Differentiation of Forms

Abstract

The goal of this section is to figure out what we mean by the derivative of a differential form. One way to think about a derivative is as a function which measures the variation of some other function. Suppose ω is a 1-form on ℝ². What do we mean by the “variation” of ω? One thing we can try is to plug in a vector field V. The result is a function from ℝ² to ℝ. We can then think about how this function varies near a point p of ℝ². But p can vary in a lot of ways, so we need to pick one. In Section 1.5, we learned how to take another vector, W, and use it to vary p. Hence, the derivative of ω, which we shall denote “dω,” is a function that acts on both V and W. In other words, it must be a 2-form!