Mathematical Analysis 2

1. Riemann integral calculus for f: R-->R
Primitive functions, the indefinite integral. Calculation techniques for indefinite integrals. Riemann-integrability for f:[a,b]-->R and the definite integral. Geometric meaning of the integral. Conditions of integration. Properties of the space of integrable functions and of the integral. The integral function and its properties. The Fundamental Theorem of Integral Calculus and its consequences. Improper integrals, convergence conditions and examples. Relations between integrals and numerical series.

2. Differential calculus for f: Rn-->Rm
Limits, continuity and related problems.
The case of real-valued functions: directional derivatives; gradient vector, links between directional differentiability and continuity. Differentiability: necessary conditions, total differential theorem, tangent hyperplane, geometric meaning of the gradient vector, class C¹ functions. Lagrange theorem.
The case of vector-valued functions: Jacobian matrix; differentiability. Composition of differentiable functions. Finite increment theorem.
Second derivatives, Hessian matrix, Schwartz theorem. The second differential as a bilinear form. Class C² functions. Partial derivatives of order k. Taylor's formula, with remainders according to Peano and Lagrange.
Free optimization for real functions: stationary, extreme and saddle points. Use of the Hessian matrix for the classification of extrema points: the role of eigenvalues.

3. Riemann integral for real functions of several real variables.
The Riemann integral for functions defined on n-dimensional intervals, and the calculation by means of iterated integrals. Measurable sets according to Peano-Jordan, the measure of P-J and its properties. Sets of zero measure, generally continuous functions and their integrability. Explicit computation via iterated integration; normal sets with respect to the coordinate axes. Diffeomorphisms of open sets of R^n. Integration by changing variables. Improper multiple integrals.