exterior derivative

Overview

The exterior derivative is a map that mimics/ generalizes the total derivative from multivariate calculus. Instead of just functions, it uses differential forms. Its definition is motivated by the idea of finding a “closest linear approximation”.

Definition

Let be a smooth manifold. The exterior derivative is the unique linear map

that satisfies the following conditions: 1.

Doing it twice gives the zero map. 2.

It doesn’t respect the grading of the graded algebra structure of 3. For a -form and a -form

Follows the Liebsnitz rule (with a graded enhancement). 4. If , then is defined by by

In other words, given any derivation of local behavior, evaluates the derivation on .

Explicit formula

The exterior derivative can be calculated using the formula

where denotes omitting those arguments.

Notes

Derivative on smooth functions

For a smooth function then

Proof

For , let

be a basis for .

For , . Therefore,

where is the dual basis of , and are the unique coefficients from basis vectors.

However, if we evaluate on the basis for we get

because of the the dual basis, and linearity.

By the definition, and 4 above