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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2019/2020

Mathematics for Mechanics

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Code Completion Credits Range Language
2011097 Z,ZK 4 3P+1C Czech
Lecturer:
Petr Sváček
Tutor:
Petr Sváček
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:
Time-table for winter semester 2019/2020:
06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Fri
Thu
roomKN:A-313
Sváček P.
14:15–16:45
(lecture parallel1)
Karlovo nám.
Učebna KA313
roomKN:A-313
Sváček P.
16:45–17:30
(lecture parallel1
parallel nr.101)

Karlovo nám.
Učebna KA313
Fri
Time-table for summer semester 2019/2020:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2019-09-17
For updated information see http://bilakniha.cvut.cz/en/predmet1891006.html