Foreword
In the “Understanding Riemannian Geometry” series, I shared some “geometric” insights into Riemannian geometry, while leaving one question unanswered: how do we actually calculate the Riemann tensor? MTW’s Gravitation mentions a method based on exterior calculus, but since I was unfamiliar with it, I decided to study it. Indeed, it was the efficient steps for calculating the curvature tensor in Gravitation that prompted me to delve deeper into exterior calculus. As it turns out, tangible benefits are the primary driving force.
This series of articles mainly shares some insights from learning exterior calculus. It has undergone multiple revisions and improvements and covers a wide range of content, such as exterior products, moving frames, exterior differentiation, and their applications in Riemannian geometry, finally including an effective method for calculating curvature.
Notation Description: In this series, bold letters represent vectors, matrices, and bases, while normal letters represent scalars. A scalar may be a scalar function or a component of a vector. Unless otherwise specified, n denotes the dimension of the space (manifold). Einstein summation convention is used throughout this article, where identical upper and lower indices indicate a summation from 1 to n, i.e., \alpha_{\mu}\beta^{\mu}=\sum_{\mu=1}^{n} \alpha_{\mu}\beta^{\mu}. By convention, the subscript is written first; for example, \alpha_{\mu}\beta^{\mu} is actually equivalent to \beta^{\mu}\alpha_{\mu}, but it is customary to write the former. Common notations include: \mu, \nu for component indices, x^{\mu} for coordinate components of a point, dx^{\mu} for components of a tangent vector (differential element), and Greek letters such as \alpha, \beta, \omega for differential forms. There are instances where symbols are reused, but their meanings are generally explained near their appearance, so they should not cause confusion.
Finally, I must admit that I still do not have a particularly strong intuition for exterior calculus. Therefore, there may be errors in these articles; I ask for the reader’s understanding and welcome corrections. Naming this series “A Brief Talk on Exterior Calculus” is not out of modesty; my understanding is indeed quite shallow, and what I have to say is equally basic.
Inspiration from Vectors
Since we first learned the concept of vectors in high school, vectors have encompassed two methods of operation: one is to establish a coordinate system and use coordinates for calculations, and the other is to operate directly using vector laws. To cope with solid geometry in the college entrance examination, students practice coordinate-based calculations—the language of components—more frequently. However, component language sometimes hinders our understanding of vectors as “objective entities,” and it is not necessarily always simpler. For example, consider the following simple problem:
Let \boldsymbol{A} and \boldsymbol{B} be two vectors with equal magnitude. Prove that \boldsymbol{A}-\boldsymbol{B} is perpendicular to \boldsymbol{A}+\boldsymbol{B}.
The standard answer to this problem should be (\boldsymbol{A}-\boldsymbol{B})\cdot (\boldsymbol{A}+\boldsymbol{B}) = \boldsymbol{A}^2 - \boldsymbol{B}^2 = 0, which leads to the conclusion that \boldsymbol{A}-\boldsymbol{B} is perpendicular to \boldsymbol{A}+\boldsymbol{B}. I believe no one would want to establish a coordinate system, provide every component, and then perform the calculation. Moreover, this conclusion holds for a space of any dimension, whereas establishing a coordinate system effectively implies fixing the dimension of the space, thereby specializing a general conclusion. In other words, there exists a purely vectorial formal language that has unique advantages in describing and operating on vectors as “objective entities.”
Note: From the perspective of tensor language, the vectors in this article and the “Understanding Riemannian Geometry” series are all contravariant vectors. In fact, I believe the concepts of covariance and contravariance are unnecessary; true vectors are all contravariant, while so-called covariant vectors are merely defined. From a geometric perspective, one can still successfully complete the required tasks without the concepts of contravariance and covariance. We do not need to know whether a quantity is covariant or contravariant; we only need to determine whether it is a geometric quantity. This approach might make deriving certain algebraic expressions more difficult, but it provides a deeper understanding.
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