# Differential Forms

• Gregory S. Chirikjian
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

This chapter introduces differential forms, exterior differentiation, and multi-vectors in a concrete and explicit way by restricting the discussion to ℝn. This is extended to more general settings later. Roughly speaking, differential forms generalize and unify the concepts of the contour integral, curl, element of surface area, divergence, and volume element that are used in statements of Stokes’ theorem and the divergence theorem. At first it may seem unnecessary to learn yet another new mathematical construction. The trouble is that without an appropriate extension of the concept of the cross product, it is difficult and messy to try to extend the theorems of vector calculus to higher dimensions, and to non-Euclidean spaces. As was illustrated in Chapter 1 in the context of heat and fluid flow problems, these theorems play a central role. Likewise, in probability flow problems involving stochastic differential equations and their associated Fokker–Planck equations, these theorems play a role in assessing how much probability density flows past a given surface. Since the problems of interest (such as the stochastic cart in Figure 1.1) will involve stochastic flows on Lie groups, understanding how to extend Stokes’ theorem and the divergence theorem to these generalized settings will be useful. The first step in achieving this goal is to understand differential forms in ℝn.

## Keywords

Tensor Product Scalar Multiplication Divergence Theorem Exterior Derivative Wedge Product
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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