pullback

(45 minutes to learn)

Summary

Pullback is a mathematical operator which represents functions or differential forms on one space in terms of the corresponding object on another space. They are used to define surface integrals of differential forms.

Context

This concept has the prerequisites:

differential forms (Pullback operates on differential forms.)
exterior derivative (Pullback commutes with the exterior derivative.)
Chain Rule (The pullback of a differential is given by the chain rule.)
evaluating multiple integrals: change of variables (When surface integrals are defined in terms of pullback, this is a generalization of the change of variables formula.)

Goals

Define the pullback of a function and of a differential form

Show that the pullback commutes with the exterior derivative

Be able to manipulate pullback, wedge products, and the exterior derivative algebraically

Define the surface integral of a differential form in terms of pullback

Core resources (read/watch one of the following)

-Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

Location: Section 8.2.3, "Pullback," pages 342-345

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