Mathematics for Mechanics
| Code | Completion | Credits | Range | Language |
|---|---|---|---|---|
| 2011097 | Z,ZK | 4 | 3P+1C | Czech |
- Garant předmětu:
- Petr Sváček
- 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 2024/2025:
-
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 Wed Thu Fri - Time-table for summer semester 2024/2025:
- Time-table is not available yet
- The course is a part of the following study plans: