molecules\_64 — a Point \[blue\] drawn in cloud (location=(2.7700746931135654, 2.6682809708639983), marker\_radius=0.05); molecules\_65 — a Point \[blue\] drawn in cloud (location=(2.52064503878355, 1.8272618183866143), marker\_radius=0.05); molecules\_66 — a Point \[blue\] drawn in cloud (location=(0.5049991838634014, 2.968048774264753), marker\_radius=0.05); molecules\_67 — a Point \[blue\] drawn in cloud (location=(2.85181645154953, 2.5314772905781866), marker\_radius=0.05); molecules\_68 — a Point \[blue\] drawn in cloud (location=(5.682313472777605, 0.636507628671825), marker\_radius=0.05); molecules\_69 — a Point \[blue\] drawn in cloud (location=(5.174944503605365, 1.9726854914799332), marker\_radius=0.05); molecules\_70 — a Point \[blue\] drawn in cloud (location=(1.504247250407934, 1.276477167569101), marker\_radius=0.05)

Actions:
- [00:26.88](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=26.880000000000003): probe is shown on the screen, written out.
- [00:27.96](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=27.960000000000004): probe\_name is shown on the screen, written out.
- [00:31.16](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=31.160000000000004): probe is redrawn as the numbers it depends on change.
- [00:31.16](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=31.160000000000004): size ticks to 0.3.

##### [00:37.98](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=37.980000000000004)

Narration: Ci fermiamo quindi a una scala delta V star: piccolissima rispetto al sistema, ma abbastanza grande da contenere moltissime molecole. La densità è il limite del rapporto tra massa e volume verso questa scala.

Board: heading — a Heading that says "Il fluido come mezzo continuo"; cloud — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); molecules — a Point \[blue\] drawn in cloud (location=(3.543513910472393, 1.2773533949628473), marker\_radius=0.05); molecules\_2 — a Point \[blue\] drawn in cloud (location=(2.0573781721293924, 2.691104256175459), marker\_radius=0.05); molecules\_3 — a Point \[blue\] drawn in cloud (location=(5.759593617916107, 0.569746738858521), marker\_radius=0.05); molecules\_4 — a Point \[blue\] drawn in cloud (location=(5.674672777205705, 3.6926537020131946), marker\_radius=0.05); molecules\_5 — a Point \[blue\] drawn in cloud (location=(1.5646392419934272, 1.3989436024799944), marker\_radius=0.05); molecules\_6 — a Point \[blue\] drawn in cloud (location=(2.116310992091894, 2.9864718889817596), marker\_radius=0.05); molecules\_7 — a Point \[blue\] drawn in cloud (location=(5.263862040638924, 3.042646126635373), marker\_radius=0.05); molecules\_8 — a Point \[blue\] drawn in cloud (location=(0.328107712417841, 0.23548282068222762), marker\_radius=0.05); molecules\_9 — a Point \[blue\] drawn in cloud (location=(2.6242713674902918, 2.798792658187449), marker\_radius=0.05); molecules\_10 — a Point \[blue\] drawn in cloud (location=(3.472802882641554, 1.0195996364578606), marker\_radius=0.05); molecules\_11 — a Point \[blue\] drawn in cloud (location=(0.24313170313835145, 2.228868296928704), marker\_radius=0.05); molecules\_12 — a Point \[blue\] drawn in cloud (location=(1.0680497772991657, 3.4583320332691074), marker\_radius=0.05); molecules\_13 — a Point \[blue\] drawn in cloud (location=(3.2867321863770487, 1.806684192456305), marker\_radius=0.05); molecules\_14 — a Point \[blue\] drawn in cloud (location=(5.607215782254934, 2.207612437568605), marker\_radius=0.05); molecules\_15 — a Point \[blue\] drawn in cloud (location=(4.724468645453453, 1.8601255910471082), marker\_radius=0.05); molecules\_16 — a Point \[blue\] drawn in cloud (location=(2.5879956759512424, 1.6183409271761775), marker\_radius=0.05); molecules\_17 — a Point \[blue\] drawn in cloud (location=(4.78323812931776, 0.6933686325326562), marker\_radius=0.05); molecules\_18 — a Point \[blue\] drawn in cloud (location=(4.873837410658598, 2.844148903153837), marker\_radius=0.05); molecules\_19 — a Point \[blue\] drawn in cloud (location=(3.747145938873291, 0.5770658971741796), marker\_radius=0.05); molecules\_20 — a Point \[blue\] drawn in cloud (location=(1.9131803937256335, 3.485881511680782), marker\_radius=0.05); molecules\_21 — a Point \[blue\] drawn in cloud (location=(3.62752515822649, 1.6940007025375963), marker\_radius=0.05); molecules\_22 — a Point \[blue\] drawn in cloud (location=(0.3415062688291073, 3.3602988159283997), marker\_radius=0.05); molecules\_23 — a Point \[blue\] drawn in cloud (location=(1.0282686859369277, 1.9277101503685117), marker\_radius=0.05); molecules\_24 — a Point \[blue\] drawn in cloud (location=(2.4912021972239016, 0.9804957179352642), marker\_radius=0.05); molecules\_25 — a Point \[blue\] drawn in cloud (location=(1.1314737662672996, 0.5815769234672189), marker\_radius=0.05); molecules\_26 — a Point \[blue\] drawn in cloud (location=(1.7342366389930246, 3.1355406304821374), marker\_radius=0.05); molecules\_27 — a Point \[blue\] drawn in cloud (location=(1.5283990204334257, 2.3158898325636987), marker\_radius=0.05); molecules\_28 — a Point \[blue\] drawn in cloud (location=(3.5826476342976092, 3.3568328989669682), marker\_radius=0.05); molecules\_29 — a Point \[blue\] drawn in cloud (location=(3.820733977854252, 1.3903731131926178), marker\_radius=0.05); molecules\_30 — a Point \[blue\] drawn in cloud (location=(0.27184358462691305, 2.0586789732798936), marker\_radius=0.05); molecules\_31 — a Point \[blue\] drawn in cloud (location=(1.7171811074018477, 3.6996844725683333), marker\_radius=0.05); molecules\_32 — a Point \[blue\] drawn in cloud (location=(2.9517165340483187, 2.5405873296782375), marker\_radius=0.05); molecules\_33 — a Point \[blue\] drawn in cloud (location=(3.550350348651409, 1.889381886459887), marker\_radius=0.05); molecules\_34 — a Point \[blue\] drawn in cloud (location=(3.2105679504573343, 1.638782142661512), marker\_radius=0.05); molecules\_35 — a Point \[blue\] drawn in cloud (location=(4.338966143131256, 0.63985914606601), marker\_radius=0.05); molecules\_36 — a Point \[blue\] drawn in cloud (location=(1.6568470068275927, 2.814435706846416), marker\_radius=0.05); molecules\_37 — a Point \[blue\] drawn in cloud (location=(4.5203869655728335, 0.20575015489012005), marker\_radius=0.05); molecules\_38 — a Point \[blue\] drawn in cloud (location=(3.2837013624608518, 2.0661196088418365), marker\_radius=0.05); molecules\_39 — a Point \[blue\] drawn in cloud (location=(1.678437355160713, 2.8983019756153228), marker\_radius=0.05); molecules\_40 — a Point \[blue\] drawn in cloud (location=(5.341340444236994, 3.253974649123848), marker\_radius=0.05); molecules\_41 — a Point \[blue\] drawn in cloud (location=(1.9915524467825887, 0.9982166269794106), marker\_radius=0.05); molecules\_42 — a Point \[blue\] drawn in cloud (location=(0.6422112278640271, 2.423099513910711), marker\_radius=0.05); molecules\_43 — a Point \[blue\] drawn in cloud (location=(5.726235002279282, 3.586461630649865), marker\_radius=0.05); molecules\_44 — a Point \[blue\] drawn in cloud (location=(5.223986519128084, 1.3136581970378756), marker\_radius=0.05); molecules\_45 — a Point \[blue\] drawn in cloud (location=(2.3008249416947364, 3.1805493080988527), marker\_radius=0.05); molecules\_46 — a Point \[blue\] drawn in cloud (location=(4.495365735143423, 0.4452061718329787), marker\_radius=0.05); molecules\_47 — a Point \[blue\] drawn in cloud (location=(2.044581374526024, 0.86378482747823), marker\_radius=0.05); molecules\_48 — a Point \[blue\] drawn in cloud (location=(0.3896881856024265, 3.1627353129908444), marker\_radius=0.05); molecules\_49 — a Point \[blue\] drawn in cloud (location=(5.777077130973339, 1.5949830004945398), marker\_radius=0.05); molecules\_50 — a Point \[blue\] drawn in cloud (location=(3.2513588540256024, 2.631937568448484), marker\_radius=0.05); molecules\_51 — a Point \[blue\] drawn in cloud (location=(0.9329057931900024, 1.1861538292840124), marker\_radius=0.05); molecules\_52 — a Point \[blue\] drawn in cloud (location=(5.380086679011583, 1.171187381260097), marker\_radius=0.05); molecules\_53 — a Point \[blue\] drawn in cloud (location=(2.9567012265324593, 2.635656803287566), marker\_radius=0.05); molecules\_54 — a Point \[blue\] drawn in cloud (location=(3.9579353354871274, 1.5077428614720703), marker\_radius=0.05); molecules\_55 — a Point \[blue\] drawn in cloud (location=(1.0934696108102797, 0.5043239461258053), marker\_radius=0.05); molecules\_56 — a Point \[blue\] drawn in cloud (location=(3.884186515957117, 1.3456657079979777), marker\_radius=0.05); molecules\_57 — a Point \[blue\] drawn in cloud (location=(0.8992339715361595, 3.4220516124740246), marker\_radius=0.05); molecules\_58 — a Point \[blue\] drawn in cloud (location=(1.0430157117545604, 2.8329598804935814), marker\_radius=0.05); molecules\_59 — a Point \[blue\] drawn in cloud (location=(5.796991211175919, 1.012361034937203), marker\_radius=0.05); molecules\_60 — a Point \[blue\] drawn in cloud (location=(5.32898049429059, 3.51542902123183), marker\_radius=0.05); molecules\_61 — a Point \[blue\] drawn in cloud (location=(1.2029816403985023, 2.9224276209250095), marker\_radius=0.05); molecules\_62 — a Point \[blue\] drawn in cloud (location=(3.1553212963044643, 1.964752626977861), marker\_radius=0.05); molecules\_63 — a Point \[blue\] drawn in cloud (location=(1.948271009325981, 2.246078999526799), marker\_radius=0.05); molecules\_64 — a Point \[blue\] drawn in cloud (location=(2.7700746931135654, 2.6682809708639983), marker\_radius=0.05); molecules\_65 — a Point \[blue\] drawn in cloud (location=(2.52064503878355, 1.8272618183866143), marker\_radius=0.05); molecules\_66 — a Point \[blue\] drawn in cloud (location=(0.5049991838634014, 2.968048774264753), marker\_radius=0.05); molecules\_67 — a Point \[blue\] drawn in cloud (location=(2.85181645154953, 2.5314772905781866), marker\_radius=0.05); molecules\_68 — a Point \[blue\] drawn in cloud (location=(5.682313472777605, 0.636507628671825), marker\_radius=0.05); molecules\_69 — a Point \[blue\] drawn in cloud (location=(5.174944503605365, 1.9726854914799332), marker\_radius=0.05); molecules\_70 — a Point \[blue\] drawn in cloud (location=(1.504247250407934, 1.276477167569101), marker\_radius=0.05); probe — a Circle \[yellow\] drawn in cloud (center=(3.0, 2.0)); probe\_name — a Point \[yellow\] labelled "Delta V" drawn in cloud (location=(3.0, 2.0), show\_marker=False)

