Mathematics for Mechanics

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Technical Mathematics

Synopsis:

Summary: Tensor calculus. Introduction to functional analysis. Calculus of variations.

Orthogonal transformation of coordinate systems.

Afinne orthogonal tensors and tensor operations.

Tensor as linear operator and bilinear form.

Metrics and metric spaces. Convergence. Completness.

Linear normed space. Banach space.

Linear space with scalar product (unitary space). Hilbert space.

Contractive operators and Banach fixed point theorem.

Function spaces in examples.

Operators and functionals. Linear, continuous and bounded operator/functional.

Derivative of a functional in the given direction. Gateaux differential and derivative.

Necessary and sufficient conditions for extremes of a functional.

Convex set and convex functional. Minimum of convex functional.

Extremes of functional of different types. Euler equation. Necessary and sufficient conditions for extrema.

Discrete methods for approximation of the minima of an functional. Ritz method.

Requirements:

Syllabus of lectures:

Orthogonal transformation of coordinate systems.

Afinne orthogonal tensors and tensor operations.

Tensor as linear operator and bilinear form.

Metrics and metric spaces. Convergence. Completness.

Linear normed space. Banach space.

Linear space with scalar product (unitary space). Hilbert space.

Contractive operators and Banach fixed point theorem.

Function spaces in examples.

Operators and functionals. Linear, continuous and bounded operator/functional.

Derivative of a functional in the given direction. Gateaux differential and derivative.

Necessary and sufficient conditions for extremes of a functional.

Convex set and convex functional. Minimum of convex functional.

Extremes of functional of different types. Euler equation. Necessary and sufficient conditions for extrema.

Discrete methods for approximation of the minima of an functional. Ritz method.

Syllabus of tutorials:

Orthogonal transformation of coordinate systems.

Afinne orthogonal tensors and tensor operations.

Tensor as linear operator and bilinear form.

Metrics and metric spaces. Convergence. Completness.

Linear normed space. Banach space.

Linear space with scalar product (unitary space). Hilbert space.

Contractive operators and Banach fixed point theorem.

Function spaces in examples.

Operators and functionals. Linear, continuous and bounded operator/functional.

Derivative of a functional in the given direction. Gateaux differential and derivative.

Necessary and sufficient conditions for extremes of a functional.

Convex set and convex functional. Minimum of convex functional.

Extremes of functional of different types. Euler equation. Necessary and sufficient conditions for extrema.

Discrete methods for approximation of the minima of an functional. Ritz method.

Study Objective:

Study materials:

I. M. Gelfand, S. V. Fomin - Calculus of Variations, Dover Books on Mathematics, 2000

E. Kreyszig: Introductory functional analysis with applications, John Willey & Sons, 1978

Limaye, Balmohan V.: Linear Functional Analysis for Scientists and Engineers, Springer Singapore, 2016

D.Lovelock, H. Rund, Tensors, Differential Forms, and Variational Principles, Dover Books on Mathematics 1989.

J.T. Oden, Applied functional analysis: a first course for students of mechanics and engineering science, 1979

A. N. Kolmogorov, S. V. Fomin , Elements of the Theory of Functions and Functional Analysis, 1999

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: