500448 - MATHEMATICAL ANALYSIS B

1. Series. Limits of real sequences. Real series: definitions and basic examples; series with positive terms (and convergence tests); absolute and simple convergence. Fundamentals of real power series. Taylor polynomials and Taylor formulas. Taylor series; Taylor series of some elementary functions. 2. Differential Equations. Introduction to ordinary differential equations. The Cauchy problem. Linear ordinary differential equation of the first order, separable equations, homogeneous equations. Linear ordinary differential equation of higher order with constant coefficients: homogeneous and complete cases. Fundamentals about boundary value problems for second order equations. 3. Differential Calculus in several real variables. Real functions of several real variables: definitions, graphs; limits and continuity; partial derivatives, gradients, and directional derivatives. Successive derivatives. Differentiability. Partial derivatives of nested functions (chain rules). Free extrema of real functions of several real variables; critical points and their classification. Fundamentals of differential calculus for vector-valued functions; jacobian matrices. 4. Multiple Integrals. Double integrals: definitions and basic properties, application in Geometry and Physics. Integration techniques: iteration formulas; change of variables; double integrals in polar coordinates. Fundamentals of volume integrals. 5. Line Integrals and Surface Integrals. Curves: tangent vectors; rectifiable curves and arc length. Surfaces: tangent planes; surface area. Line integrals with respect to the arc length. Line integrals of vector fields, and applications in Physics. Gradient fields, potentials, and path independence. The operators curl and div. Surface integrals, and applications in Physics. Green's theorem and divergence theorem in two variables. Stokes' theorem and divergence theorem in three variables.