{"version":1,"lectureId":"01M1MBSTQ7TYKCGE8MSACEFKRK","attempt":0,"publication":{"slug":"understanding-divergence-and-curl-in-vector-calculus","title":"Understanding Divergence and Curl in Vector Calculus","subject":"mathematics","summary":"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.","metaDescription":"Divergence and curl from fluid flow: flux out of a small box, circulation round a small loop, the formulas, and a worked example.","transcript":"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. 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. 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. 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. 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.","watch":{"version":1,"scenes":[{"title":"A Field of Arrows","start":0,"end":106.24533143939395,"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=<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=<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=<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=<function>, t_range=(0.0, <VariableNumber t_draw = 1.7>))","t_draw":"a VariableNumber (initial_value=0.05)"},"beats":[{"start":0,"say":"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.","live":[],"does":[[0,"head_field is shown on the screen, written out."],[3.576,"plane is shown on the screen, written out."],[7.001,"flow is shown on the screen, written out."]]},{"start":8.379,"say":"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.","live":["plane","head_field","flow"],"does":[[14.892,"point is shown on the screen, grown."],[16.960343919044956,"point is hidden from the screen."],[17.098,"point_2 is shown on the screen, grown."],[18.929746129453463,"point_2 is hidden from the screen."]]},{"start":20.6235,"say":"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.","live":null,"does":[[27.240999999999996,"streamline is shown on the screen, written out."],[27.240999999999996,"speck is shown on the screen, written out."]]},{"start":31.567999999999998,"say":"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.","live":["plane","head_field","flow","streamline","speck"],"does":[[32.346,"streamline is redrawn as the numbers it depends on change."],[32.346,"speck is redrawn as the numbers it depends on change."],[32.346,"t_draw ticks to 1.7."]]},{"start":42.559,"say":"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?","live":null,"does":[[44.277,"streamline is hidden from the screen."],[44.277,"speck is hidden from the screen."],[49.42,"point_3 is shown on the screen, grown."],[51.66707973705034,"point_3 is hidden from the screen."],[52.4855,"head_field is hidden from the screen — left the board."],[52.4855,"plane is hidden from the screen — left the board."],[52.4855,"flow is hidden from the screen — plane left the board."]]},{"start":53.6855,"say":"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.","live":[],"does":[[53.6855,"head_two is shown on the screen, written out."],[62.196000000000005,"source_plane is shown on the screen, written out."],[62.196000000000005,"source_flow is shown on the screen, written out."]]},{"start":65.791,"say":"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.","live":["source_plane","head_two","source_flow"],"does":[[69.75,"point_4 is shown on the screen, grown."],[72.09263075750282,"point_4 is hidden from the screen."],[75.05600000000001,"source_plane moves to a new place on the board."],[75.05600000000001,"spin_plane is shown on the screen, written out."],[75.05600000000001,"spin_flow is shown on the screen, written out."]]},{"start":79.4635,"say":"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.","live":["source_plane","spin_plane","head_two","source_flow","spin_flow"],"does":[[82.72600000000001,"div_label is shown on the screen, written out."],[88.194,"curl_label is shown on the screen, written out."]]},{"start":89.4795,"say":"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.","live":["div_label","source_plane","curl_label","spin_plane","head_two","source_flow","spin_flow"],"does":[[100.57800000000002,"point_5 is shown on the screen, grown."],[101.90200000000002,"point_6 is shown on the screen, grown."],[103.17523402303414,"point_5 is hidden from the screen."],[104.83693750000002,"curl_label is hidden from the screen — left the board."],[104.83693750000002,"div_label is hidden from the screen — left the board."],[104.83693750000002,"head_two is hidden from the screen — left the board."],[104.83693750000002,"source_plane is hidden from the screen — left the board."],[104.83693750000002,"source_flow is hidden from the screen — source_plane left the board."],[104.83693750000002,"spin_plane is hidden from the screen — left the board."],[104.83693750000002,"spin_flow is hidden from the screen — spin_plane left the board."],[105.20366477272727,"point_6 is hidden from the screen."]]