| Course title | Mathematics 1B |
|---|---|
| Course code | KMA/M1B-P |
| Organizational form of instruction | Lecture + Lesson |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 5 |
| Language of instruction | Czech |
| Status of course | unspecified |
| Form of instruction | Face-to-face |
| Work placements | Course does not contain work placement |
| Recommended optional programme components | None |
| Course availability | The course is available to visiting students |
| Lecturer(s) |
|---|
|
| Course content |
|
1. Metric space. Limit of sequence in metric space. 2. Function of several variables. Contours of functions. Basic planes in R3. Continuity of limit of mapping f: R2 -> R. 3. Partial and directional derivative. The total differential. Gradient. Geometrical applications. Tangential plane. 4. Chain rule. Transformation of differential equations. Partial derivative and the total differential of order n. 5. Commutation of mixed partial derivatives. Implicit Function Theorems. 6. Constrained and global extremes of functions of several variables. Study of critical points; the Hessian matrix. Conditional extreme and Lagrange multipliers. 7. Ordinary differential equations (ODE). Direction field. Cauchy problem. Existence and uniqueness of solution of the first order differential equation y' = f(x,y). Euler method of numerical solution of Cauchy problem. 8. Elemental methods for solving the first order ODE. Separation of variable method. Variation of constant. 9. Application of ODE for solving geometrical and technical problems. Orthogonal trajectory and exact differential equation. 10. Homogeneous linear ODE of order n. Fundamental system. Homogeneous linear ODE of order n with constant coefficients. Characteristic polynomial. Wronskian. 11. Heterogeneous linear ODE with constant coefficients. Variation of constants method and method of guess for special right side. 12. Series of real numbers. Convergent and divergent series, geometric series and the harmonic series. Series with non-negative terms, the comparison test, the limit comparison test and the ratio test. 13. Alternating series. Leibniz criterion of convergence. Absolute and non-absolute convergence. Function series. Pointwise and uniform convergence. Differentiation and integration of function series. 14. Power series. The radius of convergence of power series. Differentiation and integration of power series. Taylor series. Taylor series expansion of function. Practice: The material explained at the previous week lecture is practised.
|
| Learning activities and teaching methods |
Monological explanation (lecture, presentation,briefing)
|
| Learning outcomes |
|
Subject contains three thematic units: 1. basic methods for solving differential equations, linear differential equations of higher orders, especially with constant coefficients; 2. differential calculus of more variable functions: total differential and tangent plane, differentation of composite functions, extremes; 3. number series, basic convergence tests, power series, application on solution of differential equations.
Differential calculus of function of more real variable. Ordinary differential equations. Series. |
| Prerequisites |
|
Knowledge of course M1A.
|
| Assessment methods and criteria |
|
Combined examination
" Solve first order partial differential equations via the method of characteristics " Understand functions of several variables, their partial differentiation, integration, and their geometrical interpretation " evaluate the limits of a wide class of real sequences; " determine whether or not real series are convergent by comparison with standard series or using the Ratio Test; |
| Recommended literature |
|
| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester |
|---|