The exam topics

Mathematical Analysis 2

Functions of several variables, multiple integrals, curves and surfaces, series of functions.

1. Functions of several variables

   The space ℝⁿ with norm, distance and neighborhoods, subsets of the plane, functions of two or more variables with domain and level sets, limits in two variables, continuity and the Weierstrass theorem.

2. Differential calculus in several variables

   Partial derivatives and gradient, differentiability and tangent plane, directional derivatives and the gradient formula, chain rule, second derivatives and second-order Taylor formula.

3. Implicit functions

   When an equation F(x, y) = 0 defines a function y = φ(x), how to compute the derivatives of φ without knowing it, and how the result extends to surfaces and systems.

4. Free and constrained maxima and minima

   Where to look for the extrema of a function of several variables, how to classify them with the Hessian matrix, how to handle compact sets and constraints with Lagrange multipliers.

5. Curves and line integrals

   Parametric curves in the plane and in space, regularity and tangent vector, length and arc length, line integrals of the first and second kind with applications to mass, center of mass and work.

6. Vector fields and Green’s theorem

   Vector fields in the plane and in space, work along a curve, conservative fields and potential, irrotational fields and simply connected domains, Green's theorem and computation of areas.

7. Double and triple integrals

   How to integrate a function of two or three variables, how a multiple integral reduces to single integrals, how changes of variables simplify the domain, and how to compute areas, volumes, masses and centroids.

8. Surfaces, fluxes and the theorems of Gauss and Stokes

   Regular parametric surfaces, normal vector and area, surface integrals and flux of a field, divergence and curl, the divergence theorem and Stokes' theorem.

9. Sequences and series of functions

   Pointwise convergence and uniform convergence of a sequence of functions, the theorems on passing continuity, integrals and derivatives to the limit, and total convergence for series of functions.

10. Power series and Taylor series

    Where a power series converges, how to compute the radius and the set of convergence, how to differentiate and integrate term by term, and how Taylor expansions represent the elementary functions.

11. Fourier series

    How a periodic function is expanded in a series of sines and cosines, how the coefficients are computed, at which points and in what sense the series converges, and how Parseval's identity links the function to its coefficients.