Mathematical Analysis 2

The Riemann integral in one variable. Antiderivation. Definition and elementary properties of the Riemann integral on a compact interval. A fundamental characterization of the class of integrable functions. The Mean Value Theorem. Functions defined by integrals and their properties. The Fundamental Theorem of Calculus and the Fundamental Formula of Calculus in one variable. Improper integrals: definitions, examples and comparison techniques. Taylor's formula with integral remainder. Relationships between integrals and numerical series.
Differential calculus in several real variables. Limits and continuity. Directional derivatives, gradients and the Jacobian matrix. Differentiability: necessary and/or sufficiente conditions. Tangent hyperplanes. Differentiation and composition: the Chain Rule. Diffeomorphiosms in several variables. Second order derivatives, the Hessian matrix and Schwarz's theorem. Taylor's formula. Unconstrained optimization: local extrema and saddle points.
The Riemann integral in several real variables.. The Riemann integral over the cartesian product of real intervals: definition and techniques for the calculation. Brief discussion of Peano-Jordan measure. The Riemann integral on admissible domains: definition, integrability and the calculation of mutltiple integrals on simple domains. Integration in higher dimensions. Change of variables and special coordinate systems: polar, cylindrical and spherical.