Divergence measures the rate field vectors are expanding at a point.

While the gradient and curl are the fundamental “derivatives” in two dimensions, there is another useful measurement we can make. It is called divergence. It measures the rate field vectors are “expanding” at a given point.

The divergence of a vector field

Let’s state the definition:

Is the divergence of a vector field a scalar or a vector?
vector. scalar. neither a vector nor a scalar.
Consider the vector field . Compute:
Consider the vector field . Compute:

What does divergence measure?

As we’ve already said, divergence measures the rate field vectors are expanding at a point. To be more explicit, the divergence measures how the magnitude of the field vectors change as you move in the direction of the field vectors:

And:

The most obvious example of a vector field with nonzero divergence is :

On the other hand, recall that a radial vector field is a field of the form where where is a real number. The divergence of these vector fields can be surprising.
Compute the divergence of .

Now we will see a radial vector field with zero divergence.

Now let’s see a radial vector field with negative divergence.

Measuring flow across a curve

Let be a vector field, be a smooth vector valued function tracing a curve exactly once as runs from to ,

Recall that the line integral measures the accumulated flow of a vector field along a curve. We see this because measures how “aligned” field vectors are with the direction of the path . On the other hand, if we set then for any given value of , is a vector that is orthogonal to . Moreover, given a closed curve, where is parameterized with the interior on the left, points outward. Below we see a curve along with some tangent vectors and some outward normal vectors :

Since measures how “aligned” field vectors are with vectors orthogonal to the direction of the path, the integral

measures the flow of a vector field across a curve. Some folks call this a flux integral. Since and , we may write as

Consider the following vector field and curve parameterized by : Do you expect to be positive, zero, or negative?
positive zero negative
Consider the following vector field and curve parameterized by : Do you expect to be positive, zero, or negative?
positive zero negative
Consider the following vector field and curve parameterized by : Do you expect to be positive, zero, or negative?
positive zero negative

With our next example, we’ll get our hands dirty.

Connections to Green’s Theorem

Finally, note that if , then:

We also see that

this leads us to the flux form of Green’s Theorem:

Let be a vector field with . Compute:
Suppose that the curl of a vector field is constant, . If estimate:
Use Green’s Theorem.