Actions:
- [00:38.44](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=38.440000000000005): probe is redrawn as the numbers it depends on change.
- [00:38.44](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=38.440000000000005): size ticks to 1.0.
- [00:40.64](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=40.64000000000001): probe\_name: one name gives way to another over the same drawing (label\_becomes).
- [00:47.68](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=47.68000000000001): cloud moves to a new place on the board.
- [00:47.68](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=47.68000000000001): rho\_limit is shown on the screen, written out.

##### [00:51.568](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=51.5675)

Narration: Con lo stesso limite, ogni punto riceve i suoi campi: la densità in chilogrammi al metro cubo, la pressione in pascal, e la velocità in metri al secondo.

Board: heading — a Heading that says "Il fluido come mezzo continuo"; rho\_limit — a Math \[text\] that says "$rho(bold(r), t) = lim\_(Delta V arrow.r Delta V^\*) frac(Delta m, Delta V)$"; cloud — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 4.0), aspect=(6.0, 4.0)); molecules — a Point \[blue\] drawn in cloud (location=(3.543513910472393, 1.2773533949628473), marker\_radius=0.05); molecules\_2 — a Point \[blue\] drawn in cloud (location=(2.0573781721293924, 2.691104256175459), marker\_radius=0.05); molecules\_3 — a Point \[blue\] drawn in cloud (location=(5.759593617916107, 0.569746738858521), marker\_radius=0.05); molecules\_4 — a Point \[blue\] drawn in cloud (location=(5.674672777205705, 3.6926537020131946), marker\_radius=0.05); molecules\_5 — a Point \[blue\] drawn in cloud (location=(1.5646392419934272, 1.3989436024799944), marker\_radius=0.05); molecules\_6 — a Point \[blue\] drawn in cloud (location=(2.116310992091894, 2.9864718889817596), marker\_radius=0.05); molecules\_7 — a Point \[blue\] drawn in cloud (location=(5.263862040638924, 3.042646126635373), marker\_radius=0.05); molecules\_8 — a Point \[blue\] drawn in cloud (location=(0.328107712417841, 0.23548282068222762), marker\_radius=0.05); molecules\_9 — a Point \[blue\] drawn in cloud (location=(2.6242713674902918, 2.798792658187449), marker\_radius=0.05); molecules\_10 — a Point \[blue\] drawn in cloud (location=(3.472802882641554, 1.0195996364578606), marker\_radius=0.05); molecules\_11 — a Point \[blue\] drawn in cloud (location=(0.24313170313835145, 2.228868296928704), marker\_radius=0.05); molecules\_12 — a Point \[blue\] drawn in cloud (location=(1.0680497772991657, 3.4583320332691074), marker\_radius=0.05); molecules\_13 — a Point \[blue\] drawn in cloud (location=(3.2867321863770487, 1.806684192456305), marker\_radius=0.05); molecules\_14 — a Point \[blue\] drawn in cloud (location=(5.607215782254934, 2.207612437568605), marker\_radius=0.05); molecules\_15 — a Point \[blue\] drawn in cloud (location=(4.724468645453453, 1.8601255910471082), marker\_radius=0.05); molecules\_16 — a Point \[blue\] drawn in cloud (location=(2.5879956759512424, 1.6183409271761775), marker\_radius=0.05); molecules\_17 — a Point \[blue\] drawn in cloud (location=(4.78323812931776, 0.6933686325326562), marker\_radius=0.05); molecules\_18 — a Point \[blue\] drawn in cloud (location=(4.873837410658598, 2.844148903153837), marker\_radius=0.05); molecules\_19 — a Point \[blue\] drawn in cloud (location=(3.747145938873291, 0.5770658971741796), marker\_radius=0.05); molecules\_20 — a Point \[blue\] drawn in cloud (location=(1.9131803937256335, 3.485881511680782), marker\_radius=0.05); molecules\_21 — a Point \[blue\] drawn in cloud (location=(3.62752515822649, 1.6940007025375963), marker\_radius=0.05); molecules\_22 — a Point \[blue\] drawn in cloud (location=(0.3415062688291073, 3.3602988159283997), marker\_radius=0.05); molecules\_23 — a Point \[blue\] drawn in cloud (location=(1.0282686859369277, 1.9277101503685117), marker\_radius=0.05); molecules\_24 — a Point \[blue\] drawn in cloud (location=(2.4912021972239016, 0.9804957179352642), marker\_radius=0.05); molecules\_25 — a Point \[blue\] drawn in cloud (location=(1.1314737662672996, 0.5815769234672189), marker\_radius=0.05); molecules\_26 — a Point \[blue\] drawn in cloud (location=(1.7342366389930246, 3.1355406304821374), marker\_radius=0.05); molecules\_27 — a Point \[blue\] drawn in cloud (location=(1.5283990204334257, 2.3158898325636987), marker\_radius=0.05); molecules\_28 — a Point \[blue\] drawn in cloud (location=(3.5826476342976092, 3.3568328989669682), marker\_radius=0.05); molecules\_29 — a Point \[blue\] drawn in cloud (location=(3.820733977854252, 1.3903731131926178), marker\_radius=0.05); molecules\_30 — a Point \[blue\] drawn in cloud (location=(0.27184358462691305, 2.0586789732798936), marker\_radius=0.05); molecules\_31 — a Point \[blue\] drawn in cloud (location=(1.7171811074018477, 3.6996844725683333), marker\_radius=0.05); molecules\_32 — a Point \[blue\] drawn in cloud (location=(2.9517165340483187, 2.5405873296782375), marker\_radius=0.05); molecules\_33 — a Point \[blue\] drawn in cloud (location=(3.550350348651409, 1.889381886459887), marker\_radius=0.05); molecules\_34 — a Point \[blue\] drawn in cloud (location=(3.2105679504573343, 1.638782142661512), marker\_radius=0.05); molecules\_35 — a Point \[blue\] drawn in cloud (location=(4.338966143131256, 0.63985914606601), marker\_radius=0.05); molecules\_36 — a Point \[blue\] drawn in cloud (location=(1.6568470068275927, 2.814435706846416), marker\_radius=0.05); molecules\_37 — a Point \[blue\] drawn in cloud (location=(4.5203869655728335, 0.20575015489012005), marker\_radius=0.05); molecules\_38 — a Point \[blue\] drawn in cloud (location=(3.2837013624608518, 2.0661196088418365), marker\_radius=0.05); molecules\_39 — a Point \[blue\] drawn in cloud (location=(1.678437355160713, 2.8983019756153228), marker\_radius=0.05); molecules\_40 — a Point \[blue\] drawn in cloud (location=(5.341340444236994, 3.253974649123848), marker\_radius=0.05); molecules\_41 — a Point \[blue\] drawn in cloud (location=(1.9915524467825887, 0.9982166269794106), marker\_radius=0.05); molecules\_42 — a Point \[blue\] drawn in cloud (location=(0.6422112278640271, 2.423099513910711), marker\_radius=0.05); molecules\_43 — a Point \[blue\] drawn in cloud (location=(5.726235002279282, 3.586461630649865), marker\_radius=0.05); molecules\_44 — a Point \[blue\] drawn in cloud (location=(5.223986519128084, 1.3136581970378756), marker\_radius=0.05); molecules\_45 — a Point \[blue\] drawn in cloud (location=(2.3008249416947364, 3.1805493080988527), marker\_radius=0.05); molecules\_46 — a Point \[blue\] drawn in cloud (location=(4.495365735143423, 0.4452061718329787), marker\_radius=0.05); molecules\_47 — a Point \[blue\] drawn in cloud (location=(2.044581374526024, 0.86378482747823), marker\_radius=0.05); molecules\_48 — a Point \[blue\] drawn in cloud (location=(0.3896881856024265, 3.1627353129908444), marker\_radius=0.05); molecules\_49 — a Point \[blue\] drawn in cloud (location=(5.777077130973339, 1.5949830004945398), marker\_radius=0.05); molecules\_50 — a Point \[blue\] drawn in cloud (location=(3.2513588540256024, 2.631937568448484), marker\_radius=0.05); molecules\_51 — a Point \[blue\] drawn in cloud (location=(0.9329057931900024, 1.1861538292840124), marker\_radius=0.05); molecules\_52 — a Point \[blue\] drawn in cloud (location=(5.380086679011583, 1.171187381260097), marker\_radius=0.05); molecules\_53 — a Point \[blue\] drawn in cloud (location=(2.9567012265324593, 2.635656803287566), marker\_radius=0.05); molecules\_54 — a Point \[blue\] drawn in cloud (location=(3.9579353354871274, 