The idea of divergence and curl

Vector fields

We can think of a vector-valued function F : R² $\rightarrow$ R² as representing fluid flow in two dimensions, so that F(x, y) gives the velocity of a fluid at the point (x, y). In this case, we may call F(x, y) the velocity field of the fluid. More generally, we refer to a function like F(x, y) as a two-dimensional vector field. You can read more about how we can visualize the fluid flow by plotting the velocity F(x, y) as vector positioned at the point (x, y).

We can do the same thing for a three-dimensional fluid flow with velocity represented by a function F : R³$\to$R³. In this case, F(x, y, z) is the velocity of the fluid at the point (x, y, z), and we can visualize it as the vector F(x, y, z) positioned a the point (x, y, z). We refer to F(x, y, z) as a three-dimensional vector field.

Divergence

The divergence of a vector field is relatively easy to understand intuitively. Imagine that the vector field F below gives the velocity of some fluid flow. It appears that the fluid is exploding outward from the origin.

This expansion of fluid flowing with velocity field F is captured by the divergence of F, which we denote div F. The divergence of the above vector field is positive since the flow is expanding.

In contrast, the below vector field represents fluid flowing so that it compresses as it moves toward the origin. Since this compression of fluid is the opposite of expansion, the divergence of this vector field is negative.

The divergence is defined for both two-dimensional vector fields F(x, y) and three-dimensional vector fields F(x, y, z). A three-dimensional vector field F showing expansion of fluid flow is shown below. Again, because of the expansion, we can conclude that div F > 0.

Now, imagine that one placed a sphere S centered at the origin. It is clear that the fluid is flowing out of the sphere.

Later, when we introduce the divergence theorem, we will show that the divergence of a vector field and the flow out of spheres are closely related. For now, it's enough to see that if a fluid is expanding (i.e., the flow has positive divergence everywhere inside the sphere), the net flow out of a sphere will be positive.

Since the above vector field has positive divergence everywhere, the flow out of the sphere will be positive even if we move the sphere away from the origin. Can you see why flow out is still positive even when you move the sphere around using the sliders?

(Notice that the arrows continue to get longer as one moves away from the origin. Moreover, since the arrows are radiating outward, the fluid is always entering the sphere over less than half its surface and is exiting the sphere over greater than half its surface. Hence, the flow out of the sphere is always greater than the flow into the sphere.)

One last observation about the divergence: the divergence is a scalar. At a given point, the divergence of a vector field is just a single number that represents how much the flow is expanding at that point.

Care to read about some subtleties about the divergence or an example of calculating the divergence?

The curl

The curl of a vector field is slightly more complicated than the divergence. It captures the idea of how a fluid may rotate. Imagine that the below vector field F represents fluid flow. It appears that fluid is circulating around a bit. From the figure's original perspective (i.e., before you rotate the graph with your mouse), the fluid appears to circulate in a counter clockwise fashion. (If you rotate the graph, you might see dots floating along the axis of rotation. These dots are representations of vectors of zero length, as the velocity is zero there.)

This macroscopic circulation of fluid around circles (i.e., the rotation you can easily view in the above graph) isn't exactly what curl measures. But, it turns out that this vector field also has curl, which we might think of as "microscopic circulation." To test for curl, imagine that you immerse a small sphere into the fluid flow, and you fix the center of the sphere at some point so that the sphere cannot follow the fluid around. Although you fix the center of the sphere, you allow the sphere to rotate in any direction around its center point. The rotation of such a sphere is illustrated below. To see the rotation of the sphere, you must hold your mouse cursor over the figure. (If you double-click, the animation will stop; double-click again to restart the animation.) The rotation of the sphere measures the curl of the vector field F at the point in the center of the sphere. (The sphere should actually be really really small, because, remember, the curl is microscopic circulation.)

The vector field F determines both in what direction the sphere rotates, and the speed at which it rotates. We define the curl of F, denoted curl F, by a vector that points along the axis of the rotation and whose length corresponds to the speed of the rotation. As the curl a vector, it is very different from the divergence, which is a scalar.

We can draw the vector corresponding to curl F as follows. As mentioned above, the length of the vector curl F is determined by how fast the sphere is rotating. The direction of curl F points along the axis of rotation, but we need to specify in which direction along this axis the vector should point. We will (arbitrarily?) set the direction of the curl vector by using the right hand rule, as follows. To see where curl F should point, curl the fingers of your right hand in the direction the sphere is rotating; your thumb will point in the direction of curl F. For our example, curl F is shown by the green arrow. (You can rotate the graph to see the green arrow better.)

For this particular vector field, it turns out that the curl F doesn't change with position (this, of course, is not true in general). For example, if we move the sphere to another location, it will still spin in the same direction with the same speed. Can you see why the sphere spins the same way when the sphere is in the location shown below?

(Notice that the arrows continue to get longer as one moves away from the axis around which the fluid is rotating. For this reason, the fluid flow pushes the sphere more strongly on the side away from this axis, causing the sphere to spin in the same direction and speed as before. The general rotation of the flow also contributes to the sphere's spinning, as it causes the fluid to push against the sphere for a greater distance on the side away from the fluid's axis of rotation.)

You can read more about how one can determine the components of the vector curl F. You can also see an example of calculating the divergence and curl of a vector field. As usual, pictures can be deceiving; so if you want to make sure you really understand curl, check out some subtleties about the curl.

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