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Tensor Calculus (MB-20)

Announcements

We will start providing course material for “Tensor Calculus” on April 6th in the Moodle online class.

Even if the the semester start will be postponed due to the outbreak of the corona virus, we aim at providing course material, starting on April 6th. It will be done in the form of digital content and modern methods, e.g., digital lecture notes, screen casts, an online forum for discussion, etc.

Note that these announcements are made based upon the current situation, and that these arrangement may be subject to change based on new developments. At the moment, for instance, we do not know if and how examinations will take place. Please check this website and the Moodle class for updates.

The entire class will be organized and provided on Moodle by the name "Tensor Calculus” in this semester “SoSe 2020”. Joining the Moodle class is obligatory. Please write an email using the subject: "Registration Tensor Calculus" to carina.witt@tu-dortmund.de from your TU-Dortmund account (only ...@tu-dortmund.de, not ...@udo.edu) until April 24th in order to register for the Moodle classroom.

Stay Healthy!

Contents

The term "vector" is well established - the precise representation of vectors, however, requires not only the introduction of related coefficients but also of the related basis system. The vector itself can be represented with respect to different basis systems. In other words, the vector itself is independent of the particular basis system chosen which is fundamental for applications of vector (and tensor) calculus to continuum physics.

The course starts with basic vector properties, operations and transformations, whereby representations are typically referred to a Cartesian basis system. Based on this basic vector algebra, related properties and operators are introduced for second order tensors. This includes additive, multiplicative and spectral decompositions as well as the Cayley-Hamilton theorem and properties of special second order tensors. Moreover, isotropic tensor functions are discussed. By analogy with second order tensors, fourth order tensors are introduced thereafter, including Kelvin and Voigt representations.

The course continues with basic vector and tensor analysis, including directional derivatives, gradient-, divergence-, and rotation-operations. Based on this, integral theorems are discussed, which are most relevant for, e.g., transformations of balance relations in continuum physics.

Finally, general basis systems are introduced and applied to vector and tensor algebra and analysis. This includes basis systems which are neither normalised nor orthogonal and which may be either fixed in space or change in space.

In particular, the course includes the following chapters

1.   Introduction
2.   Vector Algebra
3.   Tensor Algebra of Second Order Tensors
4.   Special Second Order Tensors and Operators
5.   Eigenvalue Problems of Second Order Tensors
6.   Permutation Tensor
7.   Cayley-Hamilton Theorem
8.   Tensor Algebra of Fourth Order Tensors
9.   Special Fourth Order Tensors and Operators
10. Vector and Tensor Analysis
11. Tensor Functions
12. Integral Theorems
13. General Basis Systems

The exercises of the course include, amongst others, computational aspects and programming of selected topics of tensor calculus.

Organisation

Semester

Lecturer

Date

Location

SS 2020

Dr.-Ing. Tobias Kaiser

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

MB-Building,
Room 165, Room 163 (CIP-Pool)