# Understanding Divergence and Curl in Vector Calculus > Divergence and curl, built from the fluid they describe rather than announced as formulas. A vector field is read as water in motion, and two local questions are asked of it: is anything being created here, and is anything turning here. The first is answered by counting what crosses the four sides of a shrinking box, the second by walking round its edge, and each count leaves behind one derivative formula. Along the way a pure rotation is shown to have no divergence at all, and a flow of perfectly straight arrows is shown to have curl, so neither quantity can stand in for the other. The lecture then lifts both into three dimensions, draws the curl vector along the axis a whirlpool turns about, introduces the del notation, and finishes by computing both quantities for one field in space and for the field it opened on. - Canonical watch page: [Understanding Divergence and Curl in Vector Calculus](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Mathematics - Published: 2026-09-03T19:48:25.867Z - Updated: 2026-09-03T19:48:25.867Z - Duration: PT892S (14 minutes 52 seconds) - Chapters: 5 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M1MBSTQ7TYKCGE8MSACEFKRK/0/dark/master.m3u8) - Embed: [Player](https://academa.ai/embed/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M1MBSTQ7TYKCGE8MSACEFKRK/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M1MBSTQ7TYKCGE8MSACEFKRK/0/dark/poster.jpg) ## Description Divergence and curl from fluid flow: flux out of a small box, circulation round a small loop, the formulas, and a worked example. ## Chapters - [00:00–01:46.245 · A Field of Arrows](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=0) - [01:46.245–04:56.444 · Divergence: What Is Being Created Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=106.24533143939395) - [04:56.444–08:35.714 · Curl: What Is Turning Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=296.44393560606056) - [08:35.714–11:34.158 · Into Three Dimensions](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=515.7143731060605) - [11:34.158–14:52 · Putting Them Together](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=694.1578731060606) ## Transcript ### [00:00 · A Field of Arrows](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=0) Let's start with a picture rather than a definition. Here is a patch of the plane, and at every point of it I am going to draw one arrow. That collection is a vector field: a rule attaching a vector to every point. The arrows out here are long, the arrows near the middle are short, and they do not all point the same way. So think of the whole plane as water in motion, and the arrow at a point as the velocity of the water passing through it. Drop a speck of dye in, and it has to follow the arrows. There it goes. It is carried round and pushed outward at the same time, so its path is a spiral. Every particle of this fluid is doing something like that, all at once. Watching all of it at once is hopeless. So we ask small local questions instead: what is the fluid doing right here, in a tiny neighbourhood of one single point? There are two such questions, and between them they catch most of what a fluid can do locally. The first is about spreading. Look at this field: every arrow points away from the middle. That is the purest case of fluid being created at a point, as if from a tap somewhere under the surface. The second question is about turning. In this field nothing streams away at all: the water simply goes round. Both measurements have names. The spreading one is called the divergence of the field, and at each point it is a single number. The turning one is called the curl. And here is where we are heading. Each one is built out of derivatives, each one has a formula you can compute from the components, and each one answers a question you could in principle settle with a paddle wheel and a bottle of dye. Divergence first. ### [01:46.245 · Divergence: What Is Being Created Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=106.24533143939395) Here is the question that defines the first of the two measurements. Take a point of the plane, draw a small box around it, and ask whether more water leaves that box than enters. Give the box some dimensions. It runs from x to x plus delta x across, and from y to y plus delta y upward, and I will write the two components of the field as P and Q. Now count what crosses each side, one at a time. Through the right edge water is leaving, at a rate equal to the sideways component of the field there multiplied by the length of the edge. Through the left edge water is entering, so that one counts negative. And look at the two expressions: the very same function P, evaluated at two places a distance delta x apart. Subtract, divide by delta x, and let the box shrink. That difference quotient becomes the partial derivative of P with respect to x, so the whole x direction contributes that derivative times the area of the box. The top and bottom edges tell the same story with the other component. Out through the top, in through the bottom, and what is left over is the partial derivative of Q with respect to y, times that same area. Now add the two contributions. Rule a line underneath, and the total flux out of the box comes to this bracket, multiplied by the area. Divide by the area and every trace of the box disappears. What is left is attached to the point alone, and it has a name: the divergence of the field. Read it as a recipe. Differentiate the first component with respect to x. Differentiate the second component with respect to y. Add the two. The answer is one number for every point of the plane. Said carefully: divergence is outward flux per unit area, in the limit as the region shrinks to the point. Positive means fluid is appearing there. Negative means it is being taken away. So compute it three times. First the field that points straight out. Here P is x, so its x derivative is one; Q is y, so its y derivative is one; and the divergence comes to two, at every point. Reverse every arrow and you have the opposite, a sink. Now P is minus x and Q is minus y, both derivatives are minus one, and the divergence is minus two everywhere. And now a field that is pure rotation: P is minus y, Q is x. The x derivative of minus y is zero. The y derivative of x is zero. So the divergence of this whirlpool is zero, everywhere. Which is exactly right, and worth saying out loud. Water going round in circles is not piling up anywhere. Divergence is completely blind to rotation, and that is why one number is not enough. ### [04:56.444 · Curl: What Is Turning Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=296.44393560606056) Here is the second question. Instead of counting what crosses the sides of a small box, we walk right around its edge and ask whether the fluid helps us along the way. Take the rotating field again, and walk the loop counterclockwise. If the water pushes with us the whole way round, there is circulation, and a tiny paddle wheel sitting there would turn. Walk the bottom edge first, left to right. There the sideways component P is pushing along the direction we are travelling, so it contributes P times delta x. Along the top edge we are travelling right to left, so the same sideways component now works against us and counts negative, evaluated a height delta y further up. Subtract those two, divide by delta y, and the pair leaves minus the partial derivative of P with respect to y, times the area. The minus sign is there because the top of the loop is walked backwards. Now the two vertical edges, with the other component. Up the right edge Q pushes with us; down the left edge it pushes against us. Together they give plus the partial derivative of Q with respect to x, times that same area. Add the two pairs. The total circulation round the loop is this bracket, times the area, and notice what is inside it: the two derivatives are subtracted, not added. Divide by the area, shrink the loop, and what is left is called the curl of the field: the x derivative of Q, minus the y derivative of P. Read it as a recipe again. Differentiate the second component with respect to x. Differentiate the first component with respect to y. Subtract, in that order. In the plane the curl is a single number, and its sign carries the sense of the turning. Positive means counterclockwise, negative means clockwise, and zero means a paddle wheel dropped there would sit perfectly still. Two quick cases, then a surprise. Here is the rotating field, with P equal to minus y and Q equal to x. Its curl is one minus minus one, which comes to a definite counterclockwise spin. And the outward field, P equal to x and Q equal to y. The x derivative of y is zero, the y derivative of x is zero, so the curl is nothing at all. Water streaming straight out does not turn a wheel. So far, so intuitive: flows that look like they spin have curl, and flows that do not, do not. Now here is the field that breaks that intuition. This water all flows to the right. Nothing here circles anything. But look at the lengths of the arrows: the flow is faster higher up, and slower lower down. Now drop a paddle wheel in. Its top blade sits in the fast water and its bottom blade in the slow water, so the top gets pushed harder than the bottom, and the wheel turns clockwise. And the formula agrees with the wheel. Q is zero, so its x derivative is zero. P is two plus y, so its y derivative is one. Zero take away one is negative, and negative means clockwise. That is the lesson to keep. Curl is not about whether the streamlines look curved. It is about whether the fluid turns a small object placed in it, and a flow of perfectly straight lines can do exactly that. ### [08:35.714 · Into Three Dimensions](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=515.7143731060605) Everything so far has been flat. Real fluids move in three dimensions, so let's take both ideas up a dimension. Here is a field in space. The water circles the vertical axis, and at the same time it drifts steadily upward, which is roughly what a bath emptying looks like if you could see inside it. Divergence goes up almost unchanged. The small box becomes a small cube, with six faces now instead of four. The faces pair off along each axis, and each pair leaves behind one derivative. So with components P, Q and R, the divergence is the x derivative of P, plus the y derivative of Q, plus the z derivative of R. One number again, and it still means flux out per unit volume. Our whirlpool has divergence zero. The circling part turns without spreading, and the upward drift has the same speed at every height, so nothing anywhere inside is being created. Curl is the one that really changes. In the plane there was only one loop to walk. In space, a small loop can be laid flat in three independent ways. One lies flat, in the plane of the floor. One stands in each of the two vertical planes. Each of them has its own circulation, so each gives its own number, and those three numbers are the components of a vector. Written out, they look like this. The x component pairs the y and z derivatives. The y component pairs the z and x derivatives. And the z component pairs the x and y derivatives. Look hard at that last line. The vertical component of the curl in space is exactly the plane curl we built a few minutes ago, which is the sense in which the flat case was never really a special case. For the whirlpool the first two components vanish, and the third one is two. So the curl here is a vector, and it points straight up the axis the water is turning about. That is the general rule, and the right hand fixes the sign. Curl your fingers the way the water goes round, and your thumb points the way the curl vector points. Up, here, because seen from above this flow runs counterclockwise. Finally, the notation everybody actually writes. Collect the three partial derivative symbols into one object, called del, and agree to treat it as though it were a vector. Dot del into the field and out comes the divergence, because a dot product of two vectors is a number. Cross del into the field and out comes the curl, because a cross product of two vectors is a vector. The notation is doing real work there. It tells you the shape of each answer before you compute a single derivative. Del dot F is one number. Del cross F is three. ### [11:34.158 · Putting Them Together](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=694.1578731060606) Let's put the two side by side, because they are easy to confuse and they answer genuinely different questions. Divergence measures spreading: is fluid being created or destroyed at this point. Curl measures turning: would a small object placed at this point start to rotate. They