1.5077428614720703), marker\_radius=0.05); molecules\_55 — a Point \[blue\] drawn in cloud (location=(1.0934696108102797, 0.5043239461258053), marker\_radius=0.05); molecules\_56 — a Point \[blue\] drawn in cloud (location=(3.884186515957117, 1.3456657079979777), marker\_radius=0.05); molecules\_57 — a Point \[blue\] drawn in cloud (location=(0.8992339715361595, 3.4220516124740246), marker\_radius=0.05); molecules\_58 — a Point \[blue\] drawn in cloud (location=(1.0430157117545604, 2.8329598804935814), marker\_radius=0.05); molecules\_59 — a Point \[blue\] drawn in cloud (location=(5.796991211175919, 1.012361034937203), marker\_radius=0.05); molecules\_60 — a Point \[blue\] drawn in cloud (location=(5.32898049429059, 3.51542902123183), marker\_radius=0.05); molecules\_61 — a Point \[blue\] drawn in cloud (location=(1.2029816403985023, 2.9224276209250095), marker\_radius=0.05); molecules\_62 — a Point \[blue\] drawn in cloud (location=(3.1553212963044643, 1.964752626977861), marker\_radius=0.05); molecules\_63 — a Point \[blue\] drawn in cloud (location=(1.948271009325981, 2.246078999526799), marker\_radius=0.05); molecules\_64 — a Point \[blue\] drawn in cloud (location=(2.7700746931135654, 2.6682809708639983), marker\_radius=0.05); molecules\_65 — a Point \[blue\] drawn in cloud (location=(2.52064503878355, 1.8272618183866143), marker\_radius=0.05); molecules\_66 — a Point \[blue\] drawn in cloud (location=(0.5049991838634014, 2.968048774264753), marker\_radius=0.05); molecules\_67 — a Point \[blue\] drawn in cloud (location=(2.85181645154953, 2.5314772905781866), marker\_radius=0.05); molecules\_68 — a Point \[blue\] drawn in cloud (location=(5.682313472777605, 0.636507628671825), marker\_radius=0.05); molecules\_69 — a Point \[blue\] drawn in cloud (location=(5.174944503605365, 1.9726854914799332), marker\_radius=0.05); molecules\_70 — a Point \[blue\] drawn in cloud (location=(1.504247250407934, 1.276477167569101), marker\_radius=0.05); probe — a Circle \[yellow\] drawn in cloud (center=(3.0, 2.0)); probe\_name — a Point \[yellow\] labelled "Delta V" drawn in cloud (location=(3.0, 2.0), show\_marker=False)

Actions:
- [00:51.568](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=51.5675): fields is shown on the screen, written out.
- [00:55.32](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=55.32000000000001): fields is shown on the screen, written out.
- [00:57.36](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=57.36000000000001): fields is shown on the screen, written out.
- [00:58.88](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=58.88000000000001): fields is shown on the screen, written out.

##### [01:1.443](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=61.4425)

Narration: Da qui in avanti il fluido è un mezzo continuo: campi regolari, che possiamo derivare come qualsiasi funzione.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [01:04](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=64): rho\_limit is indicated — a transient flash.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): cloud is hidden from the screen — left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_2 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_3 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_4 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_5 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_6 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_7 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_8 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_9 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_10 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_11 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_12 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_13 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_14 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_15 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_16 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_17 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_18 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_19 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_20 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_21 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_22 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_23 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_24 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_25 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_26 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_27 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_28 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_29 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_30 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_31 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_32 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_33 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_34 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_35 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_36 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_37 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_38 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_39 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_40 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_41 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_42 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_43 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_44 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_45 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_46 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_47 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_48 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_49 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_50 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_51 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_52 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_53 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_54 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_55 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_56 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_57 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_58 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_59 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_60 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_61 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_62 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_63 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_64 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_65 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_66 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_67 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_68 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_69 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): molecules\_70 is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): probe is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): probe\_name is hidden from the screen — cloud left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): fields is hidden from the screen — left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): heading is hidden from the screen — left the board.
- [01:8.399](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=68.39939583333333): rho\_limit is hidden from the screen — left the board.

### Scene 2: [Niente taglio a riposo](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999)

Span: 01:9.441–02:24.654 (69.44106249999999s–144.65354166666668s).

