Differential Geometry Notes Spring 1991

Based on undergraduate linear algebra and multivariable calculus

if you find these notes useful please contact me!

Copyright 2007: conversion to LaTeX/PDF format and complete edit and expansion in progress: dgn1991-2008.pdf
[> 4.5MB, Table of Contents, Preface and Intro 2008]

Part 1 Algebra

dg0.pdf	Table of contents and summary of still missing motivating introduction (unrecorded for posterity: 2006 insert plus Maple visualization)	0
	CHAPTER 1:
dg1.pdf	Index conventions	1
	A vector space and its dual space	2
	Linear transformations of a vector space into itself (and tensors)	10
	Tensor product and matrix multiplication	15c
	Worked Problems	16
	Remark	19
dg2.pdf	Linear transformations of V into itself and a change of basis	20
	Linear transformations between V and V^*	27
	Invertible maps between V and V^*	28
	Problems	29
	Index shifting (and inner products)	31
	Geometric interpretation of index shifting	33
	Problem	34
	Cute fact	35
	Index shifting conventions	37
	Problem	40
	Partial evaluation of a tensor and index shifting	41
	Contraction of tensors	41
	Rⁿ with usual inner product	42
	Problem	43

	CHAPTER 2
dg3.pdf	Measure Motivation	48b
	Symmetry properties and volume (measure)	49
	The algebra of antisymmetric tensors	59
	CHAPTER 3
	Whoa! Review of what we've done so far	62
	Problem set (p.43) discussed	71
	Remark on generalized Kronecker deltas [MOVED TO CHAPTER 2]	74
	CHAPTER 4
	The wedge product	75
	Subspace orientation and antisymmetric tensors (duality operation)	81
	Exercises	85
	Inner product for antisymmetric tensors	88
	The unit n-form on an oriented vector space with inner product	89
	Linear maps	93
	Exercises	97

Part 2 Calculus

	CHAPTER 5 with extra section of flow lines of vector fields
dg4.pdf	The tangent space in multivariable calculus	1
	Some problems in 3d calculus	9
	More motivation for the re-interpretation of the tangent space	10
	Frames and dual frames	15
	More on Lie brackets	17b
	Exercises	17b
	More on tangent covectors, the differential, and vector fields	17c
dg5.pdf	Non-Cartesian coordinates on Rⁿ	18
	Cylindrical and spherical coordinates on R³	28
	Exercise	34
	Exercise answer	34b
	Exercises p.17b,c answered	34c
	Summarizing remark, exercise	35
	Exercise p.35 answer	36
	spherical coordinates in detail	37
	Exercise p.40 worked	40b
	Exercises	40c

	CHAPTER 6
dg6.pdf	Covariant derivatives on Rⁿ with Euclidean metric	44
	Notation for covariant derivatives	47
	Exercise	50
	Exercises pp.30,40,42,43 worked	52
	Duality and extension to tensors of a covariant derivative	54
	Note on class of Cartesian coordinate systems	56
	Exercise	57
	Exercise pp.49,50 worked	58
	Exercise	59
	The clever way of evaluating the components of the covariant derivative	60
	Exercises	62
	Noncoordinate frames	63
	Exercises	64

	CHAPTER 7
dg7.pdf	Laplacian and divergence and gradient	65
	Exercises	66
	Aside on determinant of differentials	68
	Exercises	69
	Second covariant derivatives	71
	Exercises	72
	More practice evaluating components of the covariant derivative	74
	Aside on orthogonal matrices	78
	Exercise p.40c worked	84
	Exercise p.57 worked	85
	Exercise p.59 worked	86
	Exercise p.62 worked	87
	Exercise p.64 worked	89
	Exercise	90
	CHAPTER 8
dg8.pdf	Covariant differentiation along a curve and parallel translation	91
	Exercise	92
	Exercise	93
	Exercise	99
dg9.pdf	Geodesics	100
	Exercise	101
	The 2-sphere of radius r_o	103
	Exercise	106
	CHAPTER 9
	Describing intrinsic curvature	107
	Exercises	110
	Interpretation of curvature	111
	CHAPTER 10 TO DO extrinsic curvature

	CHAPTER 11
dg10.pdf	Integration of differential forms	114
	Parametrized p-surface	115
	Pulling back differential forms	117
	Exercises	121
	Change of variables	123
dg11.pdf	The exterior derivative d	126
	Exercise	131
	The exterior derivative and a metric	136
	Exercise	142
dg12.pdf	Induced orientation	145
	Stokes' theorem	148
	CHAPTER 12
	Final remarks	151
dg13.pdf	Final exam	156
	Final exam worked	162
	Last page	171
	APPENDIX A. From trigonometry to hyperbolic functions and hyperbolic geometry
	APPENDIX B. Special relativity and hyperbolic geometry
	APPENDIX C. Worked problems

Index of PDF files, page totals and filesizes

Preface

These handwritten notes from a course given at Villanova University in the spring semester of 1991 were scanned and posted on the web in 2006 at http://www34.homepage.villanova.edu/robert.jantzen/notes/dg1991
and were converted to a LaTeX compuscript and revised in 2007 with the help of Hans Kuo of Taiwan through a serendipitous internet collaboration. Most undergraduate courses on differential geometry are leftovers from the early part of the last century, focusing on curves and surfaces in space, which is not very useful for the most important application of the twentieth century: general relativity and field theory in theoretical physics. Most mathematicians who teach such courses are not well versed in physics, so perhaps this is a natural consequence of the distancing of mathematics from physics, two fields which developed together in creating these ideas from Newton to Einstein and beyond.

The idea of these notes is to develop the essential tools of modern differential geometry while bypassing more abstract notions like manifolds, which although important for global questions, are not essential for local differential geometry and therefore need not steal precious time from a first course aimed at undergraduates.

Part 1 (Algebra) develops the vector space structure of $\mathbb{R}^n$ and its dual space of real-valued linear functions, and builds the tools of tensor algebra on that structure, getting the index manipulation part of tensor analysis out of the way first. Part 2 (Calculus) then develops $\mathbb{R}^n$ as a manifold first analyzed in Cartesian coordinates, beginning by redefining the tangent space of multivariable calculus to be the space of directional derivatives at a point, so that all of the tools of Part 1 then can be applied pointwise in the space. Non-Cartesian coordinates and the Euclidean metric are then used as a shortcut to what would be the consideration of more general manifolds with Riemannian metrics in a more ambitious course, followed by the covariant derivative and parallel transport, leading naturally into curvature. The exterior derivative and integration of differential forms is the final topic, showing how conventional vector analysis fits into a more elegant unified framework.

The theme of Part 1 is that one needs to distinguish the linearity properties from the inner product (``metric") properties of linear algebra. The inner product geometry governs lengths and angles, and the determinant then enables one to extend the linear measure of length to area and volume in the plane or 3-dimensional space, and to $p$-dimensional objects in $\mathbb{R}^n$. The determinant also tests linear independence of a set of vectors and hence is key to characterizing subspaces independent of the particular set of vectors we use to describe them while assigning an actual measure to the $p$-parallelepipeds formed by a particular set, once an inner product sets the length scale for orthogonal directions. By appreciating the details of these basic notions in the setting of $\mathbb{R}^n$, one is ready for the tools needed point by point in the tangent spaces to $\mathbb{R}^n$, once one understands the relationship between each tangent space and the simpler enveloping space.

The LaTeX version of these notes is in progress.