eat the same thing and hand back different things. Both take a vector field. Divergence gives you back one number; curl gives you back a whole vector. In the compact notation, one of them is a dot product and the other is a cross product, and that is exactly why one answer is a scalar and the other is a vector. And each has a word for being zero. A field whose divergence vanishes everywhere is called incompressible. A field whose curl vanishes everywhere is called irrotational. Those two words are worth keeping. Now one worked example in space, done slowly. Here is the field: x times y, then y times z, then z times x. Divergence first. Differentiate the first component with respect to x. Treat y as a constant while you do it, so x times y differentiates to y. Then the second with respect to z, and the third with respect to x. Careful, though: each component is differentiated with respect to its own variable. The second gives z, the third gives x, and adding all three the divergence is x plus y plus z. So this field is a source wherever that sum is positive, and a sink wherever it is negative. On the flat surface where the sum is exactly zero, it is neither one nor the other. Now the curl, one component at a time. The x component uses the other two components: the y derivative of z x is nothing, and the z derivative of y z is y, so the answer is minus y. The y component pairs the z and x derivatives in the same pattern, and it comes out as minus z. And the z component pairs the x and y derivatives. The x derivative of y z is nothing, the y derivative of x y is x, and so this one is minus x. Put them together and the curl is minus y, minus z, minus x. Every component is nonzero somewhere, so this fluid is turning almost everywhere, about an axis that swings around from point to point. Let me finish where we began. This was the very first field I drew: x minus y for the first component, x plus y for the second. Its divergence is one plus one, which is two, so it is spreading. Its curl is one minus minus one, which is also two, so it is turning. Both at once, which is exactly what that spiralling speck of dye was telling us. So: two derivatives of a vector field. Divergence, a number, saying how much is being created at a point. Curl, a vector, saying how the fluid turns there. Between them, they are the language the rest of vector calculus is written in. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M1MBSTQ7TYKCGE8MSACEFKRK/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: [A Field of Arrows](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=0) Span: 00:00–01:46.245 (0s–106.24533143939395s). #### Objects - curl\_label: a Tex \[text\] that says "$op("curl") arrow(F)$: how much is turning" - div\_label: a Tex \[text\] that says "$op("div") arrow(F)$: how much is spreading out" - flow: a VectorField \[blue\] drawn in plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12) - head\_field: a Heading that says "A Vector Field" - head\_two: a Heading that says "Two Questions at a Point" - plane: an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)) - point: a Point \[yellow\] drawn in plane (location=(1.8, 1.1)) - point\_2: a Point \[yellow\] drawn in plane (location=(0.4, 0.4)) - point\_3: a Point \[yellow\] drawn in plane (location=(-1.0, 0.9)) - point\_4: a Point \[yellow\] drawn in source\_plane - point\_5: a Point \[yellow\] drawn in spin\_plane (location=(1.2, 0.0)) - point\_6: a Point \[yellow\] drawn in source\_plane (location=(1.2, 0.6)) - source\_flow: a VectorField \[red\] drawn in source\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - source\_plane: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - speck: a Point \[green\] labelled "P" drawn in plane (location=(((0.35 \* exp(t\_draw)) \* cos(t\_draw)), ((0.35 \* exp(t\_draw)) \* …) - spin\_flow: a VectorField \[green\] drawn in spin\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - spin\_plane: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - streamline: a ParametricCurve \[yellow\] drawn in plane (function=\, t\_range=(0.0, \)) - t\_draw: a VariableNumber (initial\_value=0.05) #### Beats ##### [00:00](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=0) Narration: Let's start with a picture rather than a definition. Here is a patch of the plane, and at every point of it I am going to draw one arrow. Board: Empty. Actions: - [00:00](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=0): head\_field is shown on the screen, written out. - [00:3.576](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=3.576): plane is shown on the screen, written out. - [00:7.001](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=7.001): flow is shown on the screen, written out. ##### [00:8.379](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=8.379) Narration: That collection is a vector field: a rule attaching a vector to every point. The arrows out here are long, the arrows near the middle are short, and they do not all point the same way. Board: plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); head\_field — a Heading that says "A Vector Field"; flow — a VectorField \[blue\] drawn in plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12) Actions: - [00:14.892](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=14.892): point is shown on the screen, grown. - [00:16.96](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=16.960343919044956): point is hidden from the screen. - [00:17.098](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=17.098): point\_2 is shown on the screen, grown. - [00:18.93](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=18.929746129453463): point\_2 is hidden from the screen. ##### [00:20.623](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=20.6235) Narration: So think of the whole plane as water in motion, and the arrow at a point as the velocity of the water passing through it. Drop a speck of dye in, and it has to follow the arrows. Board: Unchanged from the preceding beat in this scene. Actions: - [00:27.241](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=27.240999999999996): streamline is shown on the screen, written out. - [00:27.241](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=27.240999999999996): speck is shown on the screen, written out. ##### [00:31.568](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=31.567999999999998) Narration: There it goes. It is carried round and pushed outward at the same time, so its path is a spiral. Every particle of this fluid is doing something like that, all at once. Board: plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); head\_field — a Heading that says "A Vector Field"; flow — a VectorField \[blue\] drawn in plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12); streamline — a ParametricCurve \[yellow\] drawn in plane (function=\, t\_range=(0.0, \)); speck — a Point \[green\] labelled "P" drawn in plane (location=(((0.35 \* exp(t\_draw)) \* cos(t\_draw)), ((0.35 \* exp(t\_draw)) \* …) Actions: - [00:32.346](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=32.346): streamline is redrawn as the numbers it depends on change. - [00:32.346](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=32.346): speck is redrawn as the numbers it depends on change. - [00:32.346](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=32.346): t\_draw ticks to 1.7. ##### [00:42.559](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=42.559) Narration: Watching all of it at once is hopeless. So we ask small local questions instead: what is the fluid doing right here, in a tiny neighbourhood of one single point? Board: Unchanged from the preceding beat in this scene. Actions: - [00:44.277](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=44.277): streamline is hidden from the screen. - [00:44.277](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=44.277): speck is hidden from the screen. - [00:49.42](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=49.42): point\_3 is shown on the screen, grown. - [00:51.667](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=51.66707973705034): point\_3 is hidden from the screen. - [00:52.486](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=52.4855): head\_field is hidden from the screen — left the board. - [00:52.486](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=52.4855): plane is hidden from the screen — left the board. - [00:52.486](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=52.4855): flow is hidden from the screen — plane left the board. ##### [00:53.685](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=53.6855) Narration: There are two such questions, and between them they catch most of what a fluid can do locally. The first is about spreading. Look at this field: every arrow points away from the middle. Board: Empty. Actions: - [00:53.685](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=53.6855): head\_two is shown on the screen, written out. - [01:2.196](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=62.196000000000005): source\_plane is shown on the screen, written out. - [01:2.196](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=62.196000000000005): source\_flow is shown on the screen, written out. ##### [01:5.791](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=65.791) Narration: That is the purest case of fluid being created at a point, as if from a tap somewhere under the surface. The second question is about turning. In this field nothing streams away at all: the water simply goes round. Board: source\_plane — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); head\_two — a Heading that says "Two Questions at a Point"; source\_flow — a VectorField \[red\] drawn in source\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [01:9.75](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=69.75): point\_4 is shown on the screen, grown. - [01:12.093](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=72.09263075750282): point\_4 is hidden from the screen. - [01:15.056](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=75.05600000000001): source\_plane moves to a new place on the board. - [01:15.056](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=75.05600000000001): spin\_plane is shown on the screen, written out. - [01:15.056](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=75.05600000000001): spin\_flow is shown on the screen, written out. ##### [01:19.463](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=79.4635) Narration: Both measurements have names. The spreading one is called the divergence of the field, and at each point it is a single number. The turning one is called the curl. Board: source\_plane — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); spin\_plane — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); head\_two — a Heading that says "Two Questions at a Point"; source\_flow — a VectorField \[red\] drawn in source\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); spin\_flow — a VectorField \[green\] drawn in spin\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [01:22.726](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=82.72600000000001): div\_label is shown on the screen, written out. - [01:28.194](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=88.194): curl\_label is shown on the screen, written out. ##### [01:29.48](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=89.4795) Narration: And here is where we are heading. Each one is built out of derivatives, each one has a formula you can compute from the components, and each one answers a question you could in principle settle with a paddle wheel and a bottle of dye. Divergence first. Board: div\_label — a Tex \[text\] that says "$op("div") arrow(F)$: how much is spreading out"; source\_plane — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); curl\_label — a Tex \[text\] that says "$op("curl") arrow(F)$: how much is turning"; spin\_plane — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); head\_two — a Heading that says "Two Questions at a Point"; source\_flow — a VectorField \[red\] drawn in source\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); spin\_flow — a VectorField \[green\] drawn in spin\_plane (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [01:40.578](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=100.57800000000002): point\_5 is shown on the screen, grown. - [01:41.902](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=101.90200000000002): point\_6 is shown on the screen, grown. - [01:43.175](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=103.17523402303414): point\_5 is hidden from the screen. