Modern Mathematics and Physics, Spring 1984

Notes by Robert T Jantzen

too ambitious for an undergraduate level course, but a useful reference

if you find these notes useful please contact me! See later notes and an undergraduate textbook online.

Table of Contents [about 150 8.5x11in US sized paper sheets]

0  Introduction 1
1  Linear Algebra Summary (metrics, antisymmetrization, duals) 1
2  Some more explanation for 1 (some examples) 19
3  Still more explanation for 1 (more on antisymetrization, metric index shifting, duals) 28
4  The end of linear algebra (more on wedge, metric dual) 36
5  Differential geometry on Rn using Cartesian coordinates 43
6  Examples of differential manifolds, coordinate transformations 52
7  The exterior derivative 63
8  Orientation, integration of p-forms, normals, volume elements (glitch on inner/outer orientation) 71
9  Stokes' theorem without metric, with metric 92
10  Covariant differentiation, parallel transport 102
11  Gauge fields, covariant exterior derivative* 119
12  Lie derivative, Lie dragging 132

These were scanned into PDF directly using a Canon photocopier, not quite so nice quality.
Here is a comparison test with the HP scanner scanning into Adobe Acrobat: Canon versus HP.

* on the first page of this section there is an unfortunate ambiguity of symbols, but easily distinguished since they have consistent index positioning:
the two contravariant indexed omega is the dual frame basis 2-form ωij = ωi \wedge ωj ,
the 1 contravariant, 1 covariant indexed omega is the connection 1 -form matrix ωi j .