Tensors and applications

1 Course description

Differential geometry is an important and often indispensable mathematical tool for the formulation of theories in physics. The present course introduces the concepts of differential geometry such as metric and other tensors, tangential vectors, one-forms, manifolds, and charts. In particular, the students acquire the tools necessary to follow subsequent lectures on general relativity. The course focuses on conveying the mathematical framework in modern formulation, but in basic manner for students from physics and other sciences. The concepts are illustrated and motivated throughout by applications in physics, in particular from special relativity, theoretical mechanics and electrodynamics.

2 Topics



1. Special relativity and flat space time

1.1 Spacetime

1.2 Metric tensor

1.3 Paths

1.4 Lorentz transformations

1.5 Galilei transformations

1.6 Vectors

1.7 Dual vectors

1.8 Tensors

1.9 Nonlinear coordinate transformations

2. Curved space time

2.1 Motivation

2.2 Manifolds

3 Examination

Written exam at the end of the course.

4 Prior knowledge

Theoretical mechanics, linear algebra, basic analysis, electrodynamics.

5 Literature

Sean Carroll, Spacetime and geometry, Addison Wesley, San Francisco (2004).

A first version of the book is accessible online, http://preposterousuniverse.com/grnotes/, further copies are available in the library.

6 Specifics

The course will be taught in English.

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