Tensor Calculus

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The focus of the course Tensor analysis lies mainly on acquiring an understanding of the principles and ideas of the direct (component-free) tensor language, which is widely used in many courses.


Tensor algebra, theory of tensor functions and tensor analysis are introduced. The symmetries of tensors and tensor functions are considered. The course emphasizes the application of tensor algebra and calculus to a wide variety of applied areas from engineering and physics.

Students who have completed the course must have an idea about tensor and properties of invariant operations on tensors; Hamilton nabla operator and its properties; the curvilinear coordinates. Students have to know the following items: basic concepts and methods of the tensor algebra and analysis. mathematical properties and the physical meaning of vector differential operators the basic theorem of tensor algebra and analysis;

Students have to be able to:

  • carry out calculations using both the direct tensor calculus and coordinate notation
  • perform calculations using the rules of working with such operators as gradient, curl, divergence;
  • differentiate tensor and the scalar-valued functions of tensor arguments

The course includes the following items: The tensor algebra

  • Operations with tensors of the second rank.
  • Linear display.
  • Tensor invariants.
  • Spectral and polar decomposition of tensor algebra in a non orthogonal basis.

The tensor functions

  • Tensor function. Differentiation of tensor functions
  • Tensor analysis
  • Tensor fields. Hamiltonian operator, curvilinear coordinate, orthogonal curvilinear coordinates, Gauss formula and Stokes formula.

Lectures by Elena Vilchevskaya

File:Tensor analysis. Lecture1.pdf

File:Tensor analysis. Lecture2.pdf

File:Tensor analysis. Lecture3.pdf

File:Tensor analysis. Lecture4.pdf

File:Tensor analysis. Lecture5.pdf

File:Tensor analysis. Lecture6.pdf

File:Tensor analysis. Lecture7.pdf

File:Tensor analysis. Lecture8.pdf

File:Tensor analysis. Lecture9.pdf

File:Tensor analysis. Lecture10.pdf

File:Tensor analysis. Lecture11.pdf

File:Tensor analysis. Lecture12.pdf

File:Tensor analysis. Lecture13.pdf

File:Tensor analysis. Lecture14.pdf

File:Tensor analysis. Lecture15.pdf

File:Tensor analysis. Lecture16.pdf

back to International MSc program "Mechanics and Mathematical Modeling"