The Exterior Derivative

We’re going to derive half the identities in Vector Calculus by introducing a powerful new operator.

This article is stop 16 on The Road to Quantum Mechanics.

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.

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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

• Gradient,
• Divergence,
• Curl,
• 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.

Product Rules

Prove the following examples of the product rule.

Divergence Rules

Prove the following identities for the divergence of a scalar-vector product and the cross product of two vectors.

Curl Rules

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