There will be a tutorial on Tuesdays and a lecture on Wednesdays.

A question time regarding Matlab / programming problems will be offered in the CIP pool every Tuesday between 15.00h and 16.00h, starting on May 14.

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.

Semester |
Lecturer |
Date |
Location |
---|---|---|---|

SS 2019 |
Wednesday, 12:00-13:30 |
MB-Building, |

Documents |
Contents |
---|---|

Chapter 0 | Contents (will be continually updated) |

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 |

Tensor Functions | |

General Basis Systems | |

Integral Theorems |

Semester | Date | Location |
---|---|---|

SS 2019 |
Tuesday, 13:15-14:45 and |
MB-Building, |

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 | 09.04.19 | Basis systems and transformations (Whiteboard-based tutorial in the lecture room) |
Handout |

2 | 16.04.19 + 23.04.19 |
Programming based on index notation (updated) | Handout Framework |

3 | 23.04.19 | Finite rotations | Handout Framework |

4 | 29.04.19 | Additive tensor decompositions (Whiteboard-based tutorial in the lecture room) |
Handout |

5 | 07.04.19 | Finite rotation with Euler angels | Handout Framework |

6 | 14.05.19 | Eigenvalue problems and Cayley-Hamilton theorem (Whiteboard-based tutorial in the lecture room) |
Handout |

7 | 15.05.19 | Eigen-projection with a perturbation algorithm | Handout Framework |

8 | 21.05.19 | Polar decomposition | Handout Framework |

9 | |||

10 | |||

11 | |||

12 |

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

***No lecture notes or text books are allowed. **

***You should bring your pocket calculator along.**

- J. Betten, Tensorrechnung für Ingenieure, Teubner, 1987
- R. De Boer, Vektor- und Tensorrechnung für Ingenieure, Springer, 1982
- E. Klingbeil, Tensorrechnung für Ingenieure, BI Wissenschaftsverlag, 1989
- M. Itskov, Tensor Algebra and Tensor Analysis for Engineers, Springer, 2015
- R. M. Bowen and C.-C. Wang, Introduction to Vectors and Tensors, Dover Publications, 2008
- C. Eringen, Continuum Physics, Academic Press, 1971

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