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): curl\_label is hidden from the screen — left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): div\_label is hidden from the screen — left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): head\_two is hidden from the screen — left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): source\_plane is hidden from the screen — left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): source\_flow is hidden from the screen — source\_plane left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): spin\_plane is hidden from the screen — left the board. - [01:44.837](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=104.83693750000002): spin\_flow is hidden from the screen — spin\_plane left the board. - [01:45.204](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=105.20366477272727): point\_6 is hidden from the screen. ### Scene 2: [Divergence: What Is Being Created Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=106.24533143939395) Span: 01:46.245–04:56.444 (106.24533143939395s–296.44393560606056s). #### Objects - axes: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - axes\_2: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - axes\_3: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - box: a Polygon \[yellow\] drawn in flux\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.18) - case1\_div: a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$" - case1\_field: a VectorField \[red\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - case1\_law: a Math \[text\] that says "$arrow(F) = (x, thin y)$" - case2\_div: a Math \[text\] that says "$op("div") arrow(F) = - 1 - 1 = - 2$" - case2\_field: a VectorField \[blue\] drawn in axes\_2 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - case2\_law: a Math \[text\] that says "$arrow(F) = (- x, thin - y)$" - case3\_div: a Math \[text\] that says "$op("div") arrow(F) = 0 + 0 = 0$" - case3\_field: a VectorField \[green\] drawn in axes\_3 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - case3\_law: a Math \[text\] that says "$arrow(F) = (- y, thin x)$" - comps: a Math \[text\] that says "$arrow(F) = (P(x, y), thin Q(x, y))$" - corner: a Point \[yellow\] labelled "(x, thin y)" drawn in flux\_plane (location=(0.4, 0.3)) - div\_note: a Panel that says "The divergence at a point is the outward flux per unit area of a small box around it, in the limit as the box shrinks to the point." - divergence: a Math \[text\] that says "$op("div") arrow(F) = frac(partial P, partial x) + frac(partial Q, partial y)$" - dx\_brace: a Brace \[cyan\] labelled "Delta x" drawn in flux\_plane (targets=('box',)) - dy\_brace: a Brace \[cyan\] labelled "Delta y" drawn in flux\_plane (targets=('box',), side='left') - edges: a Derivation \[text\] that says "$upright("right") &approx P(x + Delta x, y) Delta y \\ upright("left") &approx - P(x, y) Delta y \\ upright("top") &approx Q(x, y + Delta y) Delta x \\ upright("bottom") &approx - Q(x, y) Delta x$" - flux\_flow: a VectorField \[blue\] drawn in flux\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15) - flux\_plane: an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)) - flux\_sum: an Arithmetic \[text\] that says "$frac(partial P, partial x) Delta A frac(partial Q, partial y) Delta A (frac(partial P, partial x) + frac(partial Q, partial y)) Delta A$" (operator='+', operands=('frac(partial P, partial x) Delta A', 'frac(partial Q, partial…, result='(frac(partial P, partial x) + frac(partial Q, partial y)) Delt…) - head\_box: a Heading that says "Flux Out of a Small Box" - head\_cases: a Heading that says "Three Fields, Three Divergences" - head\_formula: a Heading that says "Divergence" - point: a Point \[yellow\] drawn in axes\_3 (location=(0.85, 0.0)) - question: a Panel that says "Does more fluid leave a small box around a point than enters it?" #### Beats ##### [01:46.245](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=106.24533143939395) Narration: Here is the question that defines the first of the two measurements. Take a point of the plane, draw a small box around it, and ask whether more water leaves that box than enters. Board: Empty. Actions: - [01:46.245](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=106.24533143939395): question is shown on the screen, written out. - [01:50.773](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=110.77333143939396): flux\_plane is shown on the screen, written out. - [01:50.773](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=110.77333143939396): flux\_flow is shown on the screen, written out. - [01:52.166](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=112.16633143939396): box is shown on the screen, written out. ##### [01:57.306](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=117.30633143939394) Narration: Give the box some dimensions. It runs from x to x plus delta x across, and from y to y plus delta y upward, and I will write the two components of the field as P and Q. Board: flux\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); box — a Polygon \[yellow\] drawn in flux\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.18); question — a Panel that says "Does more fluid leave a small box around a point than enters it?"; flux\_flow — a VectorField \[blue\] drawn in flux\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15) Actions: - [01:58.594](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=118.59433143939395): corner is shown on the screen, written out. - [02:2.809](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=122.80933143939396): dx\_brace is shown on the screen, written out. - [02:5.886](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=125.88633143939396): dy\_brace is shown on the screen, written out. - [02:7.859](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=127.85933143939394): flux\_plane moves to a new place on the board. - [02:7.859](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=127.85933143939394): comps is shown on the screen, written out. - [02:10.82](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=130.82033143939395): flux\_plane moves to a new place on the board. - [02:10.82](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=130.82033143939395): comps is hidden from the screen — left the board. - [02:10.82](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=130.82033143939395): question is hidden from the screen — left the board. ##### [02:11.42](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=131.42033143939395) Narration: Now count what crosses each side, one at a time. Through the right edge water is leaving, at a rate equal to the sideways component of the field there multiplied by the length of the edge. Board: flux\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); box — a Polygon \[yellow\] drawn in flux\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.18); flux\_flow — a VectorField \[blue\] drawn in flux\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15); corner — a Point \[yellow\] labelled "(x, thin y)" drawn in flux\_plane (location=(0.4, 0.3)); dx\_brace — a Brace \[cyan\] labelled "Delta x" drawn in flux\_plane (targets=('box',)); dy\_brace — a Brace \[cyan\] labelled "Delta y" drawn in flux\_plane (targets=('box',), side='left') Actions: - [02:11.42](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=131.42033143939395): head\_box is shown on the screen, written out. - [02:16.168](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=136.16833143939394): The segment (1.6, 0.3) to (1.6, 1.5) in flux\_plane is lit up. - [02:20.755](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=140.75533143939396): edges is shown on the screen, written out. - [02:22.984](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=142.98383143939395): flux\_plane: retire a lit segment (unemphasize\_line). ##### [02:23.584](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=143.58383143939395) Narration: Through the left edge water is entering, so that one counts negative. And look at the two expressions: the very same function P, evaluated at two places a distance delta x apart. Board: flux\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); box — a Polygon \[yellow\] drawn in flux\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.18); flux\_flow — a VectorField \[blue\] drawn in flux\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15); corner — a Point \[yellow\] labelled "(x, thin y)" drawn in flux\_plane (location=(0.4, 0.3)); dx\_brace — a Brace \[cyan\] labelled "Delta x" drawn in flux\_plane (targets=('box',)); dy\_brace — a Brace \[cyan\] labelled "Delta y" drawn in flux\_plane (targets=('box',), side='left'); head\_box — a Heading that says "Flux Out of a Small Box" Actions: - [02:24.42](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=144.42033143939395): The segment (0.4, 0.3) to (0.4, 1.5) in flux\_plane is lit up. - [02:27.16](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=147.16033143939396): edges is shown on the screen, written out. - [02:29.342](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=149.34233143939394): edges is emphasized. - [02:31.339](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=151.33933143939396): flux\_plane: retire a lit segment (unemphasize\_line). - [02:33.777](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=153.77733143939395): edges is no longer emphasized. - [02:33.777](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=153.77733143939395): edges is emphasized. - [02:36.215](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=156.21533143939394): edges is no longer emphasized. ##### [02:36.815](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=156.81533143939396) Narration: Subtract, divide by delta x, and let the box shrink. That difference quotient becomes the partial derivative of P with respect to x, so the whole x direction contributes that derivative times the area of the box. Board: Unchanged from the preceding beat in this scene. Actions: - [02:48.425](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=168.42533143939394): flux\_sum is shown on the screen, written out. ##### [02:52.264](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=172.26383143939393) Narration: The top and bottom edges tell the same story with the other component. Out through the top, in through the bottom, and what is left over is the partial derivative of Q with respect to y, times that same area. Board: Unchanged from the preceding beat in this scene. Actions: - [02:57.511](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=177.51133143939393): edges is shown on the screen, written out. - [02:57.511](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=177.51133143939393): The segment (0.4, 1.5) to (1.6, 1.5) in flux\_plane is lit up. - [02:58.788](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=178.78833143939391): edges is shown on the screen, written out. - [02:58.788](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=178.78833143939391): The segment (0.4, 0.3) to (1.6, 0.3) in flux\_plane is lit up. - [03:0.124](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=180.12433143939393): flux\_sum is shown on the screen, written out. - [03:5.592](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=185.59183143939396): flux\_plane: retire a lit segment (unemphasize\_line). - [03:5.592](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=185.59183143939396): flux\_plane: retire a lit segment (unemphasize\_line). ##### [03:6.192](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=186.19183143939392) Narration: Now add the two contributions. Rule a line underneath, and the total flux out of the box comes to this bracket, multiplied by the area. Board: Unchanged from the preceding beat in this scene. Actions: - [03:9.071](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=189.07133143939393): flux\_sum is shown on the screen, drawn. - [03:9.956](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=189.95623668474076): flux\_sum is shown on the screen, drawn. - [03:13.308](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=193.30833143939392): flux\_sum is shown on the screen, written out. