Course detail

Linear Transformations and Tensor Analysis

FSI-0TAAcad. year: 2024/2025

Application of linear algebra, especially matrix calculus for describing movement in space and spatial transformations. Introduction to tensors, tensor fields and tensor analysis with an emphasis on use in physics and technical sciences.

Language of instruction

Czech

Number of ECTS credits

3

Mode of study

Not applicable.

Guarantor

doc. RNDr. Miroslav Kureš, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra in the scope of the course 1m.

Rules for evaluation and completion of the course

Exam: both written and oral.

Aims

The student will learn to effectively use the tools of linear algebra, especially matrix calculus for describing movement in space and spatial transformations. He will use this knowledge especially in various tasks of mechanics. The student will also learn to work with tensors and tensor fields. 

Study aids

Supplementation of direct teaching in the form of e-learning support will be continuously used. 

Prerequisites and corequisites

Not applicable.

Basic literature

Johnston, Nathaniel. Advanced linear and matrix algebra. Springer, 2021. (EN)
Wasserman, Robert. Tensors and manifolds: with applications to physics. Oxford University Press, USA, 2004. (EN)

Recommended reading

Not applicable.

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme B-OBN-P Bachelor's 1 year of study, winter semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.

Syllabus

1. Linear mappings. Eigenvalues and eigenvectors.
2. Diagonalizability.
3. Orthogonal transformations.
4. Unitary and Hermitian transformations.
5. Matrix decompositions: QR decomposition, LU decomposition, spectral decomposition.
6. Multilinear mappings. Tensors.
7. Symmetric and antisymmetric tensors.
9. External algebra.
8. Tensors of the 2nd order in physics.
10. Covariant derivation of vector and tensor fields.

Topics planned for 1-2 weeks.

Exercise

13 hod., compulsory

Teacher / Lecturer

doc. RNDr. Miroslav Kureš, Ph.D.

Syllabus

Seminars follow topics of lectures. They are focused on calculations.

Elearning

eLearning: currently opened course