## Learning objectives

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.