Università Roma Tre

1. Functions of n real variables
Vector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn.

Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is ​​positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.

2. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces.

3. Implicit functions
The theorem of implicit and Inverse functions.
Constrained maxima and minima, Lagrange multipliers.

4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with matrix exponential).
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.

5. Riemann integral in Rn Review of the Riemann integral in one dimension. Rectangles
in R2, compact support functions, simple functions and their integral, function definition integrable according to Riemann in R2 (hence Rn).
Definition of measurable set, a set is measurable if and only if its boundary has zero measurement. Normal sets with respect to the Cartesian axes. A continuous function on a measurable and integrable set. Fubini reduction theorem.
Formula of change of variable in integrals (without size). Polar, cylindrical, spherical coordinates. Examples: calculation of some barycenters and moments of inertia.

6. Regular curves.
Regular curves in R ^ n. Tangent versor.
Two equivalent curves traveled in the same direction have the same tangent versor.
Length of a curve. It is greater than the displacement.
Two equivalent curves have the same length.
Curvilinear integrals.

7. Surfaces, flows and divergence theorem.
Recalls on the vector product. Definition of regular surface. Tangent plane and normal versor. Area of ​​a surface. Examples: graphs of functions and rotation surfaces. Surface integrals. Flow of a vector field through a surface. Examples. Statement of the divergence theorem. Demonstration of the divergence theorem (for normal domains with respect to the three Cartesian axes.

8. Differential forms and work.
1-Differential forms. Integral of a 1-differential form (work of a vector field), closed and exact forms. A form is exact if and only if the integral on any zero closed curve. Example of incorrect form closed.
Derived under the sign of integral. Starry sets; a closed form on a starred domain is exact.
Irrational and conservative fields, solenoidal and potential vector (on starry sets). The Green theorem in the plane. The Rotor theorem.

9. Series and sequence of functions

Series and sequence of functions: point, uniform and total convergence. Continuity of the limit, integration and derivation of uniformly convergent sequences of functions. Power series: convergence radius. Taylor series examples of elementary functions.

10. Fourier series
Fourier series, Fourier coefficients. Properties of Fourier coefficients, Bessel inequality, Lemem of Riemann Lebesgue. Pointwise convergence of the Fourier series (Dini test). Uniform convergence in the case of C1 functions. Equality of Parseval.

Analisi Matematica II, Giusti
Analisi Matematica II, Chierchia