28616 - Mathematical Analysis T-B

Learning outcomes

Knowing the methodological-operational aspects of mathematical analysis, with particular attention to the functions of multiple real variables and differential equations, in order to be able to use this knowledge to interpret and describe engineering problems.

Course contents

Generalized integrals for functions of a real variable. Definition generalized integral. Existence criterion; comparison criterion; absolute convergence criterion. Numerical series. Definition of numerical series; geometric series. Necessary conditions for convergence. Existence criterion; integral criterion; comparison criterion; absolute convergence criterion; Leibnitz criterion; ratio criterion. Limits and continuity for real-valued and vector-valued functions of several real variables. Topology of R^n. Real and vector functions of several real variables: generalities, limits and continuity. Weierstrass and intermediate value theorems. Differential calculus for real-valued and vector-valued functions of several real variables. Directional derivatives, partial derivatives and differentiability for functions of several variables; Jacobian matrix, gradient. Derivation rules. Higher order partial derivatives. Schwarz theorem, Hessian matrix. Taylor formulas up to second order for functions of several variables. Free global and local extremals for real functions of several variables: definitions, necessary conditions, sufficient conditions. Nature of the critical points. Constrained global and local extremums for real functions of several variables: definitions and methods; manifolds in R^n, Lagrange multiplier theorem. Integral calculus for functions of several real variables. Multiple integral over rectangles; Reduction theorem. Riemann multiple integral in bounded regions of R^2 and R^3; property of the integral. Reduction theorem on simple regions. Diffeomorphisms, Variable change theorem for multiple integrals. Polar coordinates; cylindrical coordinates; spherical coordinates. Complex numbers. The field of complex numbers; algebraic and trigonometric form of a complex number. Geometric interpretation of sum and product of complex numbers. Complex exponential. n-th roots of complex numbers. Equations in C. Ordinary differential equations. First order linear differential equations, solution formula. Homogeneous and non-homogeneous linear differential equations of higher order; general solutions and particular solutions; Existence and uniqueness theorems for the Cauchy problem; equations with constant coefficients, solution methods. Equations with separable variables, special equations. Applications to financial and economic models.

Readings/Bibliography

E. Giusti, Analisi Matematica 2

Teaching methods

Frontal lectures, frontal exercises.

Assessment methods

Written exam with exercises and questions of theory.

Teaching tools

Blackboard, frontal lectures, and the offifcial web page on Virtuale.

Office hours

See the website of Simone Ciani