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#differential #forms #integration #math #physics #gauge #theory

Building upon the previous two lectures, we discuss the construction of differential forms and how to integrate them. Basic operations such as pushforward, pullback, and contraction with tangent vectors are discussed, and explicit lowdimensional computations are done. Our discussion of differential forms is needed in order to get to the curvature tensor. We defer the discussion of Stoke's Theorem to the next lecture. Created by Timothy Nguyen.

Further reading:
John M. Lee. Introduction to Smooth Manifolds

Notes:
54:09: Note the abuse of notation. The radial coordinate r(t) is a function which is identically equal to the constant r which is the radius of the semicircle. If I had to redo, I'd use R for the radius to disambiguate with the polar coordinate function.
1:09:15: You can integrate on a nonorientable manifold by using a "density" instead of a topdegree differential form (the former is a slight modification of the latter).
1:18:06: Typo in Phi^*(dx^3), should be +rcos(theta)dtheta in second term.
1:31:40: Note that r = sqrt(x^2 + y^2)

Timestamps:
00:00:00 : Introduction
00:01:30 : Basic ingredients for differential forms
00:03:53 : Abstract, bundle description of differential forms
00:08:27 : Examples of differential forms by degree
00:13:25 : kforms allow integration against kdimensional domains
00:16:41 : Pushforward and pullback formula in general dimension
00:21:19 : Einstein notation
00:26:30 : Pushforward and pullback of differential forms
00:28:31 : Contraction operator
00:31:35 : Contraction operator extended to arbitrary degree forms
00:37:42 : Why do we care about contraction operators?
00:39:30 : Integration
00:40:02 : Integration of 1forms
00:44:06 : Orientation of curves
00:46:54 : Case of general curve via pullback to the interval
00:49:53 : Example: Integrate dtheta along semicircle
00:56:51 : Integration of 2forms
01:00:26 : Orientation of surfaces
01:05:17 : Case of general surfaces using local patches and pullback
01:10:40 : Example: integrate 2form to compute area of a disc
01:22:46 : Example: integrate 2form to compute surface area of a sphere
01:33:09 : Wrap up and look ahead

Lecture pdf: https://tinyurl.com/yht4ddfp

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