}]},{"title":"Divergence: What Is Being Created Here","start":106.24533143939395,"end":296.44393560606056,"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=<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=<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=<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=<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":[{"start":106.24533143939395,"say":"Here is the question that defines the first of the two measurements. 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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.","live":["flux_plane","box","flux_flow","corner","dx_brace","dy_brace"],"does":[[131.42033143939395,"head_box is shown on the screen, written out."],[136.16833143939394,"The segment (1.6, 0.3) to (1.6, 1.5) in flux_plane is lit up."],[140.75533143939396,"edges is shown on the screen, written out."],[142.98383143939395,"flux_plane: retire a lit segment (unemphasize_line)."]]},{"start":143.58383143939395,"say":"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.","live":["flux_plane","box","flux_flow","corner","dx_brace","dy_brace","head_box"],"does":[[144.42033143939395,"The segment (0.4, 0.3) to (0.4, 1.5) in flux_plane is lit up."],[147.16033143939396,"edges is shown on the screen, written out."],[149.34233143939394,"edges is emphasized."],[151.33933143939396,"flux_plane: retire a lit segment (unemphasize_line)."],[153.77733143939395,"edges is no longer emphasized."],[153.77733143939395,"edges is emphasized."],[156.21533143939394,"edges is no longer emphasized."]]},{"start":156.81533143939396,"say":"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.","live":null,"does":[[168.42533143939394,"flux_sum is shown on the screen, written out."]]},{"start":172.26383143939393,"say":"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.","live":null,"does":[[177.51133143939393,"edges is shown on the screen, written out."],[177.51133143939393,"The segment (0.4, 1.5) to (1.6, 1.5) in flux_plane is lit up."],[178.78833143939391,"edges is shown on the screen, written out."],[178.78833143939391,"The segment (0.4, 0.3) to (1.6, 0.3) in flux_plane is lit up."],[180.12433143939393,"flux_sum is shown on the screen, written out."],[185.59183143939396,"flux_plane: retire a lit segment (unemphasize_line)."],[185.59183143939396,"flux_plane: retire a lit segment (unemphasize_line)."]]},{"start":186.19183143939392,"say":"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.","live":null,"does":[[189.07133143939393,"flux_sum is shown on the screen, drawn."],[189.95623668474076,"flux_sum is shown on the screen, drawn."],[193.30833143939392,"flux_sum is shown on the screen, written out."],[195.86283143939391,"edges is hidden from the screen — left the board."],[195.86283143939391,"flux_plane is hidden from the screen — left the board."],[195.86283143939391,"box is hidden from the screen — flux_plane left the board."],[195.86283143939391,"flux_flow is hidden from the screen — flux_plane left the board."],[195.86283143939391,"corner is hidden from the screen — flux_plane left the board."],[195.86283143939391,"dx_brace is hidden from the screen — flux_plane left the board."],[195.86283143939391,"dy_brace is hidden from the screen — flux_plane left the board."],[195.86283143939391,"flux_sum is hidden from the screen — left the board."],[195.86283143939391,"head_box is hidden from the screen — left the board."]]},{"start":196.46283143939394,"say":"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.","live":[],"does":[[196.46283143939394,"head_formula is shown on the screen, written out."],[204.07833143939393,"divergence is shown on the screen, written out."]]},{"start":207.76683143939394,"say":"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.","live":["divergence","head_formula"],"does":[[211.3893314393939,"divergence (the \"frac(partial P, partial x)\" part) is emphasized."],[215.22033143939393,"divergence (the \"frac(partial P, partial x)\" part) is no longer emphasized."],[215.22033143939393,"divergence (the \"frac(partial Q, partial y)\" part) is emphasized."],[218.05333143939393,"divergence (the \"frac(partial Q, partial y)\" part) is no longer emphasized."]]},{"start":223.52933143939396,"say":"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.","live":null,"does":[[224.17933143939393,"div_note is shown on the screen, written out."],[231.48233143939393,"A box is drawn around divergence."],[236.95083143939394,"div_note is hidden from the screen — left the board."],[236.95083143939394,"divergence is hidden from the screen — left the board."],[236.95083143939394,"head_formula is hidden from the screen — left the board."]]