Learning objectives
<br />To provide the basic instruments of the differential calculus for multivariable functions, particular object is the optimization of multivariable functions.<br />To provide the basic instruments for the calculus of integrals of arc.<br />To provide the basic instruments for the calculus of multivariable integrals and of surface integrals.
Prerequisites
<br />Calcolo I<br />Calcolo II
Course unit content
<br />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).<br />2. Optimization : free extrema. Weierstrass' theorem (n.p.). Critical points. Sign of the quadratic forms. S.C. of relative max/min extrema. <br />3. Implicit functions. Dini's theorem (n.p.).<br />4. Optimization : extrema with side conditions. Lagrange's theorem.<br />5. Vector valued functions. Jacobian matrix.<br />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.<br />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.).<br />9. Improper integrals.<br />10. Gauss' theorem. Divergence theorem and Stokes' theorem for 2 variable functions.<br />11. Surfaces in parametric form. Regular surfaces. Change of parameter. Surface integrals and area of a surface. Orientation of a surface.<br />12. Divergence theorem and Stokes' theorem for 3 variable functions.
Full programme
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Bibliography
1. C.D. Pagani e S. Salsa, 'Analisi Matematica 2', ed. Masson.<br />2. Appunti del docente reperibili al centro fotocopie del Dipartimento di Fisica.
Teaching methods
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Assessment methods and criteria
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Other information
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