Visualizing One-, Two-, and Three-Forms

Jon Pierre Fortney

Document Type

Book Chapter

Source of Publication

A Visual Introduction to Differential Forms and Calculus on Manifolds

Publication Date

11-4-2018

Abstract

In this chapter we will introduce and discuss at length one of the ways that physicists sometimes visualize "nice" differential forms. In essence, we will be considering ways of visualizing one-forms, two-forms, and three-forms on a vector space. That is, we will find a "cartoon picture" of \(\alpha _p \in T^*_p\mathbb {R}^2\) and \(\alpha _p \in T^*_p\mathbb {R}^3\). Our picture of αp will be superimposed on the vector space \(T_p\mathbb {R}^3\). This perspective is developed extensively in Misner, Thorne, and Wheeler's giant book Gravitation. In fact, they make some efforts to develop the four-dimensional picture (for space-time) as well, however we will primarily stick to the two and three-dimensional cases here. Section one focuses on the two-dimensional case and sections two through four focuses on the three-dimensional case. Then in section five we expand our cartoon picture to general two and three-dimensional manifolds. Again, this cartoon picture really only applies to "nice" differential forms, of the kind physicists are more likely to encounter, but it is still useful for forms and manifolds that are not overly complicated. Finally, in section six we introduce the Hodge star operator. Our visualizations of forms in three dimensions provide a nice way to visualize what the Hodge star operator does, which makes this a nice place to introduce it. Despite the power and usefulness of the way of visualizing differential forms in physics developed in this chapter, it is rarely encountered in mathematics. One of the reasons for this is that in reality it is not a completely general way of considering forms; when dealing with dimensions greater than four or with more abstract manifolds or with forms that are not "nice" in some sense it breaks down.

Publisher

Springer International Publishing

First Page

151

Last Page

188

Disciplines

Physical Sciences and Mathematics

Indexed in Scopus

no

Open Access

no

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