"Differential and Integral Calculus 2a". School of Mathematical Sciences. (Ya. Yakubov)

1. Riemann integral, definition and basic properties. The Newton-Leibnitz theorem. Integration by parts, change of variable, indefinite integral, the length of a curve.
2. Uniform convergence of a sequence and a series of functions. Continuity of the limit. Uniform convergence for the integrals. Comparison test for uniform convergence for the series (M-test). Weierstrass theorem about uniform approximation by polyniomials.
3. Complex series. Multiplication of the series. Radius of convergence of the power series. Abel and Dirichlet criteria of the convergence. Abel theorem about continuity of the power series.
4. Fourier series. Riemann-Lebesgue lemma. Dirichlet kernel, Feyer kernel. Convergence of the Fourier series. Feyer theorem, mean convergence, Parseval formula.
5. Finite dimensional spaces, open and closed sets. Convergence. Continuous functions of several variables.
6. Differentiability. Partial derivatives. The chain rule. Directional derivatives, gradient, level curves. Continuous first order partial derivatives and differentiability. Mean value property.
7. Higher order partial derivatives. Mixed partial derivatives. Taylor's polynomials. Classification of critical points.