Is there a three-dimensional version of curl in which the result is a vector?

Question

Curl is an operation from a vector field to a vector field in 3D. Is there an interpretation of Divergence that doesn't produce a scalar field but rather a vector field, such that at any point $|\text{curl}'| = \text{curl}$?

Answer 1

From a higher level viewpoint divergence, curl and gradient are all the same operations; the application of the exterior derivative on a differential form. Now the div,grad and curl that we all know and love come from applying the exterior derivative to different differential forms. Without going into depth I will just say the following.

1) A 'zero form' on say $\mathbb{R}^3$ is a smooth scalar valued function. When we apply the exterior derivative to such an object we get the gradient which turns out to be a $1$-form.

2)A '$1$-form' can be viewed as a smooth vector field (but there is a more precise way to understand this namely its a scalar valued function that eats vectors). Now when we hit a $1$-form with the exterior derivative we get a $2$-form. This will coincide with the curl.

3)Finally (remember we're in 3 dimensions so this process stops when we get to $3$-forms) when we are given a $2$-form and hit it with the exterior derivative we get a $3$-form. Now how do you view a $2$-form? It's a function that eats pairs of vectors and gives us scalars (of course we need an assumption about differentiability). The $3$-form that you get is what's known as a volume form on $\mathbb{R}^3$ and this is exactly the divergence.

This might not make too much sense and I don't really expect it to unless you are familiar with differential forms. The point is, to really understand div,grad and curl you need this machinery. I recommend the book 'A Geometric Approach to Differential Forms' if this interests you.