Università Roma Tre

Sets of points in the plane or in the three-dimensional space. Vector space in two and three dimensions. Vectors and unit vectors.
Scalar product, vector product and scalar triple product with their geometrical interpretation. Matrices and determinants.

Parametric and cartesian equation of a plane. Parametric and cartesian equation of a straight line in space. Point-line distance. Line-line distance.
Intersections between stright lines and planes. Intersecting, parallel and skew lines.

Quadric surfaces. cylinders, cones, ellipsoids, paraboloids and hyperboloids. Contour lines and sections. Ruled surfaces.

Vector functions and parametric curves. Examples of parametric curves: lines, conics, spirals and cycloids. Tangent, normal and binormal unit vectors.
Frenet formulas. Curvature and torsion. Curves on surfaces. Cylindrical helix.

Functions of two variables. Domain of definition. Graph of a function. Contour lines and sections. Limits and continuity for two variables functions.
Partial derivatives. Tangent plane on a point to the function graph. Directional derivative. Differentiability. Gradient of a two variables function.
Geometric properties. Maximal slope direction. Higher order derivatives.

Critical points of a two variables function. Second order derivatives matrix and its Hessian determinant. Maxima, minima and saddle points.

Visualization of curves and surfaces using the software Mathematica or Python.

A mathematical topic at a choice, to be developed autonomously, from a given list of lectures.

R. Adams “Calcolo Differenziale 2, (funzioni di più variabili)”, quarta edizione, ed. casa editrice Ambrosiana

or a choosen textbook at university level, for example:

Bramanti-Pagani-Salsa: "Calcolo infinitesimale e algebra lineare", Seconda edizione, ed. Zanichelli

G.B. Thomas, R.L. Finney “Analisi Matematica”, ed. Zanichelli

-MATHEMATICAL MODELS FOR HANDLING 3d GEOMETRICAL SPACE:
-LINEAR ALGEBRA FROM A GEOMETRIC VIEWPOINT, VECTORS,
- PLANES, LINES, SKEW LINES, DISTANCES.
-CONICS , QUADRIC SURFACES: IDENTIFICATION, CLASSIFICATION, CONSTRUCTABILITY
IDENTIFICATION AS RULED, AS DEVELOPPABLE, AS SECTIONS...
-DIFFERENTIAL AND INTEGRAL CALCULUS OF TWO AND THREE VARIABLES.
EXTREMA AND CRITICAL POINTS OF A SURFACE GIVEN BY A FUNCTION, TANGENT PLANE.
- PARAMETRIC CURVES, Frenet–Serret frame of a curve.
- SUPERFACES IN SPACE,PARAMETRIC AND IMPLICIT FORMULATION .

- DOUBLE INTEGRALS, VOLUMES OF REGIONS BOUNDED BY REGULAR SUPERFACES.

ANY TEXT AT THE LEVEL OF THE SECOND AND THIRD SEMESTER OF A THREE-SEMESTER COLLEGE CALCULUS.
(several variables and curves)