Università Roma Tre

Linear first-order differential equations. General first order differential equations. Cauchy problem: local existence and uniqueness. Separation of variables. Systems of first-order equations: linearly independent solutions and Wronskian matrix. Variation of constants. Differential equations with constant coefficients and characteristic polynomial. Matrix exponential and computation for diagonalizable matrices. Some remarkable differential equations: Euler equation and Bernouilli equation.

Norm and distance in R^n. Functions of several variables. Continuous functions and Weierstrass theorem.
Partial derivatives, directional derivatives and gradient. Functions of class C^1 and C^2. Higher derivatives, Hessian matrix and Schwarz theorem. Differentiation of composed functions. Taylor expansion. Local maxima and minima. Method of Lagrange multipliers to compute local maxima and minima.

Riemann integration and Peano-Jordan measure. Integration of continuous functions, reduction formulae
and iterated integrals; area and volume. Change of variables in integrals and Jacobian matrix: polar, cylindrical and spherical coordinates. Gaussian integral.

Curves in R^n: parametrized and equivalent curves; length of a curve; curve integrals of scalar functions. Work and curve integrals of vector fields. Smooth surfaces in R^3: area of a surface and integrals on surfaces.

Teoria ed esercizi: Bertsch, Dal Passo, Giacomelli, Analisi Matematica , McGraw Hill, II edizione,
& Gentile, Introduzione ai Sistemi Dinamici Volume 1, Springer.
Esercizi: Marcellini, Sbordone, Esercitazioni di Analisi Matematica Due (vol. I e vol. II), Zanichelli ed.