Introduction to differential equations; differential calculus in several variables; curves and surfaces; linear differential forms, Gauss and Stoke's theorems.
1) M. Bramanti, C.D. Pagani, S. Salsa,"Analisi Matematica 2", Zanichelli
2) S. Salsa, A. Squellati, "Esercizi di Analisi Matematica" Vol.2, Zanichelli
3) C. Sbordone, N. Fusco, P. Marcellini, "Analisi Matematica Due", Liguori
Altri testi di consultazione consigliati:
3) G. Anichini, G. Conti, "Analisi Matematica 2", Pearson.
4) M. Boella, "Analisi Matematica 2 - Esercizi", Pearson
5) P. Marcellini, C. Sbordone, "Esercizi di Matematica Volume II" (4 Fascicoli), Liguori
Learning Objectives
Acquisition of mathematical tools needed for the description and understanding of physical phenomena. Strengthening and development of the attitude to both the analytical and logical deductive reasoning to identify the essential data in the analysis and synthesis of the presentation of possible problems.
Prerequisites
Differential and integral calculus for functions of one variable.
Teaching Methods
The course includes the carrying out of lectures and classroom exercises.
The exam consists of a written practice test and a written or oral theory test.
During the course there will be two learning assessment tests; the passing of such tests is equivalent to passing the written practice test.
Course program
INTRODUCTION TO DIFFERENTIAL EQUATIONS
Generality: solution, general integral, Cauchy problem. Examples of first order equations: linear equations, equations with separable variables. Second order linear equations with constant coefficients. Hints about higher order equations.
REAL FUNCTIONS OF TWO OR MORE VARIABLES
The real functions of two or more variables. Limits and continuity. Calculation of limits in multiple variables.
DIFFERENTIAL AND INTEGRAL CALCULATION FOR MORE VARIABLE FUNCTIONS.
Directional derivatives, partial derivatives, gradient. The differential of a function. Differentiable functions. Derivation of composition of functions. Higher order derivatives. Schwartz Theorem. Taylor's Formula. Local maximum and minum points. Critical points. Saddles. The Hessian matrix. Implicit functions. Dini's theorem. The study of absolute extremes. The Lagrange multipliers theorem. Riemann integral for functions of two or three variables. Reduction formula for double and triple integrals. Changing variables.
VECTOR FUNCTIONS - CURVE AND SURFACE - CURVE AND SURFACE INTEGRATION
Curves in parametric form. Regular curves. Generally regular curves. Straight tangent to a curve. Derivative of a scalar field that can be differentiated along a regular curve. Length of a curve. Integral of scalar fields along a curve. Mass center and moments of inertia of a curve. Integral of a vector fields along a curve. Concept of work. Surfaces in parametric form. Regular surfaces. Tangent plane. Orientation. Area of ??a surface. Integral of a scalar field over a surfaces. Center of mass and moments of inertia of a surface. Integral of a vector fields over a surface: flow integral. Divergence Theorem.
LINEAR DIFFERENTIAL FORMS
Linear Differential Forms. Exact differential forms (conservative fields). Closed Differential Forms. Primitive of exact differential forms (potential for a conservative field). Methods for calculating primitives of an exact form (determining the potential of a conservative field). Stokes theorem.