Improvements to differential/Jacobian calculations

[mikeingold]

I consolidated all of the previous differential functions into a single function that uses an if/else condition on the number of parametric dimensions

https://github.com/mikeingold/MeshIntegrals.jl/blob/5d92c86d07fa6af97d3df024cdf40fe58a50d16b/src/utils.jl#L62-L81

These operations (norm, norm of cross product, triple product) can be generalized as falling out of an exterior/wedge product using Geometric/Clifford Algebra. It would be nice to generalize this operation into something simple like norm(foldl(wedge, J)) that doesn't require a conditional.

It looks like DifferentialForms.jl could be used but would require some adaptation to translate between Forms and vectors.

[mikeingold]

Merging this with AD topic.

The above refers to generalizing the differential element function to arbitrary dimensions. However, the Jacobians this is built on are currently finite-difference approximations. Ideally, these partial differentials could be calculated using an automatic differentiation (AD) package.

[mikeingold]

Status Update

• Changes rolled in with v0.14.0 included merging of the previous derivative function into jacobian.
• I'm still planning to update differential hopefully soon with a generalization to any number of dimensions.
• I have no update on AD functionality. The last I checked, none of the big AD packages I tried were able to achieve AD on Meshes.jl parametric functions out-of-the-box, and defining a bunch of chain rules for them is really something that would need to be done upstream in Meshes.jl proper.

[mikeingold]

PR #101 implements a generalization of differential to any number of parametric dimensions using geometric algebra (via CliffordNumbers.jl).

The remaining part of this issue is in implementing improved jacobian calculations.

[mikeingold]

I'm re-splitting this Issue to track the AD topic in a re-opened #35