Description:

1. Functional sequences and series - convergence range, criteria of uniform convergence, continuity, limit, differentiation and integration of functional series, power series, Series Expansion, Taylor´s theorem.
2. Ordinary differential equations - equations of first order (method of integration factor, equation of Bernoulli, separation of variables, homogeneous equation and exact equation) and equations of higher order (fundamental system, reduction of order, variation of parameters, equations with constant coefficients and special right-hand side, Euler differential equation).
3. Metric spaces - metric, norm, scalar product, neighborhood, interior and exterior points, boundary point, isolated and non-isolated point, boundary of set, completeness of space, Hilbert spaces. Orthogonal polynomials. Complete orthogonal systems.
4. Fourier series - expansion of functions into Fourier series, trigonometric Fourier series and their convergence.
5. Differential calculus of functions of several variables - limit, continuity, partial and directional derivative, gradient, total derivatives and tangent plane, Taylor series, elementary terms of vector analysis, Jacobi matrix.
6. Functions defined implicitly by one or several equations.

Recommended literature:

Key references:
[1] M. L. Bittinger, D. J. Ellenbogen, S. J. Surgent: Calculus and Its Applications (11th Ed.), Pearson, 2015
[2] R. A. Adams, Calculus: A Complete Course, 1999
[3] J. E. Marsden, A. Tromba: Vector Calculus, W.H. Freeman, New York, 2013.

Recommneded references:
[4] J. Stewart: Multivariable Calculus, 8th Edition, Brooks Cole, 2015.

Media and tools: MATLAB

Keywords:

Function sequences, function series, differential equations, quadratic forms, quadratics surfaces, metric spaces, norm spaces, pre-Hilbert spaces, Hilbert spaces, Fourier series, function of several variables, implicit functions.

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar