【A Brief Introduction to Exterior Calculus】1. Introduction and Motivation

【外微分浅谈】1. 绪论与启发 苏剑林 · 2016-11-04 · Mathematics vector exterior differentiation

Preface

In the Understanding Riemannian Geometry series, I shared some "geometric" insights into Riemannian geometry, but left one question unresolved: how does one actually compute the Riemann tensor? MTW's Gravitation mentions a method based on exterior calculus, but I wasn't familiar with exterior calculus, so I went and studied it. Indeed, it was the shortcut procedure in Gravitation for computing the curvature tensor that convinced me to dig deeper into exterior calculus. As it turns out, a tangible payoff really is the strongest motivator.

This series of articles mainly shares what I've learned about exterior calculus. It has gone through many rounds of revision and refinement, and covers quite a lot of material: the exterior product, moving frames, exterior differentiation and its applications in Riemannian geometry, and eventually an efficient way of computing curvature.

Notation: Throughout this series, boldface letters denote vectors, matrices, and bases, while plain letters denote scalars — which might be a scalar function or a component of a vector. Unless otherwise noted, $n$ denotes the dimension of the space (manifold). I also use Einstein summation convention throughout, meaning that repeated upper and lower indices imply summation over $1\sim n$, i.e. $\alpha_{\mu}\beta^{\mu}=\sum_{\mu=1}^{n} \alpha_{\mu}\beta^{\mu}$. By convention the subscript index is written first, so $\alpha_{\mu}\beta^{\mu}$ is in fact equivalent to $\beta^{\mu}\alpha_{\mu}$, but it's customary to write it in the former order. Some commonly used notations: $\mu,\nu$ denotes component indices, $x^{\mu}$ denotes the coordinate components of a point, $dx^{\mu}$ denotes the components of a tangent vector (infinitesimal displacement), and Greek letters such as $\alpha,\beta,\omega$ are often used to denote differential forms. Some notation is reused across contexts, but the meaning of each symbol should be clarified near where it appears, so hopefully there won't be much confusion.

Lastly, I should say that I still don't have a fully intuitive feel for exterior calculus myself, so there may well be errors in these articles — please forgive me and point them out if you spot any. This series is titled "A Brief Introduction to Exterior Calculus" not out of modesty, but because it really is brief: my understanding is shallow, and what I say here is correspondingly shallow too~ more

Motivation from Vectors

Ever since high school, when we first learned the concept of vectors, vectors have come with two ways of doing computations: one is to set up a coordinate system and compute using coordinates, and the other is to work directly with vector rules. To handle solid geometry problems on the college entrance exam, students mostly practiced the former approach — setting up coordinate systems — which is to say, the language of components. But in fact, the language of components sometimes gets in the way of understanding a vector as an "objective entity," and it isn't always the simpler approach either. Consider, for example, the following simple problem:

Let $\boldsymbol{A},\boldsymbol{B}$ be two vectors of equal magnitude. Prove that $\boldsymbol{A}-\boldsymbol{B}$ is perpendicular to $\boldsymbol{A}+\boldsymbol{B}$.

The standard solution to this problem should go: $(\boldsymbol{A}-\boldsymbol{B})\cdot (\boldsymbol{A}+\boldsymbol{B}) = \boldsymbol{A}^2 - \boldsymbol{B}^2 = 0$, from which we conclude that $\boldsymbol{A}-\boldsymbol{B}$ is perpendicular to $\boldsymbol{A}+\boldsymbol{B}$. I doubt anyone would want to set up a coordinate system, write out every component, and then grind through the computation. Moreover, this conclusion holds in a space of any dimension, whereas as soon as you set up a coordinate system, you've effectively fixed the dimension of the space, turning a general result into a special case. In other words, there is a purely vectorial formal language that has a distinct advantage for describing and manipulating vectors as "objective entities."

Note: the vectors appearing in this article and in the Understanding Riemannian Geometry series are, in the language of tensors, all contravariant vectors. In fact, I think the distinction between covariant and contravariant is not strictly necessary — true vectors are always contravariant, and so-called covariant vectors are purely a matter of definition. From a geometric point of view, one can get everything done perfectly well without the notions of covariant and contravariant at all; there's no need to know whether a quantity is covariant or contravariant, only whether it is a genuine geometric object. This approach might make certain algebraic derivations more cumbersome, but it leads to a deeper understanding.