Differential Forms and Tensors

Joseph Mellor
15 min readJun 26

The Levi-Civita symbol allows us to take determinants and rewrite Differential Forms in terms of tensors.

This is stop 21 on the Road to Quantum Mechanics.

We’ve spent roughly five articles (or six if you count Vectors and Covectors) talking about Differential Forms, but concepts in Physics are expressed mostly in terms of tensors. If we want our progress in Differential Forms to carry over to the rest of Physics, we’ll need some way of rewriting them in terms of tensors.

Since Differential Forms are multilinear maps from k-vectors to scalars, converting them to tensors should be possible. In this article, we’re going to find ways to rewrite all the concepts in Differential Forms in terms of tensors.

Check Your Understanding

I hope you’re ready for index juggling. The only way to really understand how this stuff works is to do it for yourself.

Wedge Products

Repeat the process in the section Verifying Our Guess for the Wedge Product but for the three covectors in Higher Dimensions.

Antisymmetrization and the Wedge Product

Show that the formula

holds for the wedge product of an arbitrary two-form and an arbitrary three form in >5D space.

Triple Products

In 3D space, the cross product of two vectors can be defined as the Hodge dual of their wedge product. Use this definition to prove the following identities

  • u ⋅ (v × w) = the determinant of the matrix whose columns are the vectors u, v, and w.
  • a × (b × c) = b(ac) – c(ab)

Divergence in Arbitrary Coordinate Systems

Use the definition of the Hodge dual and the exterior derivative to derive a formula for the divergence in an arbitrary 3D coordinate system in terms of indices.

Note that the metric tensor may not be diagonal, but you can guarantee that certain terms will go to zero since they either show up in the Levi-Civita symbol or they show up in wedge products.

Joseph Mellor

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.