#### Objects

- cell: a Polygon \[blue\] drawn in element (vertices=((1.0, 0.8), (3.0, 0.8), (3.0, 2.8), (1.0, 2.8)))
- element: a Figure (x\_range=(0.0, 4.4), y\_range=(0.0, 4.2), aspect=(4.4, 4.2))
- fluid\_block: a Polygon \[blue\] drawn in fluid\_fig (vertices=((0.5, 0.0), (2.5, 0.0), ((2.5 + s\_fluid), 1.8), ((0.5 + s\_flui…)
- fluid\_fig: a Figure (x\_range=(-0.5, 5.5), y\_range=(-0.5, 3.2), aspect=(6.0, 3.7))
- fluid\_ground: a Line \[gray\] drawn in fluid\_fig (start=(-0.3, 0.0), end=(5.3, 0.0))
- fluid\_name: a Tex \[text\] that says "Fluido"
- fluid\_push: a Vector \[yellow\] labelled "bold(F)" drawn in fluid\_fig (start=((0.7 + s\_fluid), 2.3), end=((2.3 + s\_fluid), 2.3))
- heading\_shear: a Heading that says "Solido e fluido sotto taglio"
- heading\_tensor: a Heading that says "Il tensore degli sforzi"
- normal\_x: a Vector \[red\] labelled "sigma\_(x x)" drawn in element (start=(4.0, 1.8), end=(3.0, 1.8))
- normal\_y: a Vector \[red\] labelled "sigma\_(y y)" drawn in element (start=(2.0, 3.9), end=(2.0, 2.8))
- s\_fluid: a VariableNumber
- s\_solid: a VariableNumber
- shear\_x: a Vector \[yellow\] labelled "sigma\_(y x)" drawn in element (start=(3.25, 0.9), end=(3.25, 1.5))
- shear\_y: a Vector \[yellow\] labelled "sigma\_(x y)" drawn in element (start=(1.1, 3.05), end=(1.7, 3.05))
- sigma\_m: a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), sigma\_(x y), sigma\_(x z); sigma\_(y x), sigma\_(y y), sigma\_(y z); sigma\_(z x), sigma\_(z y), sigma\_(z z))$"
- solid\_block: a Polygon \[green\] drawn in solid\_fig (vertices=((0.5, 0.0), (2.5, 0.0), ((2.5 + s\_solid), 1.8), ((0.5 + s\_soli…)
- solid\_fig: a Figure (x\_range=(-0.5, 5.5), y\_range=(-0.5, 3.2), aspect=(6.0, 3.7))
- solid\_ground: a Line \[gray\] drawn in solid\_fig (start=(-0.3, 0.0), end=(5.3, 0.0))
- solid\_name: a Tex \[text\] that says "Solido"
- solid\_push: a Vector \[yellow\] labelled "bold(F)" drawn in solid\_fig (start=((0.7 + s\_solid), 2.3), end=((2.3 + s\_solid), 2.3))
- tau\_zero: a Math \[red\] that says "$tau = 0$" drawn in fluid\_fig
- traction: a Math \[text\] that says "$dif bold(F) = bold(sigma) dot bold(n) thin dif A$"

#### Beats

##### [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999)

Narration: Spingiamo parallelamente alla faccia superiore di un blocco solido. Il blocco si inclina di un certo angolo e poi si ferma: una forza tangenziale interna equilibra la spinta.

Board: Empty.

Actions:
- [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999): heading\_shear is shown on the screen, written out.
- [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999): solid\_name is shown on the screen, written out.
- [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999): solid\_fig is shown on the screen, written out.
- [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999): solid\_ground is shown on the screen, written out.
- [01:9.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=69.44106249999999): solid\_block is shown on the screen, written out.
- [01:10.161](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=70.16106249999999): solid\_push is shown on the screen, written out.
- [01:14.401](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=74.40106249999998): solid\_block is redrawn as the numbers it depends on change.
- [01:14.401](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=74.40106249999998): solid\_push is redrawn as the numbers it depends on change.
- [01:14.401](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=74.40106249999998): s\_solid ticks to 0.6.

##### [01:21.071](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=81.07106249999998)

Narration: Con un fluido la stessa spinta non trova alcuna nuova posizione di equilibrio: finché agisce, il fluido continua a scorrere.

Board: solid\_name — a Tex \[text\] that says "Solido"; solid\_fig — a Figure (x\_range=(-0.5, 5.5), y\_range=(-0.5, 3.2), aspect=(6.0, 3.7)); heading\_shear — a Heading that says "Solido e fluido sotto taglio"; solid\_ground — a Line \[gray\] drawn in solid\_fig (start=(-0.3, 0.0), end=(5.3, 0.0)); solid\_block — a Polygon \[green\] drawn in solid\_fig (vertices=((0.5, 0.0), (2.5, 0.0), ((2.5 + s\_solid), 1.8), ((0.5 + s\_soli…); solid\_push — a Vector \[yellow\] labelled "bold(F)" drawn in solid\_fig (start=((0.7 + s\_solid), 2.3), end=((2.3 + s\_solid), 2.3))

Actions:
- [01:21.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=81.64106249999999): fluid\_name is shown on the screen, written out.
- [01:21.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=81.64106249999999): fluid\_fig is shown on the screen, written out.
- [01:21.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=81.64106249999999): fluid\_ground is shown on the screen, written out.
- [01:21.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=81.64106249999999): fluid\_block is shown on the screen, written out.
- [01:22.681](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=82.68106249999998): fluid\_push is shown on the screen, written out.
- [01:25.108](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=85.10806249999999): fluid\_block is redrawn as the numbers it depends on change.
- [01:25.108](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=85.10806249999999): fluid\_push is redrawn as the numbers it depends on change.
- [01:25.108](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=85.10806249999999): s\_fluid ticks to 2.5.

##### [01:29.601](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=89.60106249999998)

Narration: Quindi, in un fluido in quiete, gli sforzi tangenziali sono nulli: tau uguale a zero. Le forze tra parti adiacenti, o contro le pareti, sono soltanto perpendicolari alle superfici.

Board: solid\_name — a Tex \[text\] that says "Solido"; solid\_fig — a Figure (x\_range=(-0.5, 5.5), y\_range=(-0.5, 3.2), aspect=(6.0, 3.7)); fluid\_name — a Tex \[text\] that says "Fluido"; fluid\_fig — a Figure (x\_range=(-0.5, 5.5), y\_range=(-0.5, 3.2), aspect=(6.0, 3.7)); heading\_shear — a Heading that says "Solido e fluido sotto taglio"; solid\_ground — a Line \[gray\] drawn in solid\_fig (start=(-0.3, 0.0), end=(5.3, 0.0)); solid\_block — a Polygon \[green\] drawn in solid\_fig (vertices=((0.5, 0.0), (2.5, 0.0), ((2.5 + s\_solid), 1.8), ((0.5 + s\_soli…); solid\_push — a Vector \[yellow\] labelled "bold(F)" drawn in solid\_fig (start=((0.7 + s\_solid), 2.3), end=((2.3 + s\_solid), 2.3)); fluid\_ground — a Line \[gray\] drawn in fluid\_fig (start=(-0.3, 0.0), end=(5.3, 0.0)); fluid\_block — a Polygon \[blue\] drawn in fluid\_fig (vertices=((0.5, 0.0), (2.5, 0.0), ((2.5 + s\_fluid), 1.8), ((0.5 + s\_flui…); fluid\_push — a Vector \[yellow\] labelled "bold(F)" drawn in fluid\_fig (start=((0.7 + s\_fluid), 2.3), end=((2.3 + s\_fluid), 2.3))

Actions:
- [01:34.081](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=94.08106249999997): tau\_zero is shown on the screen, written out.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): fluid\_fig is hidden from the screen — left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): fluid\_ground is hidden from the screen — fluid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): fluid\_block is hidden from the screen — fluid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): fluid\_push is hidden from the screen — fluid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): tau\_zero is hidden from the screen — fluid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): fluid\_name is hidden from the screen — left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): heading\_shear is hidden from the screen — left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): solid\_fig is hidden from the screen — left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): solid\_ground is hidden from the screen — solid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): solid\_block is hidden from the screen — solid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): solid\_push is hidden from the screen — solid\_fig left the board.
- [01:41.586](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=101.58606249999998): solid\_name is hidden from the screen — left the board.

##### [01:42.786](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=102.78606249999999)

Narration: Le forze interne di un mezzo continuo si descrivono con il tensore degli sforzi. Su un elemento di superficie d A, con normale n, agisce la forza sigma per n, d A.

Board: Empty.

Actions:
- [01:42.786](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=102.78606249999999): heading\_tensor is shown on the screen, written out.
- [01:46.161](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=106.16106249999999): element is shown on the screen, written out.
- [01:46.161](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=106.16106249999999): cell is shown on the screen, written out.
- [01:51.361](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=111.36106249999997): element moves to a new place on the board.
- [01:51.361](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=111.36106249999997): traction is shown on the screen, written out.

##### [01:54.729](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=114.72856249999998)

Narration: In tre dimensioni sigma è una matrice tre per tre. Sulla diagonale stanno gli sforzi normali, perpendicolari alle facce; fuori diagonale gli sforzi di taglio, paralleli alle facce.