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): edges is hidden from the screen — left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): flux\_plane is hidden from the screen — left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): box is hidden from the screen — flux\_plane left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): flux\_flow is hidden from the screen — flux\_plane left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): corner is hidden from the screen — flux\_plane left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): dx\_brace is hidden from the screen — flux\_plane left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): dy\_brace is hidden from the screen — flux\_plane left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): flux\_sum is hidden from the screen — left the board. - [03:15.863](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=195.86283143939391): head\_box is hidden from the screen — left the board. ##### [03:16.463](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=196.46283143939394) Narration: Divide by the area and every trace of the box disappears. What is left is attached to the point alone, and it has a name: the divergence of the field. Board: Empty. Actions: - [03:16.463](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=196.46283143939394): head\_formula is shown on the screen, written out. - [03:24.078](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=204.07833143939393): divergence is shown on the screen, written out. ##### [03:27.767](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=207.76683143939394) Narration: Read it as a recipe. Differentiate the first component with respect to x. Differentiate the second component with respect to y. Add the two. The answer is one number for every point of the plane. Board: divergence — a Math \[text\] that says "$op("div") arrow(F) = frac(partial P, partial x) + frac(partial Q, partial y)$"; head\_formula — a Heading that says "Divergence" Actions: - [03:31.389](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=211.3893314393939): divergence (the "frac(partial P, partial x)" part) is emphasized. - [03:35.22](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=215.22033143939393): divergence (the "frac(partial P, partial x)" part) is no longer emphasized. - [03:35.22](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=215.22033143939393): divergence (the "frac(partial Q, partial y)" part) is emphasized. - [03:38.053](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=218.05333143939393): divergence (the "frac(partial Q, partial y)" part) is no longer emphasized. ##### [03:43.529](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=223.52933143939396) Narration: Said carefully: divergence is outward flux per unit area, in the limit as the region shrinks to the point. Positive means fluid is appearing there. Negative means it is being taken away. Board: Unchanged from the preceding beat in this scene. Actions: - [03:44.179](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=224.17933143939393): div\_note is shown on the screen, written out. - [03:51.482](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=231.48233143939393): A box is drawn around divergence. - [03:56.951](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=236.95083143939394): div\_note is hidden from the screen — left the board. - [03:56.951](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=236.95083143939394): divergence is hidden from the screen — left the board. - [03:56.951](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=236.95083143939394): head\_formula is hidden from the screen — left the board. ##### [03:57.551](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=237.55083143939396) Narration: So compute it three times. First the field that points straight out. Here P is x, so its x derivative is one; Q is y, so its y derivative is one; and the divergence comes to two, at every point. Board: Empty. Actions: - [03:57.551](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=237.55083143939396): head\_cases is shown on the screen, written out. - [04:0.011](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=240.01133143939393): axes is shown on the screen, written out. - [04:0.011](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=240.01133143939393): case1\_field is shown on the screen, written out. - [04:2.496](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=242.49633143939394): axes moves to a new place on the board. - [04:2.496](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=242.49633143939394): case1\_law is shown on the screen, written out. - [04:11.064](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=251.06433143939392): case1\_div is shown on the screen, written out. ##### [04:13.986](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=253.98583143939396) Narration: Reverse every arrow and you have the opposite, a sink. Now P is minus x and Q is minus y, both derivatives are minus one, and the divergence is minus two everywhere. Board: axes — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case1\_law — a Math \[text\] that says "$arrow(F) = (x, thin y)$"; case1\_div — a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$"; head\_cases — a Heading that says "Three Fields, Three Divergences"; case1\_field — a VectorField \[red\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [04:14.334](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=254.3343314393939): axes\_2 is shown on the screen, written out. - [04:14.334](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=254.3343314393939): case2\_field is shown on the screen, written out. - [04:18.107](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=258.10733143939393): case2\_law is shown on the screen, written out. - [04:24.621](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=264.62133143939394): case2\_div is shown on the screen, written out. ##### [04:26.939](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=266.9393314393939) Narration: And now a field that is pure rotation: P is minus y, Q is x. The x derivative of minus y is zero. The y derivative of x is zero. So the divergence of this whirlpool is zero, everywhere. Board: axes — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case1\_law — a Math \[text\] that says "$arrow(F) = (x, thin y)$"; case1\_div — a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$"; axes\_2 — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case2\_law — a Math \[text\] that says "$arrow(F) = (- x, thin - y)$"; case2\_div — a Math \[text\] that says "$op("div") arrow(F) = - 1 - 1 = - 2$"; head\_cases — a Heading that says "Three Fields, Three Divergences"; case1\_field — a VectorField \[red\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); case2\_field — a VectorField \[blue\] drawn in axes\_2 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [04:27.473](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=267.4733314393939): axes\_3 is shown on the screen, written out. - [04:27.473](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=267.4733314393939): case3\_field is shown on the screen, written out. - [04:28.518](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=268.51833143939393): case3\_law is shown on the screen, written out. - [04:39.269](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=279.26933143939397): case3\_div is shown on the screen, written out. ##### [04:42.69](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=282.69033143939396) Narration: Which is exactly right, and worth saying out loud. Water going round in circles is not piling up anywhere. Divergence is completely blind to rotation, and that is why one number is not enough. Board: axes — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case1\_law — a Math \[text\] that says "$arrow(F) = (x, thin y)$"; case1\_div — a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$"; axes\_2 — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case2\_law — a Math \[text\] that says "$arrow(F) = (- x, thin - y)$"; case2\_div — a Math \[text\] that says "$op("div") arrow(F) = - 1 - 1 = - 2$"; axes\_3 — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); case3\_law — a Math \[text\] that says "$arrow(F) = (- y, thin x)$"; case3\_div — a Math \[text\] that says "$op("div") arrow(F) = 0 + 0 = 0$"; head\_cases — a Heading that says "Three Fields, Three Divergences"; case1\_field — a VectorField \[red\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); case2\_field — a VectorField \[blue\] drawn in axes\_2 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); case3\_field — a VectorField \[green\] drawn in axes\_3 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [04:47.775](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=287.77533143939394): point is shown on the screen, grown. - [04:50.281](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=290.28128593608625): point is hidden from the screen. - [04:51.618](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=291.61833143939396): case3\_div is indicated — a transient flash. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): axes is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case1\_field is hidden from the screen — axes left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): axes\_2 is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case2\_field is hidden from the screen — axes\_2 left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): axes\_3 is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case3\_field is hidden from the screen — axes\_3 left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case1\_div is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case1\_law is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case2\_div is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case2\_law is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case3\_div is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): case3\_law is hidden from the screen — left the board. - [04:55.402](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=295.40226893939393): head\_cases is hidden from the screen — left the board. ### Scene 3: [Curl: What Is Turning Here](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=296.44393560606056) Span: 04:56.444–08:35.714 (296.44393560606056s–515.7143731060605s). #### Objects - axes: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - axes\_2: an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)) - bottom\_blade: a Vector \[magenta\] drawn in shear\_plane (start=(0.0, -0.55), end=(0.62, -0.55)) - circ\_sum: an Arithmetic \[text\] that says "$- frac(partial P, partial y) Delta A frac(partial Q, partial x) Delta A (frac(partial Q, partial x) - frac(partial P, partial y)) Delta A$" (operator='+', operands=('- frac(partial P, partial y) Delta A', 'frac(partial Q, parti…, result='(frac(partial Q, partial x) - frac(partial P, partial y)) Delt…) - comps: a Math \[text\] that says "$arrow(F) = (P(x, y), thin Q(x, y))$" - curl2d: a Math \[text\] that says "$op("curl") arrow(F) = frac(partial Q, partial x) - frac(partial P, partial y)$" - curl\_note: a Panel that says "The curl of a plane field at a point is the counterclockwise circulation per unit area of a small loop around it, in the limit as the loop shrinks to the point." - head\_cases: a Heading that says "Two Clear Cases" - head\_formula: a Heading that says "The Curl of a Plane Field" - head\_loop: a Heading that says "Circulation Around a Small Loop" - head\_shear: a Heading that says "A Flow That Looks Straight" - hub: a Point \[magenta\] drawn in shear\_plane - loop: a Polygon \[yellow\] drawn in loop\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.12) - loop\_flow: a VectorField \[blue\] drawn in loop\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15) - loop\_plane: an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)) - out\_curl: a Math \[text\] that says "$op("curl") arrow(F) = 0 - 0 = 0$" - out\_field: a VectorField \[red\] drawn in axes\_2 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - out\_law: a Math \[text\] that says "$arrow(F) = (x, thin y)$" - point: a Point \[yellow\] drawn in loop\_plane (location=(1.0, 0.9)) - point\_2: a Point \[yellow\] drawn in axes (location=(1.2, 0.0)) - point\_3: a Point \[yellow\] drawn in axes\_2 (location=(1.2, 0.6)) - point\_4: a Point \[yellow\] drawn in shear\_plane (location=(-0.2, 1.2)) - point\_5: a Point \[yellow\] drawn in shear\_plane (location=(-0.2, -1.2)) - question: a Panel that says "Would a tiny paddle wheel dropped in the fluid at a point turn?" - shear\_curl: a Math \[text\] that says "$op("curl") arrow(F) = 0 - 1 = - 1$" - shear\_field: a VectorField \[blue\] drawn in shear\_plane (function=\, at=((-1.8, -1.2), (-1.8, -0.6), (-1.8, 0.0), (-1.8, 0.6), (-1.8, 1…, scale=0.16) - shear\_law: a Math \[text\] that says "$arrow(F) = (2 + y, thin 0)$" - shear\_plane: an Axes (x\_range=(-2.2, 2.2), y\_range=(-1.6, 1.6), aspect=(4.4, 3.2)) - sides: a Derivation \[text\] that says "$upright("bottom") &approx P(x, y) Delta x \\ upright("top") &approx - P(x, y + Delta y) Delta x \\ upright("right") &approx Q(x + Delta x, y) Delta y \\ upright("left") &approx - Q(x, y) Delta y$" - spin\_curl: a Math \[text\] that says "$op("curl") arrow(F) = 1 - (- 1) = 2$" - spin\_field: a VectorField \[green\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) - spin\_law: a Math \[text\] that says "$arrow(F) = (- y, thin x)$" - top\_blade: a Vector \[magenta\] drawn in shear\_plane (start=(0.0, 0.55), end=(1.1, 0.55)) - walk: an Orientation \[yellow\] drawn in loop\_plane (path=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), closed=True, arrows=4) - wheel\_turn: a CurvedArrow \[magenta\] drawn in shear\_plane (start=(0.95, 0.55), end=(0.95, -0.55)) #### Beats ##### [04:56.444](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=296.44393560606056) Narration: Here is the second question. Instead of counting what crosses the sides of a small box, we walk right around its edge and ask whether the fluid helps us along the way. Board: Empty. Actions: - [04:56.444](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=296.44393560606056): question is shown on the screen, written out. - [04:58.998](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=298.99793560606054): loop\_plane is shown on the screen, written out. - [04:58.998](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=298.99793560606054): loop\_flow is shown on the screen, written out. - [05:1.134](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=301.13393560606056): loop is shown on the screen, written out. - [05:3.155](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=303.1549356060606): walk is shown on the screen, written out. ##### [05:6.61](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=306.61043560606055) Narration: Take the rotating field again, and walk the loop counterclockwise. If the water pushes with us the whole way round, there is circulation, and a tiny paddle wheel sitting there would turn. Board: loop\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); question — a Panel that says "Would a tiny paddle wheel dropped in the fluid at a point turn?"; loop\_flow — a VectorField \[blue\] drawn in loop\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15); loop — a Polygon \[yellow\] drawn in loop\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.12); walk — an Orientation \[yellow\] drawn in loop\_plane (path=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), closed=True, arrows=4) Actions: - [05:7.946](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=307.9459356060606): loop\_plane moves to a new place on the board. - [05:7.946](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=307.9459356060606): comps is shown on the screen, written out. - [05:16.223](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=316.22293560606056): point is shown on the screen, grown. - [05:17.491](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=317.4909912388839): point is hidden from the screen. - [05:18.034](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=318.0344356060606): loop\_plane moves to a new place on the board. - [05:18.034](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=318.0344356060606): comps is hidden from the screen — left the board. - [05:18.034](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=318.0344356060606): question is hidden from the screen — left the board. ##### [05:18.634](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=318.63443560606055) Narration: Walk the bottom edge first, left to right. There the sideways component P is pushing along the direction we are travelling, so it contributes P times delta x. Board: loop\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); loop\_flow — a VectorField \[blue\] drawn in loop\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15); loop — a Polygon \[yellow\] drawn in loop\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.12); walk — an Orientation \[yellow\] drawn in loop\_plane (path=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), closed=True, arrows=4) Actions: - [05:18.634](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=318.63443560606055): head\_loop is shown on the screen, written out. - [05:19.354](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=319.3539356060606): The segment (0.4, 0.3) to (1.6, 0.3) in loop\_plane is lit up. - [05:26.912](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=326.9119356060606): sides is shown on the screen, written out. - [05:29.304](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=329.3039356060606): loop\_plane: retire a lit segment (unemphasize\_line). ##### [05:29.904](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=329.90393560606054) Narration: Along the top edge we are travelling right to left, so the same sideways component now works against us and counts negative, evaluated a height delta y further up. Board: loop\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); loop\_flow — a VectorField \[blue\] drawn in loop\_plane (function=\, at=((-2.0, -2.0), (-2.0, -1.2), (-2.0, -0.4), (-2.0, 0.4), (-2.0, …, scale=0.15); loop — a Polygon \[yellow\] drawn in loop\_plane (vertices=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), fill\_opacity=0.12); walk — an Orientation \[yellow\] drawn in loop\_plane (path=((0.4, 0.3), (1.6, 0.3), (1.6, 1.5), (0.4, 1.5)), closed=True, arrows=4); head\_loop — a Heading that says "Circulation Around a Small Loop" Actions: - [05:30.752](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=330.75193560606056): The segment (0.4, 1.5) to (1.6, 1.5) in loop\_plane is lit up. - [05:36.452](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=336.45193560606054): sides is shown on the screen, written out. - [05:40.028](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=340.02843560606055): loop\_plane: retire a lit segment (unemphasize\_line). ##### [05:40.628](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=340.6284356060606) Narration: Subtract those two, divide by delta y, and the pair leaves minus the partial derivative of P with respect to y, times the area. The minus sign is there because the top of the loop is walked backwards. Board: Unchanged from the preceding beat in this scene. Actions: - [05:47.861](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=347.8609356060606): circ\_sum is shown on the screen, written out. ##### [05:54.313](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=354.3129356060606) Narration: Now the two vertical edges, with the other component. Up the right edge Q pushes with us; down the left edge it pushes against us. Together they give plus the partial derivative of Q with respect to x, times that same area. Board: Unchanged from the preceding beat in this scene. Actions: - [05:58.667](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=358.6669356060606): sides is shown on the screen, written out. - [05:58.667](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=358.6669356060606): The segment (1.6, 0.3) to (1.6, 1.5) in loop\_plane is lit up. - [06:1.418](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=361.41793560606055): sides is shown on the screen, written out. - [06:1.418](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=361.41793560606055): The segment (0.4, 0.3) to (0.4, 1.5) in loop\_plane is lit up. - [06:4.924](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=364.9239356060606): circ\_sum is shown on the screen, written out. - [06:10.253](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=370.25293560606053): loop\_plane: retire a lit segment (unemphasize\_line). - [06:10.253](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=370.25293560606053): loop\_plane: retire a lit segment (unemphasize\_line). ##### [06:10.853](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=370.85293560606056) Narration: Add the two pairs. The total circulation round the loop is this bracket, times the area, and notice what is inside it: the two derivatives are subtracted, not added. Board: Unchanged from the preceding beat in this scene. Actions: - [06:11.329](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=371.32893560606055): circ\_sum is shown on the screen, drawn. - [06:12.365](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=372.3651144002393): circ\_sum is shown on the screen, drawn. - [06:15.787](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=375.7869356060605): circ\_sum is shown on the screen, written out. - [06:21.116](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=381.11593560606053): circ\_sum is indicated — a transient flash. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): circ\_sum is hidden from the screen — left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): head\_loop is hidden from the screen — left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): loop\_plane is hidden from the screen — left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): loop\_flow is hidden from the screen — loop\_plane left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): loop is hidden from the screen — loop\_plane left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): walk is hidden from the screen — loop\_plane left the board. - [06:23.02](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.0204356060606): sides is hidden from the screen — left the board. ##### [06:23.62](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.62043560606054) Narration: Divide by the area, shrink the loop, and what is left is called the curl of the field: the x derivative of Q, minus the y derivative of P. Board: Empty. Actions: - [06:23.62](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=383.62043560606054): head\_formula is shown on the screen, written out. - [06:28.079](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=388.07893560606055): curl2d is shown on the screen, written out. ##### [06:34.448](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=394.44843560606057) Narration: Read it as a recipe again. Differentiate the second component with respect to x. Differentiate the first component with respect to y. Subtract, in that order. Board: curl2d — a Math \[text\] that says "$op("curl") arrow(F) = frac(partial Q, partial x) - frac(partial P, partial y)$"; head\_formula — a Heading that says "The Curl of a Plane Field" Actions: - [06:37.978](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=397.97793560606056): curl2d (the "frac(partial Q, partial x)" part) is emphasized. - [06:41.472](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=401.4719356060605): curl2d (the "frac(partial P, partial y)" part) is emphasized. - [06:41.472](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=401.4719356060605): curl2d (the "frac(partial Q, partial x)" part) is no longer emphasized. - [06:45.513](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=405.5129356060605): curl2d (the "frac(partial P, partial y)" part) is no longer emphasized. ##### [06:47.03](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=407.0299356060606) Narration: In the plane the curl is a single number, and its sign carries the sense of the turning. Positive means counterclockwise, negative means clockwise, and zero means a paddle wheel dropped there would sit perfectly still. Board: Unchanged from the preceding beat in this scene. Actions: - [06:49.201](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=409.20093560606057): curl\_note is shown on the screen, written out. - [06:56.921](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=416.92093560606054): A box is drawn around curl2d. - [07:0.59](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=420.5899356060606): curl2d is hidden from the screen — left the board. - [07:0.59](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=420.5899356060606): curl\_note is hidden from the screen — left the board. - [07:0.59](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=420.5899356060606): head\_formula is hidden from the screen — left the board. ##### [07:1.19](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=421.18993560606054) Narration: Two quick cases, then a surprise. Here is the rotating field, with P equal to minus y and Q equal to x. Its curl is one minus minus one, which comes to a definite counterclockwise spin. Board: Empty. Actions: - [07:1.19](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=421.18993560606054): head\_cases is shown on the screen, written out. - [07:2.096](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=422.09593560606055): axes is shown on the screen, written out. - [07:2.096](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=422.09593560606055): spin\_field is shown on the screen, written out. - [07:5.289](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=425.2889356060606): axes moves to a new place on the board. - [07:5.289](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=425.2889356060606): spin\_law is shown on the screen, written out. - [07:12.974](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=432.97393560606054): spin\_curl