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.
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.
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.
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(a ⋅ c) – c(a ⋅ b)
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.