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