is shown on the screen, written out. ##### [07:16.18](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=436.18043560606054) Narration: And the outward field, P equal to x and Q equal to y. The x derivative of y is zero, the y derivative of x is zero, so the curl is nothing at all. Water streaming straight out does not turn a wheel. Board: spin\_law — a Math \[text\] that says "$arrow(F) = (- y, thin x)$"; spin\_curl — a Math \[text\] that says "$op("curl") arrow(F) = 1 - (- 1) = 2$"; axes — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); head\_cases — a Heading that says "Two Clear Cases"; spin\_field — a VectorField \[green\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [07:16.86](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=436.85993560606056): axes\_2 is shown on the screen, written out. - [07:16.86](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=436.85993560606056): out\_field is shown on the screen, written out. - [07:18.253](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=438.25293560606053): out\_law is shown on the screen, written out. - [07:25.974](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=445.97393560606054): out\_curl is shown on the screen, written out. ##### [07:31.101](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=451.1014356060606) Narration: So far, so intuitive: flows that look like they spin have curl, and flows that do not, do not. Now here is the field that breaks that intuition. Board: spin\_law — a Math \[text\] that says "$arrow(F) = (- y, thin x)$"; spin\_curl — a Math \[text\] that says "$op("curl") arrow(F) = 1 - (- 1) = 2$"; axes — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); out\_law — a Math \[text\] that says "$arrow(F) = (x, thin y)$"; out\_curl — a Math \[text\] that says "$op("curl") arrow(F) = 0 - 0 = 0$"; axes\_2 — an Axes (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), aspect=(1, 1)); head\_cases — a Heading that says "Two Clear Cases"; spin\_field — a VectorField \[green\] drawn in axes (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22); out\_field — a VectorField \[red\] drawn in axes\_2 (function=\, at=((-1.2, -1.2), (-1.2, -0.6), (-1.2, 0.0), (-1.2, 0.6), (-1.2, 1…, scale=0.22) Actions: - [07:34.457](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=454.45693560606054): point\_2 is shown on the screen, grown. - [07:36.594](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=456.59393560606054): point\_3 is shown on the screen, grown. - [07:36.851](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=456.85084734136166): point\_2 is hidden from the screen. - [07:39.562](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=459.56187942895986): point\_3 is hidden from the screen. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): axes is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): spin\_field is hidden from the screen — axes left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): axes\_2 is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): out\_field is hidden from the screen — axes\_2 left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): head\_cases is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): out\_curl is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): out\_law is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): spin\_curl is hidden from the screen — left the board. - [07:41.47](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=461.4699356060606): spin\_law is hidden from the screen — left the board. ##### [07:42.07](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=462.06993560606054) Narration: This water all flows to the right. Nothing here circles anything. But look at the lengths of the arrows: the flow is faster higher up, and slower lower down. Board: Empty. Actions: - [07:42.07](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=462.06993560606054): head\_shear is shown on the screen, written out. - [07:42.639](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=462.6389356060606): shear\_plane is shown on the screen, written out. - [07:42.639](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=462.6389356060606): shear\_field is shown on the screen, written out. - [07:47.956](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=467.95593560606056): shear\_plane moves to a new place on the board. - [07:47.956](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=467.95593560606056): shear\_law is shown on the screen, written out. - [07:50.394](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=470.3939356060606): point\_4 is shown on the screen, grown. - [07:51.613](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=471.61293560606055): point\_5 is shown on the screen, grown. - [07:52.286](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=472.2863918460489): point\_4 is hidden from the screen. ##### [07:53.827](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=473.8269356060606) Narration: Now drop a paddle wheel in. Its top blade sits in the fast water and its bottom blade in the slow water, so the top gets pushed harder than the bottom, and the wheel turns clockwise. Board: shear\_law — a Math \[text\] that says "$arrow(F) = (2 + y, thin 0)$"; shear\_plane — an Axes (x\_range=(-2.2, 2.2), y\_range=(-1.6, 1.6), aspect=(4.4, 3.2)); head\_shear — a Heading that says "A Flow That Looks Straight"; shear\_field — a VectorField \[blue\] drawn in shear\_plane (function=\, at=((-1.8, -1.2), (-1.8, -0.6), (-1.8, 0.0), (-1.8, 0.6), (-1.8, 1…, scale=0.16); point\_5 — a Point \[yellow\] drawn in shear\_plane (location=(-0.2, -1.2)) Actions: - [07:54.523](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=474.52264550993635): point\_5 is hidden from the screen. - [07:54.872](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=474.87193560606056): hub is shown on the screen, written out. - [07:56.695](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=476.69493560606054): top\_blade is shown on the screen, written out. - [07:58.97](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=478.9699356060606): bottom\_blade is shown on the screen, written out. - [08:4.079](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=484.07893560606055): wheel\_turn is shown on the screen, written out. ##### [08:5.805](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=485.80493560606055) Narration: And the formula agrees with the wheel. Q is zero, so its x derivative is zero. P is two plus y, so its y derivative is one. Zero take away one is negative, and negative means clockwise. Board: shear\_law — a Math \[text\] that says "$arrow(F) = (2 + y, thin 0)$"; shear\_plane — an Axes (x\_range=(-2.2, 2.2), y\_range=(-1.6, 1.6), aspect=(4.4, 3.2)); head\_shear — a Heading that says "A Flow That Looks Straight"; shear\_field — a VectorField \[blue\] drawn in shear\_plane (function=\, at=((-1.8, -1.2), (-1.8, -0.6), (-1.8, 0.0), (-1.8, 0.6), (-1.8, 1…, scale=0.16); hub — a Point \[magenta\] drawn in shear\_plane; top\_blade — a Vector \[magenta\] drawn in shear\_plane (start=(0.0, 0.55), end=(1.1, 0.55)); bottom\_blade — a Vector \[magenta\] drawn in shear\_plane (start=(0.0, -0.55), end=(0.62, -0.55)); wheel\_turn — a CurvedArrow \[magenta\] drawn in shear\_plane (start=(0.95, 0.55), end=(0.95, -0.55)) Actions: - [08:18.065](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=498.0649356060606): shear\_curl is shown on the screen, written out. ##### [08:21.358](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=501.3584356060606) Narration: That is the lesson to keep. Curl is not about whether the streamlines look curved. It is about whether the fluid turns a small object placed in it, and a flow of perfectly straight lines can do exactly that. Board: shear\_law — a Math \[text\] that says "$arrow(F) = (2 + y, thin 0)$"; shear\_curl — a Math \[text\] that says "$op("curl") arrow(F) = 0 - 1 = - 1$"; shear\_plane — an Axes (x\_range=(-2.2, 2.2), y\_range=(-1.6, 1.6), aspect=(4.4, 3.2)); head\_shear — a Heading that says "A Flow That Looks Straight"; shear\_field — a VectorField \[blue\] drawn in shear\_plane (function=\, at=((-1.8, -1.2), (-1.8, -0.6), (-1.8, 0.0), (-1.8, 0.6), (-1.8, 1…, scale=0.16); hub — a Point \[magenta\] drawn in shear\_plane; top\_blade — a Vector \[magenta\] drawn in shear\_plane (start=(0.0, 0.55), end=(1.1, 0.55)); bottom\_blade — a Vector \[magenta\] drawn in shear\_plane (start=(0.0, -0.55), end=(0.62, -0.55)); wheel\_turn — a CurvedArrow \[magenta\] drawn in shear\_plane (start=(0.95, 0.55), end=(0.95, -0.55)) Actions: - [08:22.311](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=502.31093560606064): A box is drawn around shear\_curl. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): head\_shear is hidden from the screen — left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): shear\_curl is hidden from the screen — left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): shear\_law is hidden from the screen — left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): shear\_plane is hidden from the screen — left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): shear\_field is hidden from the screen — shear\_plane left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): hub is hidden from the screen — shear\_plane left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): top\_blade is hidden from the screen — shear\_plane left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): bottom\_blade is hidden from the screen — shear\_plane left the board. - [08:34.673](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=514.672706439394): wheel\_turn is hidden from the screen — shear\_plane left the board. ### Scene 4: [Into Three Dimensions](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=515.7143731060605) Span: 08:35.714–11:34.158 (515.7143731060605s–694.1578731060606s). #### Objects - comps3: a Math \[text\] that says "$arrow(F) = (P, thin Q, thin R)$" - cube: a Solid \[yellow\] drawn in frame (upper=\, lower=\, x\_range=(0.4, 1.0)) - curl3: a Derivation \[text\] that says "$(op("curl") arrow(F))\_x &= frac(partial R, partial y) - frac(partial Q, partial z) \\ (op("curl") arrow(F))\_y &= frac(partial P, partial z) - frac(partial R, partial x) \\ (op("curl") arrow(F))\_z &= frac(partial Q, partial x) - frac(partial …$" - curl\_check: a Math \[text\] that says "$op("curl") arrow(F) = (0, thin 0, thin 2)$" - curl\_form: a Math \[text\] that says "$op("curl") arrow(F) = nabla times arrow(F)$" - curl\_vec: a Vector \[red\] labelled "op("curl") arrow(F)" drawn in frame (start=(0.0, 0.0, 0.0), end=(0.0, 0.0, 1.3)) - div3: a Math \[text\] that says "$op("div") arrow(F) = frac(partial P, partial x) + frac(partial Q, partial y) + frac(partial R, partial z)$" - div\_check: a Math \[text\] that says "$op("div") arrow(F) = 0 + 0 + 0 = 0$" - div\_form: a Math \[text\] that says "$op("div") arrow(F) = nabla dot arrow(F)$" - field3: a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…) - frame: an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)) - head\_curl3: a Heading that says "Curl in Space" - head\_del: a Heading that says "The Notation Everybody Uses" - head\_div3: a Heading that says "Divergence in Space" - head\_space: a Heading that says "A Field in Space" - loop\_xy: a Polygon \[green\] drawn in frame (vertices=((0.3, 0.3, 0.0), (1.0, 0.3, 0.0), (1.0, 1.0, 0.0), (0.3, 1.0, …, fill\_opacity=0.35) - loop\_yz: a Polygon \[red\] drawn in frame (vertices=((0.0, 0.3, 0.3), (0.0, 1.0, 0.3), (0.0, 1.0, 1.0), (0.0, 0.3, …, fill\_opacity=0.35) - loop\_zx: a Polygon \[magenta\] drawn in frame (vertices=((0.3, 0.0, 0.3), (0.3, 0.0, 1.0), (1.0, 0.0, 1.0), (1.0, 0.0, …, fill\_opacity=0.35) - nabla\_def: a Math \[text\] that says "$nabla = (frac(partial, partial x), thin frac(partial, partial y), thin frac(partial, partial z))$" #### Beats ##### [08:35.714](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=515.7143731060605) Narration: Everything so far has been flat. Real fluids move in three dimensions, so let's take both ideas up a dimension. Board: Empty. Actions: - [08:35.714](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=515.7143731060605): head\_space is shown on the screen, written out. - [08:39.963](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=519.9633731060605): frame is shown on the screen, written out. ##### [08:44](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=524.0003731060605) Narration: Here is a field in space. The water circles the vertical axis, and at the same time it drifts steadily upward, which is roughly what a bath emptying looks like if you could see inside it. