Università Roma Tre

Review of integrals. Simple differential equations solvable by direct integration. First order linear equations, homogeneous and non-homogeneous. Bernoulli equation. Equations with separable variables. Linear equations of any order. Linear equations with constant coefficients. Non-homogeneous linear equations: method of indeterminate coefficients and method of variation of constants. Special cases of differential equations with variable coefficients: Euler equation, lowering of order.
Notes on natural topology in R^n. Functions of several variables. Limits and continuity of functions of several variables. First order and second order partial derivatives. Schwarz's lemma. Relative maxima and minima of functions of several variables. Use of the Hessian determinant to determine the nature of critical points. Absolute maximum and minimum of functions of two variables in compact sets.
Notes on the Peano-Jordan measurement theory in R^n. Multiple integrals. Normal domains in the plane and in space. Reduction formula for double and triple integrals. Changes of variables in multiple integrals. Special cases: linear change, polar coordinates and generalizations, spherical coordinates and generalizations. Applications of multiple integrals: centroids, moments of inertia.
Length of an arc of a curve. Regular curves. Curve integrals of scalar and vector fields. Conservative fields. Green's theorem in the plane.
Regular surfaces. Area of ​​a regular surface. Surface integrals of scalar and vector fields. Stokes' theorem. Green's theorem in space.

Differential equations and functions of several variables: notes provided by the teacher
Integrals: B. Palumbo, "Integrali di funzioni di più variabili" (in Italian), ed. Accademica, 2nd edition (2009)