Università Roma Tre

Scalar products and symmetrical bilinear forms. The standard scalar product on R^n. Examples. The matrix associated with a symmetrical bilinear form. Positive definite matrices and the criterion of the main minors. Examples.
Euclidean vector spaces. Length and versor of a vector. Orthogonality between vectors and orthogonal space to a vector. Orthonormal bases. Gram-Schmidt procedure.
Formula of transition from one base to another for the matrix of a scalar product. Schwarz's inequality. Triangular inequality.
Convex angle between two vectors. Orthogonal space to a vector or a subspace. Projection of a vector in the direction of a non-zero vector. The notion of quadratic form. The polar symmetrical bilinear form of a quadratic form. Orthogonal operators and orthogonal matrices.
Self-oadjoint operators and properties. Quadratic form associated with a self-adjoint operator. Eigenvalues ​​of a symmetric matrix. Spectral theorem of self-adjoint operators.
Spectral theorem of self-adjoint operators: procedure for diagonalizing a symmetric matrix. Exercises. Canonical forms of quadratic forms.
Sylvester's theorem. Affine geometry: coordinate systems in the plane. Lines in the Cartesian plane: Cartesian equation and parametric equation.
Affine geometry in the plane: Intersection of straight lines, parallelism and orthogonality, straight point distance, convex angle between two straight lines, sheaves of proper and improper lines.
Geometry in Cartesian space. Vector product and mixed product. Parametric and Cartesian equations of planes and lines.
Switch from parametric to Cartesian equations of planes and lines in the Cartesian space. Plane through 3 non aligned points. Intersections plane / plane, plane / line and line / line in space. Sheaves of proper planes.
Sheaves of improper planes, Affinity and isometry in a similar space. Examples. Conics in the Cartesian plane. Examples. Affine and metrical classification of conics. Fundamental theorem of affine-like classification of conics in the plane.
Similar properties and metrics of conics. General and degenerate conics, center conics and parables. Ellipses, parables and hyperbole: examples. Affine and metric classification of conics.
Reduction to the canonical and metric forms of conics. Examples.
Symmetries in the plane with respect to a point and a line. Center of symmetry of a conic in the center. Projective spaces: definition and examples.
Real projective plan. Lines and conics in the projective plane. Polarity with respect to a conic of the projective plane.
Generalities on differential equations. Differential equations of the first order. Differential equations with separable variables.
Linear differential equations of the first order. Linear differential equations with constant coefficients of order n.
Linear differential equations with constant coefficients of the second order. Method of constants variation. Vector functions: generalities and examples. Continuous curve arch.
Derivative of a regular function and arcs of regular curves. Curves in polar coordinates and reparametrizations. Length of a regular curve arc.
Arc parameter. Line integrals and applications for the calculation of surface areas, mass, center of gravity and moment of inertia relative to an axis of a material line. Elements of differential geometry: tangent, normal, curvature, and osculating circle. Binormal and the Frenet-Serret formulas.
Osculator plane to a curve, decomposition of acceleration and proof of Frenet-Serret formulas. Plane curves and their torsion. Functions in several variables: examples of graphical representation, limits and continuity. Examples.
Elements of topology of R ^ n: internal, external, border points, open, closed and connected sets. Open and closed sets defined by continuous functions, Theorem of Wierstarss and Theorem of the zeros. Partial derivatives, derivability and gradient. Examples.
Tangent plan and differentiability. Directional derivatives. Derivatives of a higher order and Schwarz's theorem. Free optimization: critical points and the criterion of the Hessian matrix. Double integrals: integration on rectangular domains, x-simple and y-simple sets. Conditions of integration on regular domains. Examples. Change of variables in the integral and integration in polar coordinates. Calculation of the volume of the sphere.
Vector fields and line integrals of second species. Gradient, rotor and divergence. Work of a vector field along a curve and circuits. Conservative and potential fields.
Conservative and potential fields: examples. Conservative and irrotational fields. Simply connected, convex and star like sets.

F. Flamini, A. Verra: Matrici e vettori. Corso di base di geometria e algebra lineare. Carocci.
M. Bramanti, C.D. Pagani, S. Salsa: Analisi Matematica 2. Zanichelli.