Courses

Differential and Integral Calculus2 (90902)

Abstract:

Infinite series. Convergence tests. Series of f?unctions; convergence; (uniform convergence-optional). Power series; representation of f?unction by power series. Taylor series. F?unctions of several variables- limits and continuity, partial and directional derivatives. Linear approximation. Gradient. The chain rule. Higher order partial derivatives and second-degree Taylor polynomial. Relative/absolute maximum and minimum values. Lagrange multipliers. Multiple integrals. Fubini's theorem. Change of variables and Jacobian. Polar, cylindrical and spherical coordinates. Line integrals of scalar f?unctions. Line integrals of vector fields. Independence of path and Green’s theorem. Surface integrals of scalar f?unctions. Oriented surfaces and surface integrals of vector fields. The divergence theorem (Gauss-Ostrogradsky). Stokes' theorem. Applications.

Harmonic Analysis (90916)

Abstract:

Fourier series: expansion to Fourier series on a finite interval,

Fourier coefficients. Complex representation of Fourier series,

the convergence of the series, Dirichlet function, convergence in a jump discontinuity. Gibbs phenomena.

Parseval’s identity. Differentiation and integration of Fourier series.

Fourier transform, definition, properties and the transform table.

Applications of Fourier transform in signal processing and

in solutions of differential equations. Laplace transform and its applications in solving ordinary differential equations.

Solution in cases where the forcing term is a step function or a delta function.

Complex Functions (90917)

Abstract:

Complex integration : Contour Integrals. Residue theory.

Cauchy’s Residue Theorem and its applications: evaluation of integrals.