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Tensorrechnung (19/6)

Tensor Calculus (19/6)

Announcements

The examination takes place on Friday, 10th August, at 10.00 o'clock in room E21, MB-Buiding.


Contents

 

The term "vector" is well known, but for the precise definition it is not sufficient to introduce a triple of three (real) numbers. As shown in this lecture, a definition of the underlying base coordinate system is required first. For this reason, the lecture puts an emphasis on index notation, although symbolic notation is also introduced.

The most commonly used vector products are introduced in Cartesian coordinates. Co- and contra-variant bases are then introduced by making use of curvilinear coordinates and their relation to the Cartesian system. The transformation between bases and vector coordinates for the Cartesian bases, co- and contra-variant bases as well as their physical counterparts are elaborated in detail.

With the help of transformations of Cartesian coordinate systems – e.g., rotation and reflection matrices – we introduce the term "tensors". Moreover, we show that vectors can, in certain cases, be considered as first order tensors. Examples for so-called pseudo-vectors that do not comply with tensorial transformation rules are also presented.

In the course of dealing with tensor algebra, the relations obtained for vectors are generalised to higher-order tensors. New operations like the dyadic product or the (multiple) contraction of tensors in co-, contra- and mixed-variant representations are introduced. Regarding symmetric second-order tensors, we show unique decompositions of these tensors, eigenvalue problems, tensor powers, as well as selected tensor functions of interest.

The tensor calculus introduced thereafter covers differential operators applied to tensor fields. Based on the nabla operator, the corresponding operations such as the gradient, divergence, and rotation are derived. Moreover, derivatives of tensor functions with respect to tensor coordinates as well as generalised differentiation relations – i.e. the Fréchet and Gâteaux derivatives – are covered. The second fundamental topic of tensor calculus is the integration of tensor fields and the application of different integration rules.

Finally, the interconnections between the above named relations are elaborated by means of variational calculus.


Lectures

Semester

Lecturer

Date

Location

SS 2018

Prof. Dr.-Ing. Andreas Menzel

Tuesday, 13:15-14:45 and Wednesday, 12:00-13:30

MB-Building,
Room 165

Accompanying documents for the lectures

Documents

Contents

Chapter 0

Contents

Chapter 1

Introduction

Chapter 2

Vector Algebra

Chapter 3

Tensor Algebra

Chapter 4

Special Second Order Tensors
and Operators

Chapter 5

Eigenvalue Problems of Second
Order Tensors

Chapter 6

Permutation Tensor

Chapter 7

Cayley-Hamilton Theorem

Chapter 8

Tensor Algebra of Fourth Order
Tensors

Chapter 9

Special Fourth Order Tensors
and Operators

Chapter 10

Vector and Tensor Analysis

Chapter 11

Integral Theorems

Chapter 12

Tensor Functions

Chapter 13

Whiteboard-based lecture

Tutorials

 

Semester Supervisor Date Location
SS 2018 Isabelle Guschke, M. Sc.

Tuesday, 13:15-14:45 and
Wednesday, 12:00-13:30

MB-Building,
Room 163
(CIP-Pool)

Accompanying documents for the tutorials

The documents accompanying the tutorials are to be used in preparation of each exercise. All participants are requested to become acquainted with the exercises dealing with the subject of the respective tutorial. This will ensure that all participants are on the same level of knowledge for each individual lesson.

 

Tutorial No.

Date

Description

Files

1

02.05.2018

Index notation and additive tensor decomposition

(Whiteboard-based tutorial in the lecture room)

Handout

2

08.05.2018

Programming determinants based on index notation

Handout

Framework

3

09.05.2018

Finite rotation of tensorial quantities

Handout

Framework

4

22.05.2018

Finite rotation with Euler-angles and statistically homogeneous rotation matrices

Handout

Framework

5

29.05.2018

Change of Basis and Eigenprojections

(Whiteboard-based tutorial in the lecture room)

Handout
6

30.05.2018

Computation of Eigenprojections

Handout

Framework

7

12.06.2018

Polar decomposition in closed form

Handout

Framework

8

13.06.2018

Kelvin-notation and inversion of 4th order tensors

Handout

Framework

9

20.06.2018

Tensor derivatives

(Whiteboard-based tutorial in the lecture room)

Handout

10

03.07.2018 +  04.07.2018

Material model based on eigenvalues

Handout

Framework

11

Whiteboard-based tutorial


Examinations

*The final examination takes place on Friday, 10th August, at 10.00 o'clock.

*The final examination is a written test (60minutes).

*The exam will take place in room E23 MB-Building.

*No lecture notes or text books are allowed. 

*You should bring your pocket calculator along.


References