## Learning objectives

The aim of the course is to provide knowledge and abilities related to improper integrals, ordinary differential equations theory, curves theory, differential and integral calculus in several real variables.

At the end of the course the student is expected to be able

D1 - Knowledge and understanding:

To know

- improper integral theory

- ordinary differential equations theory

- curves theory

- basic notions of functions of several real variables: law, domain, zeros and sign, level sets, graph

- general equations and features of all surfaces showed in the course: plane, paraboloid of revolution, cone of revolution, half-spherical surface

- topology theory (neighbourhoods, interior, exterior, boundary of a set, open, closed, bounded and compact sets)

- continuous functions and Weierstrass Theorem

- basic notions of differential calculus for functions of several real variables: partial derivatives, gradient, tangent plane, differentiability, directional derivatives, higher order derivatives

- Total Differential and Schwartz Theorems

- definitions of local and absolute extreme points and saddle point

- free and constrained extrema theory: critical points, Fermat Theorem, sufficient conditions, Lagrangian multipliers

- multiple integral Theory: definition, geometric meaning, reduction theorems, change of variables, center of mass.

D2 - Applying knowledge:

Being able to

- evaluate the convergence or divergence of an improper integral

- solve an ordinary differential equation or a Cauchy problem

- recognize and draw the support of a plane curve, determine and draw tangent and normal vectors and unit vectors, determine tangent and normal lines equations and the length of the curve

- determine for a curve in space the tangent line equation at a point, the plane perpendicular to this line and the length of the curve

- solve a two variables inequality

- write the parametric equations of a given curve and of the boundary of a given set

- determine and draw domain, zeros, sign and level sets of a function of two real variables

- write graph equation, recognize the surface given by the graph and draw it

- draw a solid in space

- determine the interior, the exterior and the boundary of a set, recognize an open, closed, bounded or compact set

- compute partial derivatives, gradient, tangent plane, directional derivatives and higher order derivatives of a function of several real variables

- prove the differentiability of a function

- determine critical points of a function and their nature

- apply Weierstrass Theorem to prove the existence of the extrema of a function

- determine the extrema of a function

- apply Lagrangian multipliers

- compute a multiple integral and the volume of a solid

- determine the center of mass.

D3 - Making judgments:

Being able to

- understand the mathematical machinery employed in non-mathematical courses

- check the credibility of the results

- deal with a new problem and plan its solution

- organize work in a precise way.

D4 - Communicating skills:

Being able to communicate mathematical contents, even outside of an exclusively applicative context.

D5 - Learning skills:

To have acquired a good grounding in mathematical analysis to face, in the future, an autonomous analysis of possible applications in a study or in a project.