Introduction to Tensor Calculus

TU Bergakademie Freiberg | Wintersemester 2013 / 2014 Introduction to Tensor Calculus

Fundamentals of Tensor Calculus with a Primer on Manifolds

Tentative schedule (constantly updated)

Attention: bugs in homework 5 - now corrected!

Week

(cal.)

Week

(sem.)

Lecture

Thursday 14:00-15:30 KKB-1069

Homework

topics

15

1

Introduction: geometry and space, vectors as geom. objects, diff. manifolds: first contact

Euclidean geometry

16

2

vector space, lin. independence and basis, Ricci notation, change of basis, dual space

Ricci-notation, algebraic operation with vectors and dual vectors

17

3

inner product spaces, reciprocal basis

metric coefficients, inner product and orthogonal mappings

18

4

May Day

 

19

5

Affine and euclidean space

observer transformations

20

6

2nd order tensors as linear mappings

transpose of a 2nd order tensor, unity tensor, metric tensor

21

7

tensors of arbitrary order

orthogonal tensors, antisymmetric tensors

22

8

Ascension Day

 

23

9

Multivectors, multicovectors and exterior algebra

Working with multi-indices, exterior algebra

24

10

exterior algebra

exterior algebra

25

11

tensor analysis in euclidean spaces

Frechet-derivative, Gateaux-differential

26

12

tensor analysis in euclidean spaces

tangent plane to surfaces in R3

27

13

analysis on differentiable manifolds:

tangent space and push forward

example: same physical problem on a sphere with and without embedding in E3

28

14

cotangent space, differential forms

 

29

15

Covariant derivative, Integration on manifolds

 

 

 

 

Potential audience:
==================
CMS students 3rd semester
students of Mechanical Engineering specializing in computation and
simulation
researchers from engineering and physics

Lecture 2 SWS on Vector and Tensor Calculus

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