Differential Geometry Notes Spring 1991

Based on undergraduate linear algebra and multivariable calculus

by Bob Jantzen

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


dg3.pdf Measure Motivation 48b
  Symmetry properties and volume (measure) 49
  The algebra of antisymmetric tensors 59
  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
  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


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


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
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 ro 103
  Exercise 106
  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


 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.

Table of Contents