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); head\_space — a Heading that says "A Field in Space" Actions: - [08:44.766](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=524.7663731060605): field3 is shown on the screen, written out. - [08:46.949](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=526.9493731060605): frame turns in its own slot. - [08:55.343](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=535.3428731060606): frame moves to a new place on the board. - [08:55.343](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=535.3428731060606): head\_space is hidden from the screen — left the board. ##### [08:55.943](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=535.9428731060605) Narration: Divergence goes up almost unchanged. The small box becomes a small cube, with six faces now instead of four. The faces pair off along each axis, and each pair leaves behind one derivative. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…) Actions: - [08:55.943](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=535.9428731060605): head\_div3 is shown on the screen, written out. - [09:1.051](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=541.0513731060605): cube is shown on the screen, written out. ##### [09:10.44](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=550.4403731060605) Narration: So with components P, Q and R, the divergence is the x derivative of P, plus the y derivative of Q, plus the z derivative of R. One number again, and it still means flux out per unit volume. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…); head\_div3 — a Heading that says "Divergence in Space"; cube — a Solid \[yellow\] drawn in frame (upper=\, lower=\, x\_range=(0.4, 1.0)) Actions: - [09:11.09](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=551.0903731060605): comps3 is shown on the screen, written out. - [09:13.528](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=553.5283731060605): div3 is shown on the screen, written out. ##### [09:24.903](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=564.9028731060605) Narration: Our whirlpool has divergence zero. The circling part turns without spreading, and the upward drift has the same speed at every height, so nothing anywhere inside is being created. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…); comps3 — a Math \[text\] that says "$arrow(F) = (P, thin Q, thin R)$"; div3 — a Math \[text\] that says "$op("div") arrow(F) = frac(partial P, partial x) + frac(partial Q, partial y) + frac(partial R, partial z)$"; head\_div3 — a Heading that says "Divergence in Space"; cube — a Solid \[yellow\] drawn in frame (upper=\, lower=\, x\_range=(0.4, 1.0)) Actions: - [09:27.073](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=567.0733731060606): div\_check is shown on the screen, written out. - [09:31.322](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=571.3223731060605): cube is hidden from the screen. - [09:36.187](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.1873731060605): comps3 is hidden from the screen — left the board. - [09:36.187](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.1873731060605): div3 is hidden from the screen — left the board. - [09:36.187](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.1873731060605): div\_check is hidden from the screen — left the board. - [09:36.187](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.1873731060605): head\_div3 is hidden from the screen — left the board. ##### [09:36.787](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.7873731060605) Narration: Curl is the one that really changes. In the plane there was only one loop to walk. In space, a small loop can be laid flat in three independent ways. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…) Actions: - [09:36.787](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=576.7873731060605): head\_curl3 is shown on the screen, written out. - [09:45.274](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=585.2743731060605): loop\_xy is shown on the screen, written out. ##### [09:47.581](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=587.5808731060605) Narration: One lies flat, in the plane of the floor. One stands in each of the two vertical planes. Each of them has its own circulation, so each gives its own number, and those three numbers are the components of a vector. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…); head\_curl3 — a Heading that says "Curl in Space"; loop\_xy — a Polygon \[green\] drawn in frame (vertices=((0.3, 0.3, 0.0), (1.0, 0.3, 0.0), (1.0, 1.0, 0.0), (0.3, 1.0, …, fill\_opacity=0.35) Actions: - [09:51.435](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=591.4353731060605): loop\_yz is shown on the screen, written out. - [09:52.689](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=592.6893731060605): loop\_zx is shown on the screen, written out. - [09:55.545](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=595.5453731060605): loop\_xy is hidden from the screen. - [09:57.576](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=597.5763731060605): loop\_yz is hidden from the screen. - [09:57.998](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=597.9975510179348): loop\_zx is hidden from the screen. ##### [10:1.695](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=601.6948731060605) Narration: Written out, they look like this. The x component pairs the y and z derivatives. The y component pairs the z and x derivatives. And the z component pairs the x and y derivatives. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…); head\_curl3 — a Heading that says "Curl in Space" Actions: - [10:4.91](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=604.9103731060605): curl3 is shown on the screen, written out. - [10:8.509](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=608.5093731060605): curl3 is shown on the screen, written out. - [10:12.271](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=612.2713731060605): curl3 is shown on the screen, written out. ##### [10:15.484](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=615.4838731060605) Narration: Look hard at that last line. The vertical component of the curl in space is exactly the plane curl we built a few minutes ago, which is the sense in which the flat case was never really a special case. Board: Unchanged from the preceding beat in this scene. Actions: - [10:16.505](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=616.5053731060605): curl3 is emphasized. - [10:25.549](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=625.5493731060606): curl3 is no longer emphasized. ##### [10:28.123](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=628.1228731060605) Narration: For the whirlpool the first two components vanish, and the third one is two. So the curl here is a vector, and it points straight up the axis the water is turning about. Board: Unchanged from the preceding beat in this scene. Actions: - [10:31.687](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=631.6873731060605): curl\_check is shown on the screen, written out. - [10:34.682](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=634.6823731060605): curl\_vec is shown on the screen, written out. ##### [10:39.207](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=639.2068731060605) Narration: That is the general rule, and the right hand fixes the sign. Curl your fingers the way the water goes round, and your thumb points the way the curl vector points. Up, here, because seen from above this flow runs counterclockwise. Board: frame — an Axes3D (x\_range=(-1.6, 1.6), y\_range=(-1.6, 1.6), z\_range=(-1.6, 1.6)); field3 — a VectorField \[blue\] drawn in frame (function=\, at=((-1.0, -1.0, -1.0), (-1.0, -1.0, 0.0), (-1.0, -1.0, 1.0), (-1.…); curl\_check — a Math \[text\] that says "$op("curl") arrow(F) = (0, thin 0, thin 2)$"; head\_curl3 — a Heading that says "Curl in Space"; curl\_vec — a Vector \[red\] labelled "op("curl") arrow(F)" drawn in frame (start=(0.0, 0.0, 0.0), end=(0.0, 0.0, 1.3)) Actions: - [10:46.683](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=646.6833731060606): curl\_vec is indicated — a transient flash. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): curl3 is hidden from the screen — left the board. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): curl\_check is hidden from the screen — left the board. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): frame is hidden from the screen — left the board. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): field3 is hidden from the screen — frame left the board. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): curl\_vec is hidden from the screen — frame left the board. - [10:54.126](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.1258731060605): head\_curl3 is hidden from the screen — left the board. ##### [10:54.726](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.7258731060605) Narration: Finally, the notation everybody actually writes. Collect the three partial derivative symbols into one object, called del, and agree to treat it as though it were a vector. Board: Empty. Actions: - [10:54.726](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=654.7258731060605): head\_del is shown on the screen, written out. - [10:58.719](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=658.7193731060605): nabla\_def is shown on the screen, written out. ##### [11:6.959](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=666.9588731060605) Narration: Dot del into the field and out comes the divergence, because a dot product of two vectors is a number. Cross del into the field and out comes the curl, because a cross product of two vectors is a vector. Board: nabla\_def — a Math \[text\] that says "$nabla = (frac(partial, partial x), thin frac(partial, partial y), thin frac(partial, partial z))$"; head\_del — a Heading that says "The Notation Everybody Uses" Actions: - [11:7.365](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=667.3653731060606): div\_form is shown on the screen, written out. - [11:13.959](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=673.9593731060605): curl\_form is shown on the screen, written out. ##### [11:20.701](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=680.7013731060605) Narration: The notation is doing real work there. It tells you the shape of each answer before you compute a single derivative. Del dot F is one number. Del cross F is three. Board: nabla\_def — a Math \[text\] that says "$nabla = (frac(partial, partial x), thin frac(partial, partial y), thin frac(partial, partial z))$"; div\_form — a Math \[text\] that says "$op("div") arrow(F) = nabla dot arrow(F)$"; curl\_form — a Math \[text\] that says "$op("curl") arrow(F) = nabla times arrow(F)$"; head\_del — a Heading that says "The Notation Everybody Uses" Actions: - [11:29.78](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=689.7803731060606): div\_form is indicated — a transient flash. - [11:32.137](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=692.1373731060605): curl\_form is indicated — a transient flash. - [11:33.116](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=693.1162064393939): curl\_form is hidden from the screen — left the board. - [11:33.116](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=693.1162064393939): div\_form is hidden from the screen — left the board. - [11:33.116](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=693.1162064393939): head\_del is hidden from the screen — left the board. - [11:33.116](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=693.1162064393939): nabla\_def is hidden from the screen — left the board. ### Scene 5: [Putting Them Together](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=694.1578731060606) Span: 11:34.158–14:52.374 (694.1578731060606s–892.3744356060606s). #### Objects - close\_curl: a Math \[text\] that says "$op("curl") arrow(F) = 1 - (- 1) = 2$" - close\_div: a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$" - close\_flow: a VectorField \[blue\] drawn in close\_plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12) - close\_law: a Math \[text\] that says "$arrow(F) = (x - y, thin x + y)$" - close\_plane: an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)) - compare: a Table \[text\] that says "divergence curl what it measures spreading turning what you feed it a vector field a vector field what comes back a number a vector compact form $nabla dot arrow(F)$ $nabla times arrow(F)$ zero everywhere incompressible irrotational" (rows=(('', 'divergence', 'curl'), ('what it measures', 'spreading', …, header=True) - curl\_answer: a Math \[text\] that says "$op("curl") arrow(F) = (- y, thin - z, thin - x)$" - curl\_work: a Derivation \[text\] that says "$(op("curl") arrow(F))\_x &= frac(partial, partial y)(z