The previous post showed that if a vector field F over a simply connected domain has zero curl, then there is a potential function φ whose gradient is F.
An analogous result says that if the vector field F has zero divergence, again over a simply connected domain, then there is a vector potential Φ whose curl is F.
These are both special cases of Poincaré’s lemma.
This post will outline how to calculate Φ. First of all, Φ is far from unique. Any vector field with zero curl can be added to Φ without changing its curl. So if
Φ = (Φ1, Φ2, Φ3)
then we can assume that one of the components, say Φ3, is zero by adding the right curl-free component. If you find that argument less than convincing, look at it this way: we’re going to solve a harder problem than simply find Φ such that ∇×Φ = F by giving ourselves the additional requirement that the last component of Φ must be zero.
Now if Φ = (Φ1, Φ2, 0) and ∇×Φ = F, then
-∂z Φ2= F1
∂z Φ1 = F2
∂x Φ2 – ∂y Φ1 = F3
We can solve the first equation by integrating F1 with respect to z and adding a function of x and y to be determined later. We can solve the second equation similarly, then use the third equation to determine the functions of x and y left over from solving the first two equations.
***
This post is the third, and probably last, in a series of posts looking at vector calculus from a more advanced perspective. The first post in the series looked at applying {grad, curl, div} to {grad, curl, div}: seeing which combinations are defined, and which combinations are always 0. To illustrate the general pattern, the post dips into differential forms and the Hodge star operator.
The second post looked at finding the (scalar) potential function for a curl-free vector field, and mentions connections to topology and differential equations.
The present post is a little terse, but it makes more sense if you’ve gone through the previous post. The method of solution here is analogous to the method in the previous post, and that post goes into a little more detail.