Differential Forms and Tensors

Joseph Mellor
15 min readJun 26, 2023

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



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/.