x) - frac(partial, partial z)(y z) = - y \\ (op("curl") arrow(F))\_y &= frac(partial, partial z)(x y) - frac(partial, partial x)(z x) = - z \\ (op("curl") arrow(F))\_z &= frac(partial, part…$" - div\_answer: a Math \[text\] that says "$op("div") arrow(F) = x + y + z$" - div\_work: a Derivation \[text\] that says "$op("div") arrow(F) &= frac(partial, partial x)(x y) + frac(partial, partial y)(y z) + frac(partial, partial z)(z x) \\ &= y + z + x$" - head\_close: a Heading that says "Back Where We Started" - head\_cmp: a Heading that says "Two Different Questions" - point: a Point \[yellow\] drawn in close\_plane (location=(1.1, 0.4)) - problem: a Tex \[text\] that says "For $arrow(F) = (x y, thin y z, thin z x)$, find the divergence and the curl." #### Beats ##### [11:34.158](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=694.1578731060606) Narration: Let's put the two side by side, because they are easy to confuse and they answer genuinely different questions. Board: Empty. Actions: - [11:34.158](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=694.1578731060606): head\_cmp is shown on the screen, written out. - [11:35.052](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=695.0518731060606): compare is shown on the screen, written out. ##### [11:41.085](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=701.0853731060606) Narration: Divergence measures spreading: is fluid being created or destroyed at this point. Curl measures turning: would a small object placed at this point start to rotate. Board: head\_cmp — a Heading that says "Two Different Questions" Actions: - [11:42.618](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=702.6178731060606): compare is shown on the screen, written out. - [11:48.005](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=708.0048731060606): compare is indicated — a transient flash. ##### [11:53.132](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=713.1323731060606) Narration: They eat the same thing and hand back different things. Both take a vector field. Divergence gives you back one number; curl gives you back a whole vector. Board: Unchanged from the preceding beat in this scene. Actions: - [11:53.69](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=713.6898731060606): compare is shown on the screen, written out. - [12:0.656](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=720.6558731060607): compare is shown on the screen, written out. ##### [12:4.193](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=724.1928731060606) Narration: In the compact notation, one of them is a dot product and the other is a cross product, and that is exactly why one answer is a scalar and the other is a vector. Board: Unchanged from the preceding beat in this scene. Actions: - [12:5.424](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=725.4238731060606): compare is shown on the screen, written out. - [12:8.825](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=728.8248731060606): compare is indicated — a transient flash. ##### [12:15.195](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=735.1953731060606) Narration: And each has a word for being zero. A field whose divergence vanishes everywhere is called incompressible. A field whose curl vanishes everywhere is called irrotational. Those two words are worth keeping. Board: Unchanged from the preceding beat in this scene. Actions: - [12:17.239](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=737.2388731060606): compare is shown on the screen, written out. - [12:28.64](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=748.6398731060606): compare is indicated — a transient flash. - [12:29.51](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=749.5103731060606): compare is hidden from the screen — left the board. - [12:29.51](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=749.5103731060606): head\_cmp is hidden from the screen — left the board. ##### [12:30.11](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=750.1103731060606) Narration: Now one worked example in space, done slowly. Here is the field: x times y, then y times z, then z times x. Divergence first. Board: Empty. Actions: - [12:31.283](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=751.2828731060606): problem is shown on the screen, written out. ##### [12:42.692](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=762.6923731060606) Narration: Differentiate the first component with respect to x. Treat y as a constant while you do it, so x times y differentiates to y. Then the second with respect to z, and the third with respect to x. Board: problem — a Tex \[text\] that says "For $arrow(F) = (x y, thin y z, thin z x)$, find the divergence and the curl." Actions: - [12:43.041](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=763.0408731060606): div\_work is shown on the screen, written out. ##### [12:57.851](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=777.8513731060606) Narration: Careful, though: each component is differentiated with respect to its own variable. The second gives z, the third gives x, and adding all three the divergence is x plus y plus z. Board: Unchanged from the preceding beat in this scene. Actions: - [13:6.663](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=786.6628731060606): div\_work is shown on the screen, written out. - [13:7.568](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=787.5678731060606): div\_answer is shown on the screen, written out. ##### [13:11.373](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=791.3728731060606) Narration: So this field is a source wherever that sum is positive, and a sink wherever it is negative. On the flat surface where the sum is exactly zero, it is neither one nor the other. Board: div\_answer — a Math \[text\] that says "$op("div") arrow(F) = x + y + z$"; problem — a Tex \[text\] that says "For $arrow(F) = (x y, thin y z, thin z x)$, find the divergence and the curl." Actions: - [13:12.615](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=792.6148731060606): A box is drawn around div\_answer. - [13:22.982](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=802.9823731060605): div\_answer is hidden from the screen — left the board. - [13:22.982](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=802.9823731060605): div\_work is hidden from the screen — left the board. ##### [13:23.582](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=803.5823731060606) Narration: Now the curl, one component at a time. The x component uses the other two components: the y derivative of z x is nothing, and the z derivative of y z is y, so the answer is minus y. Board: problem — a Tex \[text\] that says "For $arrow(F) = (x y, thin y z, thin z x)$, find the divergence and the curl." Actions: - [13:35.436](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=815.4358731060606): curl\_work is shown on the screen, written out. ##### [13:37.209](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=817.2088731060607) Narration: The y component pairs the z and x derivatives in the same pattern, and it comes out as minus z. Board: Unchanged from the preceding beat in this scene. Actions: - [13:41.923](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=821.9228731060606): curl\_work is shown on the screen, written out. ##### [13:44.426](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=824.4263731060606) Narration: And the z component pairs the x and y derivatives. The x derivative of y z is nothing, the y derivative of x y is x, and so this one is minus x. Board: Unchanged from the preceding beat in this scene. Actions: - [13:53.726](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=833.7258731060606): curl\_work is shown on the screen, written out. ##### [13:56.52](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=836.5203731060606) Narration: Put them together and the curl is minus y, minus z, minus x. Every component is nonzero somewhere, so this fluid is turning almost everywhere, about an axis that swings around from point to point. Board: Unchanged from the preceding beat in this scene. Actions: - [13:57.925](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=837.9248731060607): curl\_answer is shown on the screen, written out. - [14:5.17](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=845.1698731060607): A box is drawn around curl\_answer. - [14:10.012](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=850.0118731060606): curl\_answer is hidden from the screen — left the board. - [14:10.012](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=850.0118731060606): curl\_work is hidden from the screen — left the board. - [14:10.012](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=850.0118731060606): problem is hidden from the screen — left the board. ##### [14:10.612](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=850.6118731060606) Narration: Let me finish where we began. This was the very first field I drew: x minus y for the first component, x plus y for the second. Board: Empty. Actions: - [14:10.612](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=850.6118731060606): head\_close is shown on the screen, written out. - [14:14.153](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=854.1528731060606): close\_plane is shown on the screen, written out. - [14:14.153](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=854.1528731060606): close\_flow is shown on the screen, written out. - [14:16.695](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=856.6948731060606): close\_plane moves to a new place on the board. - [14:16.695](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=856.6948731060606): close\_law is shown on the screen, written out. ##### [14:20.14](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=860.1398731060606) Narration: Its divergence is one plus one, which is two, so it is spreading. Its curl is one minus minus one, which is also two, so it is turning. Both at once, which is exactly what that spiralling speck of dye was telling us. Board: close\_law — a Math \[text\] that says "$arrow(F) = (x - y, thin x + y)$"; close\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); head\_close — a Heading that says "Back Where We Started"; close\_flow — a VectorField \[blue\] drawn in close\_plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12) Actions: - [14:23.96](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=863.9598731060606): close\_div is shown on the screen, written out. - [14:28.534](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=868.5338731060606): close\_curl is shown on the screen, written out. - [14:31.866](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=871.8658731060606): point is shown on the screen, grown. - [14:33.661](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=873.6608832030554): point is hidden from the screen. ##### [14:34.95](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=874.9503731060606) Narration: So: two derivatives of a vector field. Divergence, a number, saying how much is being created at a point. Curl, a vector, saying how the fluid turns there. Between them, they are the language the rest of vector calculus is written in. Board: close\_law — a Math \[text\] that says "$arrow(F) = (x - y, thin x + y)$"; close\_div — a Math \[text\] that says "$op("div") arrow(F) = 1 + 1 = 2$"; close\_curl — a Math \[text\] that says "$op("curl") arrow(F) = 1 - (- 1) = 2$"; close\_plane — an Axes (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), aspect=(1, 1)); head\_close — a Heading that says "Back Where We Started"; close\_flow — a VectorField \[blue\] drawn in close\_plane (function=\, at=((-1.8, -1.8), (-1.8, -1.1), (-1.8, -0.4), (-1.8, 0.4), (-1.8, …, scale=0.12) Actions: - [14:36.669](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=876.6688731060606): close\_curl is indicated — a transient flash. - [14:39.257](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=879.2568731060605): close\_div is indicated — a transient flash. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): close\_curl is hidden from the screen — left the board. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): close\_div is hidden from the screen — left the board. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): close\_law is hidden from the screen — left the board. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): close\_plane is hidden from the screen — left the board. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): close\_flow is hidden from the screen — close\_plane left the board. - [14:51.333](https://academa.ai/@sina/lectures/understanding-divergence-and-curl-in-vector-calculus?t=891.3327689393939): head\_close is hidden from the screen — left the board.