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#math #physics #geometry #differential #tangent #cotangent #vector #pushforward #pullback

In this lecture, we illustrate the concepts fundamental to differential geometry in the familiar and simple setting of polar coordinates. We discuss tangent and cotangent vectors, pushforward and pullback, emphasizing both the intuition and the elegant rigidity behind these notions. The way in which these concepts are unified can be summarized succinctly via the principle "What it is = What is does = How it transforms". Created by Timothy Nguyen.

Prerequisites: Multivariable calculus.

Further Reading (for a traditional math treatment):
Lee, John M. Introduction to Smooth Manifolds

Correction:
01:08:38 : Phi_*(v) not Phi^*(v)

Outline:
00:00:00 : Introduction
00:01:47 : Polar Coordinates
00:04:23 : Coordinate chart
00:12:30 : Transition maps
00:18:11 : Why coordinate charts?
00:20:21 : Pullback of functions (Change of Variables)
00:26:11 : Pullback refines notion of identity
00:27:18 : Pullback concrete example
00:32:51 : Tangent vector (definition)
00:41:55 : Tangent vectors at a point form a vector space
00:45:00 : Tangent vector concrete example
00:46:53 : Vector field
00:50:28 : Pushforward of tangent vectors (Chain rule)
00:51:41 : Pushforward of d/dr
00:55:53 : Geometric picture of pushforward
1:03:37 : Pushforward of d/dtheta
1:06:57 : Relationship between pushforward and pullback
1:11:50 : Cotangent vector (definition)
1:13:52 : Example: Braket formalism of quantum mechanics
1:16:28 : Example: Legendre transform
1:19:24 : Dual basis for cotangent vectors
1:24:15 : 1form (definition)
1:29:07 : Pullback of cotangent vectors (Expansion of Differentials)
1:33:01 : Pullback of cotangent vector (definition)
1:34:10 : Pullback of 1form concrete example
1:38:05 : ***The Fundamental Concept***: "What it is = What it does = How it transforms"
1:43:38 : Summary

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

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