The curl of a vector field gives an indication of how a vector field tends to curve or rotate. We use the idea of the cross product to calculate the curl. Before we get started on the details of the curl, let's watch a video to get started.

To calculate the curl of a vector field \( \vec{F}(x,y,z) = F_i\vhat{i} + F_j\vhat{j} + F_k\vhat{k}\), we use the del operator, \(\displaystyle{ \vec{ \nabla } = \frac{\partial }{\partial x}\vhat{i} + \frac{\partial }{ \partial y}\vhat{j} + \frac{\partial }{ \partial z}\vhat{k} }\) and the cross product to give us
\(\displaystyle{ \vec{ \nabla } \times \vec{F} = }\) \(\displaystyle{ \left[ \frac{\partial }{\partial x}\vhat{i} + \right. }\) \(\displaystyle{ \frac{\partial }{ \partial y}\vhat{j} + }\) \(\displaystyle{ \left. \frac{\partial }{ \partial z}\vhat{k} \right] \times \left[ F_i\vhat{i} + F_j\vhat{j} + F_k\vhat{k} \right] = }\) \( \begin{vmatrix} \vhat{i} & \vhat{j} & \vhat{k} \\ \displaystyle{\frac{\partial }{\partial x}} & \displaystyle{\frac{\partial }{\partial y}} & \displaystyle{\frac{\partial }{\partial z}} \\ F_i & F_j & F_k \end{vmatrix} \)

Notes
1. You should know enough about linear algebra to be able to evaluate the determinant.
2. Although we call the curl a cross product and use the idea of the determinant, neither of these terms are exactly correct since del is an operator, not a true vector. However, for our purposes we follow the lead of most mathematicians and stretch the notation a bit to fit our requirements.
3. The curl is also written \( curl ~ \vec{F} \), i.e. \( curl ~ \vec{F} = \vec{ \nabla } \times \vec{F} \).
4. The curl of two vector fields is another vector field, which is perpendicular to both of the original vector fields.

Okay, time for a video. Here is a quick, no-frills introduction to calculating the curl of vector fields. It contains some detail when evaluating the determinant, which may be helpful to you if you need a reminder on how to do that.

This video clip explains the interpretation and properties of the curl, as well as the circulation density and the scalar curl. This is important to watch to get a full understanding of the curl.

This next video clip gives the big picture of the curl of a vector field. The list above is adapted from the notes of Dr Chris Tisdell. In this video clip, he discusses these items. [We highly recommend that you go to his website and download the notes for this topic.]

As you would expect, you can calculate the curl of a vector field in cylindrical coordinates. Here is a video explaining the equations and how to use them.

Calculus: Early Transcendental Functions 3rd Edition