Differential forms are powerful tools, but a lot of the geometric intuition is hidden behind the formalism. Let’s fix that.
With the exception of The Charged Particle Lagrangian, we’ve done little to no Vector Calculus. Sure we’ve talked about a few operators from Vector Calculus, but we’ve pulled formulas and identities out of thin air with no explanation. Over the next few articles, we’re going to fix that problem.
To do so, we’re going to introduce the concept of a differential form. Differential forms give us one unified approach to Vector Calculus and Integration in arbitrary coordinates. We’ll be able to derive several important theorems and rewrite Maxwell’s Equations in ways that are more practical in different situations. For this article, we’re going to talk about two of the core operators of Differential Forms: the wedge product ∧ and the Hodge star operator ⋆.
Check Your Understanding
This article can’t cover everything we would need to do all the cool stuff, so I only have a few questions for you.
Surface Area of a Sphere
Use a differential two-form to find the surface area of a sphere by finding the area element corresponding to dθ∧dφ
- using the Jacobian
- using the normal vector along with the Hodge star operator
- and any other method you can think of.
Algebraic Proof that Determinant is Volume Element
Use the algebraic properties of the wedge product ∧ to prove that the volume element is equal to the determinant of the matrix whose columns are the vectors describing the parallelotope. You’ll probably want to use either the Laplace Expansion or the Leibniz Formula for the determinant. You might also want to use a proof by induction if you’re working with a recursive formula for the determinant. Lastly, you’ll probably want to look up the Levi-Civita symbol or the sign of a permutation.