Mathematics for Mechanics
Code  Completion  Credits  Range  Language 

2011097  Z,ZK  4  3P+1C  Czech 
 Garant předmětu:
 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 timetable has been prepared for this course
 The course is a part of the following study plans: