# 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 R^{n} 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 ω^{ij} =
ω^{i}
\wedge
ω^{j} ,

the 1 contravariant, 1 covariant indexed omega is the connection 1 -form matrix ω^{i} _{j
.}