},{"start":237.55083143939396,"say":"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.","live":[],"does":[[237.55083143939396,"head_cases is shown on the screen, written out."],[240.01133143939393,"axes is shown on the screen, written out."],[240.01133143939393,"case1_field is shown on the screen, written out."],[242.49633143939394,"axes moves to a new place on the board."],[242.49633143939394,"case1_law is shown on the screen, written out."],[251.06433143939392,"case1_div is shown on the screen, written out."]]},{"start":253.98583143939396,"say":"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.","live":["axes","case1_law","case1_div","head_cases","case1_field"],"does":[[254.3343314393939,"axes_2 is shown on the screen, written out."],[254.3343314393939,"case2_field is shown on the screen, written out."],[258.10733143939393,"case2_law is shown on the screen, written out."],[264.62133143939394,"case2_div is shown on the screen, written out."]]},{"start":266.9393314393939,"say":"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.","live":["axes","case1_law","case1_div","axes_2","case2_law","case2_div","head_cases","case1_field","case2_field"],"does":[[267.4733314393939,"axes_3 is shown on the screen, written out."],[267.4733314393939,"case3_field is shown on the screen, written out."],[268.51833143939393,"case3_law is shown on the screen, written out."],[279.26933143939397,"case3_div is shown on the screen, written out."]]},{"start":282.69033143939396,"say":"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.","live":["axes","case1_law","case1_div","axes_2","case2_law","case2_div","axes_3","case3_law","case3_div","head_cases","case1_field","case2_field","case3_field"],"does":[[287.77533143939394,"point is shown on the screen, grown."],[290.28128593608625,"point is hidden from the screen."],[291.61833143939396,"case3_div is indicated — a transient flash."],[295.40226893939393,"axes is hidden from the screen — left the board."],[295.40226893939393,"case1_field is hidden from the screen — axes left the board."],[295.40226893939393,"axes_2 is hidden from the screen — left the board."],[295.40226893939393,"case2_field is hidden from the screen — axes_2 left the board."],[295.40226893939393,"axes_3 is hidden from the screen — left the board."],[295.40226893939393,"case3_field is hidden from the screen — axes_3 left the board."],[295.40226893939393,"case1_div is hidden from the screen — left the board."],[295.40226893939393,"case1_law is hidden from the screen — left the board."],[295.40226893939393,"case2_div is hidden from the screen — left the board."],[295.40226893939393,"case2_law is hidden from the screen — left the board."],[295.40226893939393,"case3_div is hidden from the screen — left the board."],[295.40226893939393,"case3_law is hidden from the screen — left the board."],[295.40226893939393,"head_cases is hidden from the screen — left the board."]]}]},{"title":"Curl: What Is Turning Here","start":296.44393560606056,"end":515.7143731060605,"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=<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=<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=<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=<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":[{"start":296.44393560606056,"say":"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.","live":[],"does":[[296.44393560606056,"question is shown on the screen, written out."],[298.99793560606054,"loop_plane is shown on the screen, written out."],[298.99793560606054,"loop_flow is shown on the screen, written out."],[301.13393560606056,"loop is shown on the screen, written out."],[303.1549356060606,"walk is shown on the screen, written out."]]},{"start":306.61043560606055,"say":"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.","live":["loop_plane","question","loop_flow","loop","walk"],"does":[[307.9459356060606,"loop_plane moves to a new place on the board."],[307.9459356060606,"comps is shown on the screen, written out."],[316.22293560606056,"point is shown on the screen, grown."],[317.4909912388839,"point is hidden from the screen."],[318.0344356060606,"loop_plane moves to a new place on the board."],[318.0344356060606,"comps is hidden from the screen — left the board."],[318.0344356060606,"question is hidden from the screen — left the board."]]},{"start":318.63443560606055,"say":"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.","live":["loop_plane","loop_flow","loop","walk"],"does":[[318.63443560606055,"head_loop is shown on the screen, written out."],[319.3539356060606,"The segment (0.4, 0.3) to (1.6, 0.3) in loop_plane is lit up."],[326.9119356060606,"sides is shown on the screen, written out."],[329.3039356060606,"loop_plane: retire a lit segment (unemphasize_line)."]]