Mathematics for Mechanics
Code | Completion | Credits | Range | Language |
---|---|---|---|---|
2013054 | Z | 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:
-
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:
-
- 13 136 NSTI MMT 2012 základ (compulsory course in the program)
- 14 141 NSTI AME 2012 základ (compulsory course in the program)
- 15 141 NSTI MCH 2012 základ (compulsory course in the program)
- 16 151 NSTI BLP 2012 základ (compulsory course in the program)