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]

dg0.pdf | Table of contents and summary of still missing motivating introduction (unrecorded for posterity: 2006 insert plus Maple visualization) | 0 |

CHAPTER 1: |
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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 | |

R^{n} with usual inner product |
42 | |

Problem | 43 |

CHAPTER 2 |
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dg3.pdf | Measure Motivation | 48b |

Symmetry properties and volume (measure) | 49 | |

The algebra of antisymmetric tensors | 59 | |

CHAPTER 3 |
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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 |
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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 |

CHAPTER 5 with extra section of flow lines of vector fields |
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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 R^{n} |
18 |

Cylindrical and spherical coordinates on R^{3} |
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 |
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dg6.pdf | Covariant derivatives on R^{n} 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 |
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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 |
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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 |
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Describing intrinsic curvature | 107 | |

Exercises | 110 | |

Interpretation of curvature | 111 | |

CHAPTER 10 TO DO extrinsic curvature |

CHAPTER 11 |
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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 |
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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 |
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APPENDIX B. Special relativity and hyperbolic geometry |
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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.