Board: traction — a Math \[text\] that says "$dif bold(F) = bold(sigma) dot bold(n) thin dif A$"; element — a Figure (x\_range=(0.0, 4.4), y\_range=(0.0, 4.2), aspect=(4.4, 4.2)); heading\_tensor — a Heading that says "Il tensore degli sforzi"; cell — a Polygon \[blue\] drawn in element (vertices=((1.0, 0.8), (3.0, 0.8), (3.0, 2.8), (1.0, 2.8)))

Actions:
- [01:55.321](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=115.32106249999998): sigma\_m is shown on the screen, written out.
- [01:58.481](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=118.48106249999998): sigma\_m (the "diagonal=main" part) is emphasized.
- [02:0.441](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=120.44106249999997): normal\_x is shown on the screen, written out.
- [02:0.656](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=120.65578551662958): normal\_y is shown on the screen, written out.
- [02:2.761](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=122.76106249999998): sigma\_m (the "diagonal=main" part) is no longer emphasized.
- [02:4.481](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=124.48106249999998): shear\_x is shown on the screen, written out.
- [02:4.792](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=124.79164400619635): shear\_y is shown on the screen, written out.

##### [02:7.069](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=127.06856249999998)

Narration: A riposo il taglio è nullo, quindi tutti i termini fuori diagonale si annullano. Restano soltanto i tre sforzi normali.

Board: traction — a Math \[text\] that says "$dif bold(F) = bold(sigma) dot bold(n) thin dif A$"; sigma\_m — a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), sigma\_(x y), sigma\_(x z); sigma\_(y x), sigma\_(y y), sigma\_(y z); sigma\_(z x), sigma\_(z y), sigma\_(z z))$"; element — a Figure (x\_range=(0.0, 4.4), y\_range=(0.0, 4.2), aspect=(4.4, 4.2)); heading\_tensor — a Heading that says "Il tensore degli sforzi"; cell — a Polygon \[blue\] drawn in element (vertices=((1.0, 0.8), (3.0, 0.8), (3.0, 2.8), (1.0, 2.8))); normal\_x — a Vector \[red\] labelled "sigma\_(x x)" drawn in element (start=(4.0, 1.8), end=(3.0, 1.8)); normal\_y — a Vector \[red\] labelled "sigma\_(y y)" drawn in element (start=(2.0, 3.9), end=(2.0, 2.8)); shear\_x — a Vector \[yellow\] labelled "sigma\_(y x)" drawn in element (start=(3.25, 0.9), end=(3.25, 1.5)); shear\_y — a Vector \[yellow\] labelled "sigma\_(x y)" drawn in element (start=(1.1, 3.05), end=(1.7, 3.05))

Actions:
- [02:8.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=128.64106249999998): shear\_x is hidden from the screen.
- [02:8.641](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=128.64106249999998): shear\_y is hidden from the screen.
- [02:11.521](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=131.52106249999997): sigma\_m becomes "$bold(sigma) = mat(sigma\_(x x), 0, 0; 0, sigma\_(y y), 0; 0, 0, sigma\_(z z))$".
- [02:12.881](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=132.88106249999998): sigma\_m (the "diagonal=main" part) is emphasized.
- [02:14.961](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=134.96106249999997): sigma\_m (the "diagonal=main" part) is no longer emphasized.

##### [02:15.561](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=135.5610625)

Narration: Sono uguali fra loro? Per ora sappiamo soltanto che sono compressioni. Per deciderlo ci serve un piccolo prisma di fluido.

Board: traction — a Math \[text\] that says "$dif bold(F) = bold(sigma) dot bold(n) thin dif A$"; sigma\_m — a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), sigma\_(x y), sigma\_(x z); sigma\_(y x), sigma\_(y y), sigma\_(y z); sigma\_(z x), sigma\_(z y), sigma\_(z z))$"; element — a Figure (x\_range=(0.0, 4.4), y\_range=(0.0, 4.2), aspect=(4.4, 4.2)); heading\_tensor — a Heading that says "Il tensore degli sforzi"; cell — a Polygon \[blue\] drawn in element (vertices=((1.0, 0.8), (3.0, 0.8), (3.0, 2.8), (1.0, 2.8))); normal\_x — a Vector \[red\] labelled "sigma\_(x x)" drawn in element (start=(4.0, 1.8), end=(3.0, 1.8)); normal\_y — a Vector \[red\] labelled "sigma\_(y y)" drawn in element (start=(2.0, 3.9), end=(2.0, 2.8))

Actions:
- [02:19.161](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=139.16106249999999): normal\_x is indicated — a transient flash.
- [02:19.161](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=139.16106249999999): normal\_y is indicated — a transient flash.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): element is hidden from the screen — left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): cell is hidden from the screen — element left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): normal\_x is hidden from the screen — element left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): normal\_y is hidden from the screen — element left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): heading\_tensor is hidden from the screen — left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): sigma\_m is hidden from the screen — left the board.
- [02:23.612](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=143.611875): traction is hidden from the screen — left the board.

### Scene 3: [La pressione è uno scalare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=144.65354166666668)

Span: 02:24.654–04:19.021 (144.65354166666668s–259.021125s).

#### Objects

- body: a Polygon \[blue\] drawn in prism (vertices=((0.0, 0.0), ((3.0 \* k), 0.0), (0.0, (2.0 \* k))), fill\_opacity=0.2)
- delta\_def: a Math \[text\] that says "$delta\_(i j) = cases(1 & upright("se") thin i = j, 0 & upright("se") thin i eq.not j)$"
- heading\_law: a Heading that says "Il tensore idrostatico"
- heading\_prism: a Heading that says "La pressione è uno scalare"
- hyp: a Line \[gray\] labelled "dif s" drawn in prism (start=((3.0 \* k), 0.0), end=(0.0, (2.0 \* k)))
- iso: a Math \[text\] that says "$P\_x = P\_y = P\_n = P$"
- k: a VariableNumber (initial\_value=1.0)
- law: a Math \[text\] that says "$sigma\_(i j) = -P delta\_(i j)$"
- leg\_x: a Line \[gray\] labelled "dif x" drawn in prism (end=((3.0 \* k), 0.0))
- leg\_y: a Line \[gray\] labelled "dif y" drawn in prism (end=(0.0, (2.0 \* k)))
- p\_n: a Vector \[red\] labelled "P\_n" drawn in prism (start=(((1.5 \* k) + 0.6656399999999999), (k + 0.9985199999999999)), end=((1.5 \* k), \<VariableNumber k = 0.45\>))
- p\_x: a Vector \[red\] labelled "P\_x" drawn in prism (start=(-1.3, \<VariableNumber k = 0.45\>), end=(0.0, \<VariableNumber k = 0.45\>))
- p\_y: a Vector \[red\] labelled "P\_y" drawn in prism (start=((1.5 \* k), -1.3), end=((1.5 \* k), 0.0))
- pdef: a Math \[text\] that says "$P = frac(dif F\_perp, dif S)$"
- prism: a Figure (x\_range=(-1.6, 4.0), y\_range=(-1.6, 3.2), aspect=(5.6, 4.8))
- prism\_work: a Derivation \[text\] that says "$dif s sin theta &= dif y, quad dif s cos theta = dif x \\ P\_x dif y - P\_n dif s sin theta &= 0 \\ P\_x &= P\_n \\ P\_y - P\_n - frac(1, 2) rho g dif y &= 0 \\ P\_y &= P\_n$"
- sigma\_d: a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), 0, 0; 0, sigma\_(y y), 0; 0, 0, sigma\_(z z))$"
- theta: an Angle \[yellow\] labelled "theta" drawn in prism (vertex=((3.0 \* k), 0.0), sides=((0.0, (2.0 \* k)), (0.0, 0.0)))
- weight: a Vector \[green\] labelled "dif m thin bold(g)" drawn in prism (start=(\<VariableNumber k = 0.45\>, (0.9 \* k)), end=(\<VariableNumber k = 0.45\>, (0.3 \* k)))

#### Beats

##### [02:24.654](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=144.65354166666668)

Narration: La pressione in un punto è il rapporto tra la forza normale e la superficie su cui agisce. La forza è un vettore; la pressione, invece, non dipende da come orientiamo la superficie. Dimostriamolo.

Board: Empty.

Actions:
- [02:24.654](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=144.65354166666668): heading\_prism is shown on the screen, written out.
- [02:26.334](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=146.3335416666667): pdef is shown on the screen, written out.

##### [02:37.366](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=157.3660416666667)

Narration: Prendiamo un prisma triangolare di fluido, con cateti d x e d y e ipotenusa d s. La profondità d z è comune a tutte le facce, e la semplifichiamo fin da subito.

Board: pdef — a Math \[text\] that says "$P = frac(dif F\_perp, dif S)$"; heading\_prism — a Heading that says "La pressione è uno scalare"

Actions:
- [02:38.174](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=158.1735416666667): prism is shown on the screen, written out.
- [02:38.174](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=158.1735416666667): body is shown on the screen, written out.
- [02:39.974](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=159.97354166666668): leg\_x is shown on the screen, written out.
- [02:40.269](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=160.269361602358): leg\_y is shown on the screen, written out.
- [02:41.974](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=161.97354166666668): hyp is shown on the screen, written out.

