The examination takes place on Friday, 10th August, at 10.00 o'clock in room E21, MBBuiding.
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 contravariant 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 contravariant 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 socalled pseudovectors 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 higherorder tensors. New operations like the dyadic product or the (multiple) contraction of tensors in co, contra and mixedvariant representations are introduced. Regarding symmetric secondorder 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.
Semester 
Lecturer 
Date 
Location 

SS 2018 
Tuesday, 13:1514:45 and Wednesday, 12:0013:30 
MBBuilding, 
Documents 
Contents 

Contents  
Introduction  
Vector Algebra  
Tensor Algebra  
Special Second Order Tensors and Operators 

Eigenvalue Problems of Second Order Tensors 

Permutation Tensor  
CayleyHamilton Theorem  
Tensor Algebra of Fourth Order Tensors 

Special Fourth Order Tensors and Operators 

Vector and Tensor Analysis  
Integral Theorems  
Tensor Functions  
Chapter 13 
Whiteboardbased lecture 
Semester  Supervisor  Date  Location 

SS 2018  Isabelle Guschke, M. Sc. 
Tuesday, 13:1514:45 and 
MBBuilding, 
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 (Whiteboardbased tutorial in the lecture room) 

2 
08.05.2018 
Programming determinants based on index notation 

3 
09.05.2018 
Finite rotation of tensorial quantities 

4 
22.05.2018 
Finite rotation with Eulerangles and statistically homogeneous rotation matrices 

5 
29.05.2018 
Change of Basis and Eigenprojections (Whiteboardbased tutorial in the lecture room) 
Handout 
6 
30.05.2018 
Computation of Eigenprojections 

7 
12.06.2018 
Polar decomposition in closed form 

8 
13.06.2018 
Kelvinnotation and inversion of 4th order tensors 

9 
20.06.2018 
Tensor derivatives (Whiteboardbased tutorial in the lecture room) 

10 
03.07.2018 + 04.07.2018 
Material model based on eigenvalues 

11 
Whiteboardbased tutorial 
*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 MBBuilding.
*No lecture notes or text books are allowed.
*You should bring your pocket calculator along.