Continue the basic foundation in mathematical analysis with respect to differential and integral calculus in one and several variables.
Expected learning outcomes
Knowledge of the principal results concerning the Riemann integral in one variable, differential calculus in several variables and multiple integrals. Acquisition of the relevant techniques for computations and the ability to apply the theoretical tools in practical situations.
The target The aim of the course is to provide basic notions and tools in the setting of the classical integral calculus for real functions of one as well as several real variables and of the differential calculus for functions of several real variables.
The content The Riemann integral in one variable. Definition and elementary properties of the Riemann integral on a bounded interval. A fundamental characterization of the class of integrable functions. Mean value integral and 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. Primitives over an interval and their calculation. 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. Directional derivatives, gradients and the Jacobian matrix. Differentiability and the differerntial, tangent hyperplanes and the theorem on the total differential. Differentiability of composite functions and the chain rule. Higher order derivatives, the Hessian matrix and Schwarz's theorem. Higher order differentiability and Taylor's formula with Lagrange and Peano remainders. Unconstrained optimizzation: necessary and sufficient conditions for having local extrema.
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.