Understanding Divergence and Curl in Vector Calculus

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.