Introduction to Forms

Abstract

A differential form is simply this: an integrand. In other words, it is a thing which can be integrated over some (often complicated) domain. For example, consider the following integral: $\int\limits_0^1 {x^2 dx}$. This notation indicates that we are integrating x² over the interval [0, 1]. In this case, x²dx is a differential form. If you have had no exposure to this subject this may make you a little uncomfortable. After all, in calculus we are taught that x² is the integrand. The symbol “dx” is only there to delineate when the integrand has ended and what variable we are integrating with respect to. However, as an object in itself, we are not taught any meaning for “dx.” Is it a function? Is it an operator on functions? Some professors call it an “infinitesimal” quantity. This is very tempting. After all, $\int\limits_0^1 {x^2 dx}$ is defined to be the limit, as $n \to \infty , of \sum\limits_{i = 1}^n {x_i^2 } \Delta x$, where {x_i} are n evenly spaced points in the interval [0, 1], and Δx = 1/n. When we take the limit, the symbol “Σ” becomes “∫,”, and the symbol “Δx” becomes “dx.” This implies that dx = lim_Δx→0 Δx, which is absurd. lim_Δx→0 Δx = 0!! We are not trying to make the argument that the symbol “dx” should be eliminated. It does have meaning. This is one of the many mysteries that this book will reveal.