Visualizing Divergence and Curl

Introduction

Two key concepts in vector calculus are divergence and curl, the latter of which is sometimes called circulation.   Basically, divergence has to do with how a vector field changes its magnitude in the neighborhood of a point, and curl has to do with how its direction changes.   Look at the graph of Field 1 below.  It has both zero divergence and zero curl, because it doesn’t change in either magnitude or direction in the neighborhood of any point.  For instance, if you look near the origin, you’ll see that all the little arrows have the same direction and the same length.  (Actually, since this is a constant field, it will be true near any point you pick, but in general fields can have different properties at different points.)

Field 1

Divergence

Now look at Field 2:

Field 2

At the origin, all arrows point radially outward.  Go out about half a unit along any radius, and you’ll see that the arrows are still pointing in the same direction, but they’re getting longer.  Since the length of an arrow in the plot is proportional to the strength of the field in the neighborhood of the arrow, we can see that as we go farther away from the origin along any radius, the direction of the field isn’t changing, but its strength is.  The graph in Field 2 above is actually a plot of the vector field F(x,y) = {x, y}.. For this vector field, the divergence of F, written div F(x,y), is equal to 2. Symbolically, div F = 2.

What if you’re not at the origin?  Well, then, you just pick some other point and look at the net change in the field at that point.  Pick (1,1) for instance.  What you’ll see there is that the arrows heading toward the point are shorter than those heading away from the point, so the change in the strength of the field is positive—in other words, the divergence of the field is positive.  You’ll see the same result at any point—it’s just easier to see at the origin where there are no arrows heading in, so that all you have to do is look at the outward-pointing ones.

Now, although Field 2 is changing in both the x and y directions, you can have one like Field 3 below, where the arrows are only changing in length in one direction/component and not the other:

Field 3

In this case, the y-component of the length of each arrow is always 1, but the x-component gets larger in magnitude as we get farther away from the origin.  And in fact, the field that produced this plot is   Its divergence is 1.

We can, of course, also have fields with negative divergence:

Field 4

The basic difference here is that the arrows are all pointing toward the origin instead of away from it. The divergence of the field in Field 4 is -2.  Or, if you pick some point other than the origin, notice that the lengths of the arrows heading into that point are longer than the lengths of the arrows heading away from the point.  Pick (1,1), for instance, to make sure you see this.

Of course, the divergence of a field doesn’t have to be constant, either.  The divergence of Field 5 for instance is 2x + 2y:

Field 5

And here’s one last field whose divergence you should consider:

Field 6

For Field 6 above:

1.      Decide whether the divergence is positive, negative, or zero in each of the four quadrants.  Explain your reasoning.

Curl

All the vector fields you’ve seen so far have had zero curl. It’s time to look at some that don’t.

Field 7

If you look at the origin, where the field seems to “circulating” around the point, you can see why curl used to be called circulation.  Notice that at any distance from the origin, say at 1 unit, the arrows have constant length but constantly changing direction.  Since their length is constant, the divergence of Field 7 is zero; since the direction is always changing, the curl is non-zero. To put it another way, the vectors all curl around a central point. (Clever nomenclature, eh?) The vector field of which Field 7 is a plot is actually G(x,y) = {-y, x}, and it has div G(x,y) = 0 and curl G(x,y) = 2.  

 

 A field with negative curl, btw, will simply curl in the opposite (clockwise) direction, like Field 8 below:

Field 8

 

You might think from these examples that all vector fields with a non-zero curl show this “curling around a point” behavior.  That would be a reasonable conclusion given the examples you’ve seen, but it’s actually incorrect.  Here’s an example where it looks like the vectors all fall on diagonal lines, with no perceptible curving, yet there is a non-zero curl:

 

Field 9

 

The way to think about curl then is a little more general than just vectors curling around a point.  Still, some idea of “circulation” must be involved since that’s what this phenomenon used to be called.  A good way to think about curl, it turns out, is to think of a vector field as representing a fluid that actually flows and then to imagine a little paddlewheel placed at a point in the flow.  If the paddlewheel will turn, then the field has a non-zero curl at that point.  In the case of Field 9, put a little paddlewheel at the origin.  Then the arrows on the left will push up on the paddles on the left and the arrows on the right will push down on the paddles on the right, which will produce a net imbalance of forces and hence a torque, and thus the paddle will spin.  Hence, the field has a nonzero curl at the origin.

You don’t even have to have direction changes anywhere for a field to have nonzero curl, though.  Let’s look at a field where the arrows are everywhere pointing in the same direction:

Field 10

 

There is still a non-zero curl in Field 10—in fact, a positive one.  You can see that by imagining your paddlewheel at (1,1).  The arrows on the right-hand side of the wheel are longer than those on the left-hand side, though, so there will be more force pushing the wheels on the right.  This imbalance of forces leads to a torque and thus to the wheel’s spinning counterclockwise and hence to a positive curl.  In fact, Field 10 is the plot of  and curl G = 1.

Now, technically, since curl represents a direction, it has to be represented by a vector, and it’s more correct to say that .  If you’re conversant with torque, you know that it acts in a direction perpendicular to the plane of the forces causing the spin, so it wouldn’t surprise you that curl G is in the z-direction for Field 10.

1.      Although curl is a vector quantity, div is not—it’s a scalar.  Go back to what you learned about divergence in the first part of this lab to explain why in fact you should expect it to be a scalar.

 

Now, let’s move on to two more complicated examples:

Here is a vector field with both divergence and curl positive:

Field 11

Field 11 is a plot of the vector field  {x - y, x + y}. Notice that it has both "counterclockwise curling" and "outward streaming" and hence both positive curl and positive divergence.

Finally, look at Field 12 below, whose description is {x – y2 + 2, xy + y}:

Field 12

Things here are quite complicated, but to simplify just a bit, look in the immediate neighborhood of the red point. What you see there are positive divergence and negative curl. Every one of the nearby vectors points outward, so there is no question about the divergence’s being positive. As for the curl, notice that all of the closest vectors are rotated clockwise from pointing straight out, so there is no question about the curl’s being negative either..

What makes this particular field more complex than the others is that both the divergence and the curl depend on the values of  x and y at the point of interest.  So, though the curl near the red point is negative, the curl in the neighborhood of the blue point is positive.  Without doing any calculations, you can see that the curl of this vector field depends on the sign of its y-coordinate at any point. 

Now, divergence and curl are measures of how a vector field changes, so we know they’re going to involve derivatives.  And that observation takes us to the next lab.