What I learned from Terry Tao's paper on differential forms and integration
I came across this awesome writeup from Terry Tao explaining differential forms and integration.
With the help of this paper, I think I finally understand the basic idea of why differential forms are fundamental to integrals. I think the central idea is:
A one-dimensional signed definite integral \( \int_\gamma \omega \) is a map from a path \(\gamma\) to a real number. The same concept is generalized to higher dimensional integrals over surfaces/volumes/etc.
We are used to obsessing about the “\(f(t)\)” in the integral \(\int f(t) dt \) because of all the drills we did in math class, where we learned a bunch of rules to calculate the integral based on what \(f(t)\) is. But if we view the integral from this new perspective, then it’s clear how differential forms play a fundamental role. A differential form (at a given point on the manifold) is a map from one or more vectors to a real number. And of course, here the relevant vectors are defined by the integration path/surface/etc. (A path maps to a series of tangent vectors, a surface maps to a series of vector pairs, etc.) So this is how differential forms enable signed definite integrals to map from paths/surfaces/etc. to a real number.
Double-clicking for a bit more detail, focusing on one-dimensional integrals:
- At a given point on a differential manifold, a vector at that point can be considered to be the velocity vector of a parametrized path \(\gamma(t)\) going through that point, where \(t\) is in an interval on the real line.
- Therefore, a given \(\gamma\) induces a set of tangent vectors along its path.
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A given differential 1-form \(\omega\) maps each point along the path to a functional, which in turn maps the tangent vector to a real number. As Tao’s paper explains (even as he expresses his disapproval of this notation), for every 1-form there is a unique vector field \(F:\mathbb{R}^n \rightarrow \mathbb{R}^n\) such that its dot product with the tangent vector is equivalent to the action of the 1-form on the tangent vector. Note that the integral of the dot product of a vector field with the tangent vector (the velocity vector of the path) is precisely the definition of the line integral of a vector field.
- The result is that for a given vector field \(F\) that is equivalent to a 1-form \(\omega\), you can map each value of \(t\) (representing a point along the path \(\gamma\)) to a real number, and integrating this over \(t\) gives us the line integral of \(F\).