30. Divergence and Curl

There are two derivatives for vector fields, the divergence and the curl. We compute these by taking the dot and cross product of our vector field with a "vector" consisting of derivatives: $\large \displaystyle \vec{\nabla}=\left<\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right>$.

Divergence

The divergence of $\large \vec{F} = \left<F_1,F_2,F_3\right>$ is:

$$\large \vec{\nabla}\cdot\vec{F} = \left< \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} \right> \cdot \left< F_1,F_2,F_3 \right> = \frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$$

Notice that the divergence of $\large \vec{F}$ is a scalar multivariable function.

Curl

The curl of $\large \vec{F} = \left&lt;F_1,F_2,F_3\right&gt;$ is:#

$$\large \begin{split} \vec{\nabla}\times\vec{F}&=\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1&F_2& F_3 \end{bmatrix}\\\ &=\hat{i}\left( \frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)-\hat{j}\left(\frac{\partial F_3}{\partial x}-\frac{\partial F_1}{\partial z}\right)+\hat{k}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\\\ &=\left< \frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z},\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x},\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right> \end{split}$$

Notice that the curl of $\large \vec{F}$ is itself a vector field.

Here's a video illustrating how to find the divergence and curl of a vector field (warning: there is a rather dumb mistake in the divergence calculation, as the derivative of $\large xyz$ with respect to $\large y$ is $\large xz$ and not $\large xy$ as stated in the video):

Significance of the Divergence and Curl

These definitions seem a bit out of the blue, but these are combinations of derivatives that are very common and useful in various fields, particularly physics, fluid dynamics, and electromagnetism.

The divergence $\large \vec{\nabla}\cdot\vec{F}$ measures the rate of change of $\large \vec{F}$ flowing out of (or into) a given point. The curl $\large \vec{\nabla}\times\vec{F}$ measures the amount of rotation in the vector field (indeed, in fluid dynamics, the curl of the velocity field is known as the vorticity).

The following plots illustrate two extremes, a field with large divergence and no curl and a field with no divergence but large curl:

On the left: an irrotational field, $\large \vec{\nabla}\cdot\vec{F}$ is large $\large \vec{\nabla}\times\vec{F}=\vec{0}$. On the right: a divergence free field, $\large \vec{\nabla}\cdot\vec{F}=0$, $\large \left|\vec{\nabla}\times\vec{F}\right|$ is large

Now that we have divergence and curl, you may want to check out the list of vector calculus identities in the Wikipedia: https://en.wikipedia.org/wiki/Vector_calculus_identities. This includes various combinations of the gradient, divergence, and curl on functions.

Tomorrow's class:

• A quick look at some famous equations that involve divergence, curl, an gradient.
• Practice computing divergence and curl.

Section Outcomes:

I can:

• Compute the divergence and curl of a vector field
• Recognize regions where either divergence or curl is large

Other resources

For more on these integrals, see https://openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl