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.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
1. The Riemann integral in one variable. Indefinite integrals and their calculation. Definition and elementary properties of the Riemann integral. Classes 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. Improper integrals: definitions, examples and comparison techniques. Taylor's formula with integral remainder. Brief discussion on the relation between series and integrals.
2. Differential calculus in several variables. Directional and partial 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.
3. Multiple integrals. The Riemann integral for rectangles: definition, integrability and techniques for their calculation. Brief discussion of Peano-Jordan measure and the integrability on rectangles of generally continuous functions. 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.