},{"start":329.90393560606054,"say":"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.","live":["loop_plane","loop_flow","loop","walk","head_loop"],"does":[[330.75193560606056,"The segment (0.4, 1.5) to (1.6, 1.5) in loop_plane is lit up."],[336.45193560606054,"sides is shown on the screen, written out."],[340.02843560606055,"loop_plane: retire a lit segment (unemphasize_line)."]]},{"start":340.6284356060606,"say":"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.","live":null,"does":[[347.8609356060606,"circ_sum is shown on the screen, written out."]]},{"start":354.3129356060606,"say":"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.","live":null,"does":[[358.6669356060606,"sides is shown on the screen, written out."],[358.6669356060606,"The segment (1.6, 0.3) to (1.6, 1.5) in loop_plane is lit up."],[361.41793560606055,"sides is shown on the screen, written out."],[361.41793560606055,"The segment (0.4, 0.3) to (0.4, 1.5) in loop_plane is lit up."],[364.9239356060606,"circ_sum is shown on the screen, written out."],[370.25293560606053,"loop_plane: retire a lit segment (unemphasize_line)."],[370.25293560606053,"loop_plane: retire a lit segment (unemphasize_line)."]]},{"start":370.85293560606056,"say":"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.","live":null,"does":[[371.32893560606055,"circ_sum is shown on the screen, drawn."],[372.3651144002393,"circ_sum is shown on the screen, drawn."],[375.7869356060605,"circ_sum is shown on the screen, written out."],[381.11593560606053,"circ_sum is indicated — a transient flash."],[383.0204356060606,"circ_sum is hidden from the screen — left the board."],[383.0204356060606,"head_loop is hidden from the screen — left the board."],[383.0204356060606,"loop_plane is hidden from the screen — left the board."],[383.0204356060606,"loop_flow is hidden from the screen — loop_plane left the board."],[383.0204356060606,"loop is hidden from the screen — loop_plane left the board."],[383.0204356060606,"walk is hidden from the screen — loop_plane left the board."],[383.0204356060606,"sides is hidden from the screen — left the board."]]},{"start":383.62043560606054,"say":"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.","live":[],"does":[[383.62043560606054,"head_formula is shown on the screen, written out."],[388.07893560606055,"curl2d is shown on the screen, written out."]]},{"start":394.44843560606057,"say":"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.","live":["curl2d","head_formula"],"does":[[397.97793560606056,"curl2d (the \"frac(partial Q, partial x)\" part) is emphasized."],[401.4719356060605,"curl2d (the \"frac(partial P, partial y)\" part) is emphasized."],[401.4719356060605,"curl2d (the \"frac(partial Q, partial x)\" part) is no longer emphasized."],[405.5129356060605,"curl2d (the \"frac(partial P, partial y)\" part) is no longer emphasized."]]},{"start":407.0299356060606,"say":"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.","live":null,"does":[[409.20093560606057,"curl_note is shown on the screen, written out."],[416.92093560606054,"A box is drawn around curl2d."],[420.5899356060606,"curl2d is hidden from the screen — left the board."],[420.5899356060606,"curl_note is hidden from the screen — left the board."],[420.5899356060606,"head_formula is hidden from the screen — left the board."]]},{"start":421.18993560606054,"say":"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.","live":[],"does":[[421.18993560606054,"head_cases is shown on the screen, written out."],[422.09593560606055,"axes is shown on the screen, written out."],[422.09593560606055,"spin_field is shown on the screen, written out."],[425.2889356060606,"axes moves to a new place on the board."],[425.2889356060606,"spin_law is shown on the screen, written out."],[432.97393560606054,"spin_curl is shown on the screen, written out."]]},{"start":436.18043560606054,"say":"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.","live":["spin_law","spin_curl","axes","head_cases","spin_field"],"does":[[436.85993560606056,"axes_2 is shown on the screen, written out."],[436.85993560606056,"out_field is shown on the screen, written out."],[438.25293560606053,"out_law is shown on the screen, written out."],[445.97393560606054,"out_curl is shown on the screen, written out."]]},{"start":451.1014356060606,"say":"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.","live":["spin_law","spin_curl","axes","out_law","out_curl","axes_2","head_cases","spin_field","out_field"],"does":[[454.45693560606054,"point_2 is shown on the screen, grown."],[456.59393560606054,"point_3 is shown on the screen, grown."],[456.85084734136166,"point_2 is hidden from the screen."],[459.56187942895986,"point_3 is hidden from the screen."],[461.4699356060606,"axes is hidden from the screen — left the board."],[461.4699356060606,"spin_field is hidden from the screen — axes left the board."],[461.4699356060606,"axes_2 is hidden from the screen — left the board."],[461.4699356060606,"out_field is hidden from the screen — axes_2 left the board."],[461.4699356060606,"head_cases is hidden from the screen — left the board."],[461.4699356060606,"out_curl is hidden from the screen — left the board."],[461.4699356060606,"out_law is hidden from the screen — left the board."],[461.4699356060606,"spin_curl is hidden from the screen — left the board."],[461.4699356060606,"spin_law is hidden from the screen — left the board."]]