##### [02:48.866](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=168.8660416666667)

Narration: Su ogni faccia la pressione spinge verso il fluido. Chiamiamo theta l'angolo alla base: allora d s seno theta vale d y, e d s coseno theta vale d x.

Board: pdef — a Math \[text\] that says "$P = frac(dif F\_perp, dif S)$"; prism — a Figure (x\_range=(-1.6, 4.0), y\_range=(-1.6, 3.2), aspect=(5.6, 4.8)); heading\_prism — a Heading that says "La pressione è uno scalare"; body — a Polygon \[blue\] drawn in prism (vertices=((0.0, 0.0), ((3.0 \* k), 0.0), (0.0, (2.0 \* k))), fill\_opacity=0.2); leg\_x — a Line \[gray\] labelled "dif x" drawn in prism (end=((3.0 \* k), 0.0)); leg\_y — a Line \[gray\] labelled "dif y" drawn in prism (end=(0.0, (2.0 \* k))); hyp — a Line \[gray\] labelled "dif s" drawn in prism (start=((3.0 \* k), 0.0), end=(0.0, (2.0 \* k)))

Actions:
- [02:50.734](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=170.73354166666667): p\_x is shown on the screen, written out.
- [02:50.97](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=170.97021395893313): p\_y is shown on the screen, written out.
- [02:51.207](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=171.2068862511996): p\_n is shown on the screen, written out.
- [02:52.574](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=172.57354166666667): theta is shown on the screen, written out.
- [02:56.014](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=176.01354166666667): prism\_work is shown on the screen, written out.

##### [03:1.136](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=181.1360416666667)

Narration: Lungo x: P x per d y, meno P n per d s seno theta, fa zero. Con la geometria i due prodotti hanno lo stesso fattore d y, e resta P x uguale a P n.

Board: pdef — a Math \[text\] that says "$P = frac(dif F\_perp, dif S)$"; prism — a Figure (x\_range=(-1.6, 4.0), y\_range=(-1.6, 3.2), aspect=(5.6, 4.8)); heading\_prism — a Heading that says "La pressione è uno scalare"; body — a Polygon \[blue\] drawn in prism (vertices=((0.0, 0.0), ((3.0 \* k), 0.0), (0.0, (2.0 \* k))), fill\_opacity=0.2); leg\_x — a Line \[gray\] labelled "dif x" drawn in prism (end=((3.0 \* k), 0.0)); leg\_y — a Line \[gray\] labelled "dif y" drawn in prism (end=(0.0, (2.0 \* k))); hyp — a Line \[gray\] labelled "dif s" drawn in prism (start=((3.0 \* k), 0.0), end=(0.0, (2.0 \* k))); p\_x — a Vector \[red\] labelled "P\_x" drawn in prism (start=(-1.3, \<VariableNumber k = 0.45\>), end=(0.0, \<VariableNumber k = 0.45\>)); p\_y — a Vector \[red\] labelled "P\_y" drawn in prism (start=((1.5 \* k), -1.3), end=((1.5 \* k), 0.0)); p\_n — a Vector \[red\] labelled "P\_n" drawn in prism (start=(((1.5 \* k) + 0.6656399999999999), (k + 0.9985199999999999)), end=((1.5 \* k), \<VariableNumber k = 0.45\>)); theta — an Angle \[yellow\] labelled "theta" drawn in prism (vertex=((3.0 \* k), 0.0), sides=((0.0, (2.0 \* k)), (0.0, 0.0)))

Actions:
- [03:1.414](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=181.41354166666667): prism\_work is shown on the screen, written out.
- [03:7.774](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=187.7735416666667): p\_x is indicated — a transient flash.
- [03:11.094](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=191.09354166666668): prism\_work is shown on the screen, written out.

##### [03:14.026](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=194.02604166666669)

Narration: Lungo y compare anche il peso del prisma, proporzionale alla sua area. Dividendo per d x resta P y meno P n meno un mezzo rho g d y, uguale a zero.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:16.014](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=196.0135416666667): weight is shown on the screen, written out.
- [03:18.774](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=198.7735416666667): prism\_work is shown on the screen, written out.

##### [03:25.999](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=205.99854166666668)

Narration: Ora restringiamo il prisma verso un punto. Il peso è di ordine superiore e svanisce con d y, quindi anche P y è uguale a P n.

Board: pdef — a Math \[text\] that says "$P = frac(dif F\_perp, dif S)$"; prism — a Figure (x\_range=(-1.6, 4.0), y\_range=(-1.6, 3.2), aspect=(5.6, 4.8)); heading\_prism — a Heading that says "La pressione è uno scalare"; body — a Polygon \[blue\] drawn in prism (vertices=((0.0, 0.0), ((3.0 \* k), 0.0), (0.0, (2.0 \* k))), fill\_opacity=0.2); leg\_x — a Line \[gray\] labelled "dif x" drawn in prism (end=((3.0 \* k), 0.0)); leg\_y — a Line \[gray\] labelled "dif y" drawn in prism (end=(0.0, (2.0 \* k))); hyp — a Line \[gray\] labelled "dif s" drawn in prism (start=((3.0 \* k), 0.0), end=(0.0, (2.0 \* k))); p\_x — a Vector \[red\] labelled "P\_x" drawn in prism (start=(-1.3, \<VariableNumber k = 0.45\>), end=(0.0, \<VariableNumber k = 0.45\>)); p\_y — a Vector \[red\] labelled "P\_y" drawn in prism (start=((1.5 \* k), -1.3), end=((1.5 \* k), 0.0)); p\_n — a Vector \[red\] labelled "P\_n" drawn in prism (start=(((1.5 \* k) + 0.6656399999999999), (k + 0.9985199999999999)), end=((1.5 \* k), \<VariableNumber k = 0.45\>)); theta — an Angle \[yellow\] labelled "theta" drawn in prism (vertex=((3.0 \* k), 0.0), sides=((0.0, (2.0 \* k)), (0.0, 0.0))); weight — a Vector \[green\] labelled "dif m thin bold(g)" drawn in prism (start=(\<VariableNumber k = 0.45\>, (0.9 \* k)), end=(\<VariableNumber k = 0.45\>, (0.3 \* k)))

Actions:
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): body is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): leg\_x is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): leg\_y is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): hyp is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): p\_x is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): p\_y is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): p\_n is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): theta is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): weight is redrawn as the numbers it depends on change.
- [03:26.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=206.69354166666668): k ticks to 0.45.
- [03:29.214](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=209.21354166666669): prism\_work (the "frac(1, 2) rho g dif y" part) is emphasized.
- [03:30.694](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=210.69354166666668): prism\_work (the "frac(1, 2) rho g dif y" part) is no longer emphasized.
- [03:32.334](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=212.3335416666667): prism\_work is shown on the screen, written out.

##### [03:35.674](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=215.67354166666667)

Narration: Dunque P x, P y e P n coincidono, qualunque sia l'angolo theta: la pressione è la stessa in tutte le direzioni. È uno scalare.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [03:38.494](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=218.49354166666666): iso is shown on the screen, written out.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): iso moves to a new place on the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): heading\_prism is hidden from the screen — left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): pdef is hidden from the screen — left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): prism is hidden from the screen — left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): body is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): leg\_x is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): leg\_y is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): hyp is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): p\_x is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): p\_y is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): p\_n is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): theta is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): weight is hidden from the screen — prism left the board.
- [03:45.314](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=225.31354166666668): prism\_work is hidden from the screen — left the board.

##### [03:46.514](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=226.5135416666667)

Narration: Torniamo al tensore, che a riposo era diagonale. Ogni sforzo normale è una compressione di modulo P, quindi tutti e tre valgono meno P.

Board: iso — a Math \[text\] that says "$P\_x = P\_y = P\_n = P$"

Actions:
- [03:46.514](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=226.5135416666667): heading\_law is shown on the screen, written out.
- [03:47.294](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=227.29354166666667): sigma\_d is shown on the screen, written out.
- [03:53.934](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=233.93354166666666): sigma\_d becomes "$bold(sigma) = mat(-P, 0, 0; 0, -P, 0; 0, 0, -P)$".

##### [03:55.736](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=235.73604166666667)

Narration: Raccogliendo meno P resta la matrice identità, che in notazione indiciale è la delta di Kronecker, delta i j.

