Università Roma Tre

- COURSE SYLLABUS 2017-18

-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.

-HANDS-ON: CONSTRUCTION OF POLYHEDRA, RULED SURFACES, paper models

ANY TEXT AT THE LEVEL OF THE SECOND AND THIRD SEMESTER OF A THREE-SEMESTER COLLEGE CALCULUS.

Linear algebra
Two-dimensional and three-dimensional vector spaces. Vectors and versients in the three-dimensional plan and space. Scalar product, vector product, mixed product, with their geometric meanings.
Straight and plane equations in space in parametric and Cartesian form; parallel lines, accidents, sghembe; intersections between straight and plane; distance between points in R3, point-to-point distances, point-plane, straight-plane, parallelism between floors. Matrices of order greater than 2: sum of matrices, product lines for columns between matrices of any size or between matrix and vector, algebraic complements, determining a square matrix of any order (Laplace development), inverse matrix.
Curves and surfaces in space
Cartesian form (explicit and implicit) of the equation of a surface. Parametric Equations of Curves and Surfaces. Examples of curves in space, planes and sghembe. Striped surfaces, with particular regard to cones and cylinders. Rotation surfaces. Quadriche: quadruple cones and cylinders, spheres, ellipsoids, paraboloids, hyperboloids. Cartesian equations, sections, level curves; reconstruction of a quadruple from the sections. Double-sided surfaces. Parametric equations of curves in space give as intersection of two surfaces.

Infinite calculus for functions of multiple variables
Real functions of two or more real variables. Domain; function flat representations z = f (x, y): level curves, sections, and their design. Surfaces with free (or cylindrical) variables. Open and closed sets, internal, external, border, isolated blocks.
Limits and continuity for functions of multiple variables. Counterexamples. First-order and superior order partial derivatives. Directional derivative. Differentiability. Normal tangent and straight plane. Gradient of a function, relationship between the gradient and other geometric aspects of the surface: level curves, tangent plane, maximum slope direction. Narrate Taylor's formula in multiple variables. Study of the nature of the critical points of functions of two variables by the Hessian determinant: maximum, relative minimum and saddle points.
Integral calculation for functions of multiple variables
Multiple integers in Rn; normal domains in the plane (ie "vertically and horizontally simple"). Reduction formula for double integers on normal domains; inversion of integration order; applications for area and volume calculation; applications for calculating masses and barycentre of flat lamina.
You can add another one to develop yourself, such as curve and surface insights, rigid surface construction, other differential and integral calculation applications for multiple variable functions, with the teacher's agreement.
Google Traduttore per le aziende:Translator ToolkitTraduttore di siti web

R. Adams: Calcolo Differenziale 2 (funzioni di più variabili), quinta edizione, CEA - Casa Editrice Ambrosiana, Milano, 2014 (va bene però anche un'edizione precedente).
B. Palumbo: Integrali di funzioni di più variabili (per la parte di curve e superfici nello spazio e per gli integrali doppi).