},{"start":462.06993560606054,"say":"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.","live":[],"does":[[462.06993560606054,"head_shear is shown on the screen, written out."],[462.6389356060606,"shear_plane is shown on the screen, written out."],[462.6389356060606,"shear_field is shown on the screen, written out."],[467.95593560606056,"shear_plane moves to a new place on the board."],[467.95593560606056,"shear_law is shown on the screen, written out."],[470.3939356060606,"point_4 is shown on the screen, grown."],[471.61293560606055,"point_5 is shown on the screen, grown."],[472.2863918460489,"point_4 is hidden from the screen."]]},{"start":473.8269356060606,"say":"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.","live":["shear_law","shear_plane","head_shear","shear_field","point_5"],"does":[[474.52264550993635,"point_5 is hidden from the screen."],[474.87193560606056,"hub is shown on the screen, written out."],[476.69493560606054,"top_blade is shown on the screen, written out."],[478.9699356060606,"bottom_blade is shown on the screen, written out."],[484.07893560606055,"wheel_turn is shown on the screen, written out."]]},{"start":485.80493560606055,"say":"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.","live":["shear_law","shear_plane","head_shear","shear_field","hub","top_blade","bottom_blade","wheel_turn"],"does":[[498.0649356060606,"shear_curl is shown on the screen, written out."]]},{"start":501.3584356060606,"say":"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.","live":["shear_law","shear_curl","shear_plane","head_shear","shear_field","hub","top_blade","bottom_blade","wheel_turn"],"does":[[502.31093560606064,"A box is drawn around shear_curl."],[514.672706439394,"head_shear is hidden from the screen — left the board."],[514.672706439394,"shear_curl is hidden from the screen — left the board."],[514.672706439394,"shear_law is hidden from the screen — left the board."],[514.672706439394,"shear_plane is hidden from the screen — left the board."],[514.672706439394,"shear_field is hidden from the screen — shear_plane left the board."],[514.672706439394,"hub is hidden from the screen — shear_plane left the board."],[514.672706439394,"top_blade is hidden from the screen — shear_plane left the board."],[514.672706439394,"bottom_blade is hidden from the screen — shear_plane left the board."],[514.672706439394,"wheel_turn is hidden from the screen — shear_plane left the board."]]}]},{"title":"Into Three Dimensions","start":515.7143731060605,"end":694.1578731060606,"objects":{"comps3":"a Math [text] that says \"$arrow(F) = (P, thin Q, thin R)$\"","cube":"a Solid [yellow] drawn in frame (upper=<function>, lower=<function>, 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=<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":[{"start":515.7143731060605,"say":"Everything so far has been flat. Real fluids move in three dimensions, so let's take both ideas up a dimension.","live":[],"does":[[515.7143731060605,"head_space is shown on the screen, written out."],[519.9633731060605,"frame is shown on the screen, written out."]]},{"start":524.0003731060605,"say":"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.","live":["frame","head_space"],"does":[[524.7663731060605,"field3 is shown on the screen, written out."],[526.9493731060605,"frame turns in its own slot."],[535.3428731060606,"frame moves to a new place on the board."],[535.3428731060606,"head_space is hidden from the screen — left the board."]]},{"start":535.9428731060605,"say":"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.","live":["frame","field3"],"does":[[535.9428731060605,"head_div3 is shown on the screen, written out."],[541.0513731060605,"cube is shown on the screen, written out."]]},{"start":550.4403731060605,"say":"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.","live":["frame","field3","head_div3","cube"],"does":[[551.0903731060605,"comps3 is shown on the screen, written out."],[553.5283731060605,"div3 is shown on the screen, written out."]]},{"start":564.9028731060605,"say":"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.","live":["frame","field3","comps3","div3","head_div3","cube"],"does":[[567.0733731060606,"div_check is shown on the screen, written out."],[571.3223731060605,"cube is hidden from the screen."],[576.1873731060605,"comps3 is hidden from the screen — left the board."],[576.1873731060605,"div3 is hidden from the screen — left the board."],[576.1873731060605,"div_check is hidden from the screen — left the board."],[576.1873731060605,"head_div3 is hidden from the screen — left the board."]]