Board: iso — a Math \[text\] that says "$P\_x = P\_y = P\_n = P$"; sigma\_d — a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), 0, 0; 0, sigma\_(y y), 0; 0, 0, sigma\_(z z))$"; heading\_law — a Heading that says "Il tensore idrostatico"

Actions:
- [03:55.974](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=235.97354166666668): sigma\_d becomes "$bold(sigma) = -P mat(1, 0, 0; 0, 1, 0; 0, 0, 1)$".
- [03:58.934](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=238.93354166666666): law is shown on the screen, written out.
- [04:0.774](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=240.7735416666667): delta\_def is shown on the screen, written out.

##### [04:3.306](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=243.30604166666666)

Narration: In una sola riga, sigma i j uguale a meno P delta i j dice due cose: se i è diverso da j lo sforzo è nullo, niente taglio; se i è uguale a j vale meno P, la stessa in ogni direzione.

Board: iso — a Math \[text\] that says "$P\_x = P\_y = P\_n = P$"; sigma\_d — a Math \[text\] that says "$bold(sigma) = mat(sigma\_(x x), 0, 0; 0, sigma\_(y y), 0; 0, 0, sigma\_(z z))$"; law — a Math \[text\] that says "$sigma\_(i j) = -P delta\_(i j)$"; delta\_def — a Math \[text\] that says "$delta\_(i j) = cases(1 & upright("se") thin i = j, 0 & upright("se") thin i eq.not j)$"; heading\_law — a Heading that says "Il tensore idrostatico"

Actions:
- [04:4.214](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=244.21354166666666): A box is drawn around law.
- [04:9.934](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=249.93354166666666): delta\_def (the "i eq.not j" part) is emphasized.
- [04:13.934](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=253.93354166666666): delta\_def (the "i = j" part) is emphasized.
- [04:13.934](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=253.93354166666666): delta\_def (the "i eq.not j" part) is no longer emphasized.
- [04:16.014](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=256.0135416666667): delta\_def (the "i = j" part) is no longer emphasized.
- [04:17.979](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=257.97945833333335): delta\_def is hidden from the screen — left the board.
- [04:17.979](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=257.97945833333335): heading\_law is hidden from the screen — left the board.
- [04:17.979](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=257.97945833333335): iso is hidden from the screen — left the board.
- [04:17.979](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=257.97945833333335): law is hidden from the screen — left the board.
- [04:17.979](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=257.97945833333335): sigma\_d is hidden from the screen — left the board.

### Scene 4: [L'equazione fondamentale e le superfici isobare](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=259.021125)

Span: 04:19.021–06:23.759 (259.021125s–383.7585s).

#### Objects

- balance: a Derivation \[text\] that says "$dif F\_(p,x) &= P(x) dif y dif z - P(x + dif x) dif y dif z \\ &= - frac(partial P, partial x) dif V \\ dif bold(F)\_p &= - nabla P thin dif V \\ dif bold(F)\_v &= rho bold(f) thin dif V \\ - nabla P thin dif V + rho bold(f) thin dif V &= bold(0)$"
- body\_force: a Vector \[green\] labelled "rho bold(f) thin dif V" drawn in cube (start=(2.2, 2.4), end=(2.2, 1.4))
- consequence: a Math \[text\] that says "$P = P(U), quad rho = - frac(dif P, dif U)$"
- cube: a Figure (x\_range=(0.0, 4.4), y\_range=(0.3, 3.7), aspect=(4.4, 3.4))
- force\_field: a VectorField \[gray\] drawn in planet (function=\<function\>, at=((-2.25, -2.25), (-2.25, -1.5), (-2.25, -0.75), (-2.25, 0.0), (…, exclude=((0.0, 0.0),))
- fundamental: a Math \[text\] that says "$nabla P = rho bold(f)$"
- grad\_p: a Vector \[red\] labelled "nabla P" drawn in planet (start=(1.685, 1.414), end=(0.996, 0.835))
- grad\_rho: a Vector \[magenta\] labelled "nabla rho" drawn in planet (start=(-1.386, 0.8), end=(-0.78, 0.45))
- grad\_u: a Vector \[blue\] labelled "nabla U" drawn in planet (start=(1.685, 1.414), end=(2.374, 1.993))
- heading\_balance: a Heading that says "Equilibrio di un cubetto"
- heading\_surfaces: a Heading that says "Isobare, equipotenziali, isopicne"
- levels: a LevelCurves \[green\] drawn in planet (function=\<function\>, values=(1.0, 1.6, 2.2, 2.8))
- planet: an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0))
- push\_left: a Vector \[red\] labelled "P(x)" drawn in cube (start=(0.1, 2.0), end=(1.2, 2.0))
- push\_right: a Vector \[red\] labelled "P(x + dif x)" drawn in cube (start=(4.3, 2.0), end=(3.2, 2.0))
- square: a Polygon \[blue\] drawn in cube (vertices=((1.2, 1.0), (3.2, 1.0), (3.2, 3.0), (1.2, 3.0)))
- surfaces: a Derivation \[text\] that says "$bold(f) &= - nabla U \\ nabla P &= - rho nabla U \\ nabla times nabla P = bold(0) &= - nabla times (rho nabla U) \\ nabla times (rho nabla U) &= nabla rho times nabla U + rho nabla times nabla U \\ nabla rho times nabla U &= bold(0)$"
- tilted: a Vector \[gray\] labelled "nabla rho" drawn in planet (start=(1.685, 1.414), end=(1.685, 0.514))
- width: a Line \[gray\] labelled "dif x" drawn in cube (start=(1.2, 1.0), end=(3.2, 1.0))

#### Beats

##### [04:19.021](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=259.021125)

Narration: Ora mettiamo in equilibrio un cubetto di fluido di volume d V. Su di esso agiscono forze di superficie, dovute alla pressione, e forze di volume, proporzionali alla massa.

Board: Empty.

Actions:
- [04:19.021](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=259.021125): heading\_balance is shown on the screen, written out.
- [04:20.754](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=260.754125): cube is shown on the screen, written out.
- [04:20.754](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=260.754125): square is shown on the screen, written out.
- [04:24.701](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=264.701125): push\_left is shown on the screen, written out.
- [04:24.923](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=264.9234297084002): push\_right is shown on the screen, written out.
- [04:28.701](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=268.701125): body\_force is shown on the screen, written out.

##### [04:29.976](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=269.97612499999997)

Narration: Lungo x la faccia sinistra riceve P di x per d y d z, quella destra P di x più d x per la stessa area. La differenza è meno la derivata parziale di P rispetto a x, per d V.

Board: cube — a Figure (x\_range=(0.0, 4.4), y\_range=(0.3, 3.7), aspect=(4.4, 3.4)); heading\_balance — a Heading that says "Equilibrio di un cubetto"; square — a Polygon \[blue\] drawn in cube (vertices=((1.2, 1.0), (3.2, 1.0), (3.2, 3.0), (1.2, 3.0))); push\_left — a Vector \[red\] labelled "P(x)" drawn in cube (start=(0.1, 2.0), end=(1.2, 2.0)); push\_right — a Vector \[red\] labelled "P(x + dif x)" drawn in cube (start=(4.3, 2.0), end=(3.2, 2.0)); body\_force — a Vector \[green\] labelled "rho bold(f) thin dif V" drawn in cube (start=(2.2, 2.4), end=(2.2, 1.4))

Actions:
- [04:30.261](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=270.261125): width is shown on the screen, written out.
- [04:30.261](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=270.261125): balance is shown on the screen, written out.
- [04:38.901](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=278.901125): balance is shown on the screen, written out.

##### [04:43.951](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=283.951125)

Narration: Lo stesso accade lungo y e lungo z: la forza netta di pressione è meno il gradiente di P per d V. La forza di volume è rho f d V, dove f è la forza per unità di massa.

Board: cube — a Figure (x\_range=(0.0, 4.4), y\_range=(0.3, 3.7), aspect=(4.4, 3.4)); heading\_balance — a Heading that says "Equilibrio di un cubetto"; square — a Polygon \[blue\] drawn in cube (vertices=((1.2, 1.0), (3.2, 1.0), (3.2, 3.0), (1.2, 3.0))); push\_left — a Vector \[red\] labelled "P(x)" drawn in cube (start=(0.1, 2.0), end=(1.2, 2.0)); push\_right — a Vector \[red\] labelled "P(x + dif x)" drawn in cube (start=(4.3, 2.0), end=(3.2, 2.0)); body\_force — a Vector \[green\] labelled "rho bold(f) thin dif V" drawn in cube (start=(2.2, 2.4), end=(2.2, 1.4)); width — a Line \[gray\] labelled "dif x" drawn in cube (start=(1.2, 1.0), end=(3.2, 1.0))

Actions:
- [04:49.021](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=289.021125): balance is shown on the screen, written out.
- [04:52.061](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=292.06112499999995): balance is shown on the screen, written out.
- [04:55.021](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=295.021125): body\_force is indicated — a transient flash.

