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CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2024/2025
NOTICE: Study plans for the following academic year are available.

Mathematics for Mechanics

The course is not on the list Without time-table
Code Completion Credits Range Language
2013054 Z 4 3P+1C Czech
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:

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.

Syllabus of tutorials:

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.

Study Objective:

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.

Study materials:
Note:
Further information:
No time-table has been prepared for this course
The course is a part of the following study plans:
Data valid to 2025-04-03
For updated information see http://bilakniha.cvut.cz/en/predmet1895506.html