Learning outcomes of the course unit
To provide the basic instruments of the differential calculus for multivariable functions, particular object is the optimization of multivariable functions.
To provide the basic instruments for the calculus of integrals of arc.
To provide the basic instruments for the calculus of multivariable integrals and of surface integrals.
Course contents summary
1. Multivariable real functions. Topology. Limits. Continuity. Partial derivatives, directional derivatives. Differentiability. Tangent plane, normal versor. Higher-order partial derivatives. Schwarz's theorem (n.p). Higher-order differentiability. Hessian matrix. Taylor's formula (n.p).
2. Optimization : free extrema. Weierstrass' theorem (n.p.). Critical points. Sign of the quadratic forms. S.C. of relative max/min extrema.
3. Implicit functions. Dini's theorem (n.p.).
4. Optimization : extrema with side conditions. Lagrange's theorem.
5. Vector valued functions. Jacobian matrix.
6. Curves in parametric form. Equivalence of paths. Change of parameter. Regular paths. Rectifiable paths and arc lenght. Line integral of a scalar function. Line integral of a first order differential form. Exact forms.
7.8. Riemann's integrals for functions of 2or 3 variables. Jordan's measurable sets. Evaluation of a multiple integral by iterated integrations (n.p). Change of variables in a multiple integral (n.p.).
9. Improper integrals.
10. Gauss' theorem. Divergence theorem and Stokes' theorem for 2 variable functions.
11. Surfaces in parametric form. Regular surfaces. Change of parameter. Surface integrals and area of a surface. Orientation of a surface.
12. Divergence theorem and Stokes' theorem for 3 variable functions.
1. C.D. Pagani e S. Salsa, 'Analisi Matematica 2', ed. Masson.
2. Appunti del docente reperibili al centro fotocopie del Dipartimento di Fisica.