##### [04:56.296](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=296.29612499999996)

Narration: In equilibrio la somma è nulla. Dividendo per d V si ottiene l'equazione fondamentale della statica dei fluidi: il gradiente di pressione è uguale a rho f.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [04:57.461](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=297.461125): balance is shown on the screen, written out.
- [05:0.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=300.98112499999996): fundamental is shown on the screen, written out.
- [05:4.661](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=304.66112499999997): A box is drawn around fundamental.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): fundamental moves to a new place on the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): balance is hidden from the screen — left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): cube is hidden from the screen — left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): square is hidden from the screen — cube left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): push\_left is hidden from the screen — cube left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): push\_right is hidden from the screen — cube left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): body\_force is hidden from the screen — cube left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): width is hidden from the screen — cube left the board.
- [05:5.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=305.98112499999996): heading\_balance is hidden from the screen — left the board.

##### [05:7.181](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=307.18112499999995)

Narration: Supponiamo ora che la forza sia conservativa: f è meno il gradiente di un potenziale U. Qui U cresce con la distanza dal centro, le sue superfici di livello sono questi cerchi, e le frecce di forza puntano verso il centro.

Board: fundamental — a Math \[text\] that says "$nabla P = rho bold(f)$"

Actions:
- [05:7.181](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=307.18112499999995): heading\_surfaces is shown on the screen, written out.
- [05:7.181](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=307.18112499999995): The box around fundamental is lifted.
- [05:9.021](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=309.021125): surfaces is shown on the screen, written out.
- [05:13.141](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=313.141125): planet is shown on the screen, written out.
- [05:17.381](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=317.381125): levels is shown on the screen, written out.
- [05:18.301](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=318.30112499999996): force\_field is shown on the screen, written out.

##### [05:21.259](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=321.258625)

Narration: Sostituendo, il gradiente di P vale meno rho per il gradiente di U. Ora applichiamo il rotore: il rotore di un gradiente è nullo, quindi anche il rotore di rho per il gradiente di U deve annullarsi.

Board: fundamental — a Math \[text\] that says "$nabla P = rho bold(f)$"; planet — an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0)); heading\_surfaces — a Heading that says "Isobare, equipotenziali, isopicne"; levels — a LevelCurves \[green\] drawn in planet (function=\<function\>, values=(1.0, 1.6, 2.2, 2.8)); force\_field — a VectorField \[gray\] drawn in planet (function=\<function\>, at=((-2.25, -2.25), (-2.25, -1.5), (-2.25, -0.75), (-2.25, 0.0), (…, exclude=((0.0, 0.0),))

Actions:
- [05:21.501](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=321.501125): surfaces is shown on the screen, written out.
- [05:26.381](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=326.381125): surfaces is shown on the screen, written out.

##### [05:33.929](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=333.928625)

Narration: Sviluppando il prodotto, il termine con il rotore del gradiente di U sparisce, e resta: gradiente di rho, vettore gradiente di U, uguale a zero.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:34.181](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=334.181125): surfaces is shown on the screen, written out.
- [05:37.541](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=337.54112499999997): surfaces (the "rho nabla times nabla U" part) is slashed through — it cancels.
- [05:38.421](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=338.42112499999996): surfaces is shown on the screen, written out.

##### [05:43.299](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=343.298625)

Narration: Un prodotto vettoriale è nullo solo se i due gradienti sono paralleli. In questo punto il gradiente di U punta in fuori, quello di P verso il centro, sulla stessa retta.

Board: Unchanged from the preceding beat in this scene.

Actions:
- [05:49.341](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=349.341125): grad\_u is shown on the screen, written out.
- [05:51.341](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=351.341125): grad\_p is shown on the screen, written out.

##### [05:53.656](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=353.656125)

Narration: Un gradiente di densità inclinato, come questo, darebbe un prodotto diverso da zero: non è ammesso. La densità può variare solo lungo la stessa direzione, ovunque.

Board: fundamental — a Math \[text\] that says "$nabla P = rho bold(f)$"; planet — an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0)); heading\_surfaces — a Heading that says "Isobare, equipotenziali, isopicne"; levels — a LevelCurves \[green\] drawn in planet (function=\<function\>, values=(1.0, 1.6, 2.2, 2.8)); force\_field — a VectorField \[gray\] drawn in planet (function=\<function\>, at=((-2.25, -2.25), (-2.25, -1.5), (-2.25, -0.75), (-2.25, 0.0), (…, exclude=((0.0, 0.0),)); grad\_u — a Vector \[blue\] labelled "nabla U" drawn in planet (start=(1.685, 1.414), end=(2.374, 1.993)); grad\_p — a Vector \[red\] labelled "nabla P" drawn in planet (start=(1.685, 1.414), end=(0.996, 0.835))

Actions:
- [05:54.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=354.981125): tilted is shown on the screen, written out.
- [05:58.941](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=358.941125): tilted is hidden from the screen.
- [06:0.581](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=360.581125): grad\_rho is shown on the screen, written out.

##### [06:4.291](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=364.29112499999997)

Narration: Quindi pressione, potenziale e densità sono costanti sulle stesse superfici: isobare, equipotenziali e isopicne coincidono, e sono ortogonali alle linee di forza.

Board: fundamental — a Math \[text\] that says "$nabla P = rho bold(f)$"; planet — an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0)); heading\_surfaces — a Heading that says "Isobare, equipotenziali, isopicne"; levels — a LevelCurves \[green\] drawn in planet (function=\<function\>, values=(1.0, 1.6, 2.2, 2.8)); force\_field — a VectorField \[gray\] drawn in planet (function=\<function\>, at=((-2.25, -2.25), (-2.25, -1.5), (-2.25, -0.75), (-2.25, 0.0), (…, exclude=((0.0, 0.0),)); grad\_u — a Vector \[blue\] labelled "nabla U" drawn in planet (start=(1.685, 1.414), end=(2.374, 1.993)); grad\_p — a Vector \[red\] labelled "nabla P" drawn in planet (start=(1.685, 1.414), end=(0.996, 0.835)); grad\_rho — a Vector \[magenta\] labelled "nabla rho" drawn in planet (start=(-1.386, 0.8), end=(-0.78, 0.45))

Actions:
- [06:7.421](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=367.421125): consequence is shown on the screen, written out.
- [06:11.981](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=371.981125): A box is drawn around surfaces.
- [06:13.341](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=373.34112500000003): grad\_p is indicated — a transient flash.

##### [06:15.834](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=375.833625)

Narration: In un campo di gravità uniforme queste superfici diventano piani orizzontali, come il pelo libero di un lago in quiete.

Board: fundamental — a Math \[text\] that says "$nabla P = rho bold(f)$"; consequence — a Math \[text\] that says "$P = P(U), quad rho = - frac(dif P, dif U)$"; planet — an Axes (x\_range=(-3.0, 3.0), y\_range=(-3.0, 3.0), aspect=(1.0, 1.0)); heading\_surfaces — a Heading that says "Isobare, equipotenziali, isopicne"; levels — a LevelCurves \[green\] drawn in planet (function=\<function\>, values=(1.0, 1.6, 2.2, 2.8)); force\_field — a VectorField \[gray\] drawn in planet (function=\<function\>, at=((-2.25, -2.25), (-2.25, -1.5), (-2.25, -0.75), (-2.25, 0.0), (…, exclude=((0.0, 0.0),)); grad\_u — a Vector \[blue\] labelled "nabla U" drawn in planet (start=(1.685, 1.414), end=(2.374, 1.993)); grad\_p — a Vector \[red\] labelled "nabla P" drawn in planet (start=(1.685, 1.414), end=(0.996, 0.835)); grad\_rho — a Vector \[magenta\] labelled "nabla rho" drawn in planet (start=(-1.386, 0.8), end=(-0.78, 0.45))

Actions:
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): consequence is hidden from the screen — left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): fundamental is hidden from the screen — left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): heading\_surfaces is hidden from the screen — left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): planet is hidden from the screen — left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): levels is hidden from the screen — planet left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): force\_field is hidden from the screen — planet left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): grad\_u is hidden from the screen — planet left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): grad\_p is hidden from the screen — planet left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): grad\_rho is hidden from the screen — planet left the board.
- [06:22.717](https://academa.ai/@laul-handle/lectures/fondamenti-di-statica-dei-fluidi-e-tensore-degli-sforzi?t=382.71683333333334): surfaces is hidden from the screen — left the board.