Mathematics for Mechanics

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:

• 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)