},{"start":576.7873731060605,"say":"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.","live":["frame","field3"],"does":[[576.7873731060605,"head_curl3 is shown on the screen, written out."],[585.2743731060605,"loop_xy is shown on the screen, written out."]]},{"start":587.5808731060605,"say":"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.","live":["frame","field3","head_curl3","loop_xy"],"does":[[591.4353731060605,"loop_yz is shown on the screen, written out."],[592.6893731060605,"loop_zx is shown on the screen, written out."],[595.5453731060605,"loop_xy is hidden from the screen."],[597.5763731060605,"loop_yz is hidden from the screen."],[597.9975510179348,"loop_zx is hidden from the screen."]]},{"start":601.6948731060605,"say":"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.","live":["frame","field3","head_curl3"],"does":[[604.9103731060605,"curl3 is shown on the screen, written out."],[608.5093731060605,"curl3 is shown on the screen, written out."],[612.2713731060605,"curl3 is shown on the screen, written out."]]},{"start":615.4838731060605,"say":"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.","live":null,"does":[[616.5053731060605,"curl3 is emphasized."],[625.5493731060606,"curl3 is no longer emphasized."]]},{"start":628.1228731060605,"say":"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.","live":null,"does":[[631.6873731060605,"curl_check is shown on the screen, written out."],[634.6823731060605,"curl_vec is shown on the screen, written out."]]},{"start":639.2068731060605,"say":"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.","live":["frame","field3","curl_check","head_curl3","curl_vec"],"does":[[646.6833731060606,"curl_vec is indicated — a transient flash."],[654.1258731060605,"curl3 is hidden from the screen — left the board."],[654.1258731060605,"curl_check is hidden from the screen — left the board."],[654.1258731060605,"frame is hidden from the screen — left the board."],[654.1258731060605,"field3 is hidden from the screen — frame left the board."],[654.1258731060605,"curl_vec is hidden from the screen — frame left the board."],[654.1258731060605,"head_curl3 is hidden from the screen — left the board."]]},{"start":654.7258731060605,"say":"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.","live":[],"does":[[654.7258731060605,"head_del is shown on the screen, written out."],[658.7193731060605,"nabla_def is shown on the screen, written out."]]},{"start":666.9588731060605,"say":"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.","live":["nabla_def","head_del"],"does":[[667.3653731060606,"div_form is shown on the screen, written out."],[673.9593731060605,"curl_form is shown on the screen, written out."]]},{"start":680.7013731060605,"say":"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.","live":["nabla_def","div_form","curl_form","head_del"],"does":[[689.7803731060606,"div_form is indicated — a transient flash."],[692.1373731060605,"curl_form is indicated — a transient flash."],[693.1162064393939,"curl_form is hidden from the screen — left the board."],[693.1162064393939,"div_form is hidden from the screen — left the board."],[693.1162064393939,"head_del is hidden from the screen — left the board."],[693.1162064393939,"nabla_def is hidden from the screen — left the board."]]}]},{"title":"Putting Them Together","start":694.1578731060606,"end":892.3744356060606,"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=<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":[{"start":694.1578731060606,"say":"Let's put the two side by side, because they are easy to confuse and they answer genuinely different questions.","live":[],"does":[[694.1578731060606,"head_cmp is shown on the screen, written out."],[695.0518731060606,"compare is shown on the screen, written out."]]},{"start":701.0853731060606,"say":"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.","live":["head_cmp"],"does":[[702.6178731060606,"compare is shown on the screen, written out."],[708.0048731060606,"compare is indicated — a transient flash."]]},{"start":713.1323731060606,"say":"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.","live":null,"does":[[713.6898731060606,"compare is shown on the screen, written out."],[720.6558731060607,"compare is shown on the screen, written out."]]},{"start":724.1928731060606,"say":"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.","live":null,"does":[[725.4238731060606,"compare is shown on the screen, written out."],[728.8248731060606,"compare is indicated — a transient flash."]]},{"start":735.1953731060606,"say":"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.","live":null,"does":[[737.2388731060606,"compare is shown on the screen, written out."],[748.6398731060606,"compare is indicated — a transient flash."],[749.5103731060606,"compare is hidden from the screen — left the board."],[749.5103731060606,"head_cmp is hidden from the screen — left the board."]]},{"start":750.1103731060606,"say":"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.","live":[],"does":[[751.2828731060606,"problem is shown on the screen, written out."]]},{"start":762.6923731060606,"say":"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.","live":["problem"],"does":[[763.0408731060606,"div_work is shown on the screen, written out."]]},{"start":777.8513731060606,"say":"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.","live":null,"does":[[786.6628731060606,"div_work is shown on the screen, written out."],[787.5678731060606,"div_answer is shown on the screen, written out."]]},{"start":791.3728731060606,"say":"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.","live":["div_answer","problem"],"does":[[792.6148731060606,"A box is drawn around div_answer."],[802.9823731060605,"div_answer is hidden from the screen — left the board."],[802.9823731060605,"div_work is hidden from the screen — left the board."]]},{"start":803.5823731060606,"say":"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.","live":["problem"],"does":[[815.4358731060606,"curl_work is shown on the screen, written out."]]},{"start":817.2088731060607,"say":"The y component pairs the z and x derivatives in the same pattern, and it comes out as minus z.","live":null,"does":[[821.9228731060606,"curl_work is shown on the screen, written out."]]},{"start":824.4263731060606,"say":"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.","live":null,"does":[[833.7258731060606,"curl_work is shown on the screen, written out."]]},{"start":836.5203731060606,"say":"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.","live":null,"does":[[837.9248731060607,"curl_answer is shown on the screen, written out."],[845.1698731060607,"A box is drawn around curl_answer."],[850.0118731060606,"curl_answer is hidden from the screen — left the board."],[850.0118731060606,"curl_work is hidden from the screen — left the board."],[850.0118731060606,"problem is hidden from the screen — left the board."]]},{"start":850.6118731060606,"say":"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.","live":[],"does":[[850.6118731060606,"head_close is shown on the screen, written out."],[854.1528731060606,"close_plane is shown on the screen, written out."],[854.1528731060606,"close_flow is shown on the screen, written out."],[856.6948731060606,"close_plane moves to a new place on the board."],[856.6948731060606,"close_law is shown on the screen, written out."]]},{"start":860.1398731060606,"say":"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.","live":["close_law","close_plane","head_close","close_flow"],"does":[[863.9598731060606,"close_div is shown on the screen, written out."],[868.5338731060606,"close_curl is shown on the screen, written out."],[871.8658731060606,"point is shown on the screen, grown."],[873.6608832030554,"point is hidden from the screen."]]},{"start":874.9503731060606,"say":"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.","live":["close_law","close_div","close_curl","close_plane","head_close","close_flow"],"does":[[876.6688731060606,"close_curl is indicated — a transient flash."],[879.2568731060605,"close_div is indicated — a transient flash."],[891.3327689393939,"close_curl is hidden from the screen — left the board."],[891.3327689393939,"close_div is hidden from the screen — left the board."],[891.3327689393939,"close_law is hidden from the screen — left the board."],[891.3327689393939,"close_plane is hidden from the screen — left the board."],[891.3327689393939,"close_flow is hidden from the screen — close_plane left the board."],[891.3327689393939,"head_close is hidden from the screen — left the board."]]}]}]},"durationSeconds":892,"chapters":[{"title":"A Field of Arrows","startSeconds":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. 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."},{"title":"Divergence: What Is Being Created Here","startSeconds":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. 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."},{"title":"Curl: What Is Turning Here","startSeconds":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. 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."},{"title":"Into Three Dimensions","startSeconds":515.7143731060605,"narration":"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."},{"title":"Putting Them Together","startSeconds":694.1578731060606,"narration":"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."}]}}