Essentials of High School Math Course

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Students are introduced to mathematical concepts and methods used in the introductory physics course.

Requirements:

No prerequisities.

Syllabus of lectures:

Coordinate systems, position (2D, 3D) (c.s. Cartesian, cylindrical, spherical,..), description of curves (conic,..). Translation, rotation around an axis.

Einstein summation convention, linear transformations, orthogonal transformations, matrices, operations with rows and columns of matrices, determinants, lin. vect. space (the existence of solutions, permittivity, permeability., flexibility, strength ...), eigenvectors, eigenvalues, matrix diagonalization, quadratic forms (moment of inertia)

Scalars - vectors - tensors, additive - the non-additive quantity, size of units, field scalar x vector. Points, vectors, forms and operators. Scalar and vector products.

Differential calculus: total, partial derivatives, curve length, curvature. Derivatives of elementary functions, derivative of product and composite functions. Functions of several variables, differential. (speed, acceleration).

Introduction to integral calculus, Rieman integral, geometric and physical meaning. Differential equations (differential eq. - numerical solution)

Approximate solutions, series, limits, approximation of functions, Taylor expansion of function (approximation for the general potential, LHO)

Complex numbers, the Euler's formula, the Moivre's formula (oscillations, ..)

The geometry of curves - the distance, tangent vector, normal, osculating circle, the radius of curvature. Description of the area, interior space coordinates, normal. (normal and tangential acceleration)

Vector field: differential operator nabla, Laplace operator, operations with operators Gauss., Stokes. theorem, geometric meaning (potential energy, conservative force, work around a closed curve)

Basis: 2D 3D ND Continuous, Fourier Transform

Mechanics of continuous media, systems of partial differential equations (continuity equation, equation, perturbation, perturbations, dispersion relations, ..)

Functional, strength, other ways to determine the trajectory, the calculus of variations

Statistics, probability distribution function

Syllabus of tutorials:

Coordinate systems, position (2D, 3D) (c.s. Cartesian, cylindrical, spherical,..), description of curves (conic,..). Translation, rotation around an axis.

Einstein summation convention, linear transformations, orthogonal transformations, matrices, operations with rows and columns of matrices, determinants, lin. vect. space (the existence of solutions, permittivity, permeability., flexibility, strength ...), eigenvectors, eigenvalues, matrix diagonalization, quadratic forms (moment of inertia)

Scalars - vectors - tensors, additive - the non-additive quantity, size of units, field scalar x vector. Points, vectors, forms and operators. Scalar and vector products.

Differential calculus: total, partial derivatives, curve length, curvature. Derivatives of elementary functions, derivative of product and composite functions. Functions of several variables, differential. (speed, acceleration).

Introduction to integral calculus, Rieman integral, geometric and physical meaning. Differential equations (differential eq. - numerical solution)

Approximate solutions, series, limits, approximation of functions, Taylor expansion of function (approximation for the general potential, LHO)

Complex numbers, the Euler's formula, the Moivre's formula (oscillations, ..)

The geometry of curves - the distance, tangent vector, normal, osculating circle, the radius of curvature. Description of the area, interior space coordinates, normal. (normal and tangential acceleration)

Vector field: differential operator nabla, Laplace operator, operations with operators Gauss., Stokes. theorem, geometric meaning (potential energy, conservative force, work around a closed curve)

Basis: 2D 3D ND Continuous, Fourier Transform

Mechanics of continuous media, systems of partial differential equations (continuity equation, equation, perturbation, perturbations, dispersion relations, ..)

Functional, strength, other ways to determine the trajectory, the calculus of variations

Statistics, probability distribution function

Study Objective:

Knowledge: learn the basic procedures for solving of simple physical problems

Abilities: applying of new abstract concepts on the description and solution of real physical situations and phenomena

Study materials:

Povinná literatura: [1] Kvasnica J.: Matematický aparát fyziky, Academia, Praha, 1989, 1997

Doporučená literatura: [2] Mathematical Physics, Sadri Hassani, Springer 2000, ISBN 978-0-387-98579-4

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: