We’re going to derive half the identities in Vector Calculus by introducing a powerful new operator.
In the previous article, I promised that we could use Differential Forms to prove a lot of important theorems, derive a lot of identities, and evaluate some integrals in arbitrary coordinate systems.
Before we could do any of that, though, we had to introduce two operators: the wedge product ∧ and the Hodge star operator ⋆. Introducing those operators took up the entire article, so we were only able to derive the formula for the volume element and some other formulas related to the wedge product and Hodge dual. In this article, we’re going to come up with the last operator needed to do all of Vector Calculus: the exterior derivative.
Check Your Understanding
While we don’t have enough to derive every identity in Vector Calculus, we can still do some of the simpler ones.
Vector Calculus in Spherical Coordinates
Derive a formula for the
- and Laplacian
in spherical coordinates using their definitions given in this article. You may need to refer back to the previous article to work with the Hodge star and the wedge operators.
Prove the following examples of the product rule.
Prove the following identities for the divergence of a scalar-vector product and the cross product of two vectors.
Prove the following identities involving the curl of various products.
Gradient of Dot Product
Prove the identity for the gradient of the dot product of two vector fields