Mathematical Analysis 1:

Number sets. Real and complex numbers. Topology of the real line. Limits of sequences and series.. Limits of functions and continuity. Continuous functions and topology. Differentiable functions and their properties. Differentiation rules. Fundamental theorems of the differential calculus. Taylor formula and second order conditions to determine the nature of a stationary point. Convexity. Study of the graph of a function. Riemann integral: definition of integrable function and properties of the integral. Classes of integrable functions. The fundamental theorem of calculus. Integration methods and computation of definite integrals. Improper integrals.

Mathematical Analysis 2:

1) Metric spaces, complete metric spaces, sequentially compact sets and their properties, continuous functions.
2) Contraction theorem.
3) Normed spaces, linear operators between normed spaces. Equivalent norms.
4) Functions from R ^ n in R ^ m. Continuity and differentiation. Partial derivatives, gradient and Jacobian matrix. Sufficient condition for differentiation. Chain rule.
5) Mean value theorem. Functions with null differential on connected sets.
6) Second partial derivatives.
7) Taylor’s theorem in several variables.
8) Implicit functions. Local theorem of existence and uniqueness.
9) Maximum and minimum. First order conditions.
10) Hessian matrix and sufficient conditions for extrema.
11) Extrema with constraints. The Lagrange Multipliers method.
12) Peano measure in R ^ n and measurable sets.
13) Integration on rectangles. Iterated integrals. Riemann integral on measurable sets.
14) First order differential equations and systems. Solutions.
15) The local existence and uniqueness theorem for the Cauchy problem.
16) Maximal solutions.
17) Sufficient conditions for existence on a interval.
18) Differential Equations of order n. Linear equations of order n. Independent solutions and solution space.
19) Linear equations of order n with constant coefficients.