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. Equivalence 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), one-dimensional conservative systems.
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.

Analisi Matematica II, Giusti- Analisi Matematica II, Chierchia

The numbers of the paragraphs and theorems refer to the book of Chierchia or the book of Giusti
(in this case indicated with [G]).

1. Riemann integral in Rn
Review of the Riemann integral in one dimension ([G], par. 12.1). Rectangles in R2,
functions with compact support, definition of
integrable function according to Riemann in R2 (hence Rn).
Definition of measurable set ([G], Def. 12.3), a set is measurable if its
frontier measures nothing ([G], Prop. 12.1). Normal sets with respect to the Cartesian axes.
A continuous function on a measurable set is integrable ([G], Teo. 12.1). Theorem
Fubini ([G], Teo. 12.2).
Formula of the change of the variable in the integrals (scheme of proof) Polar coordinates,
cylindrical, spherical. Examples: calculation of some centers of gravity and moments of inertia.

2. Curves, surfaces, flows and divergence theorem.
Recall on the vector product. Examples of varieties. Regular curves and regular surfaces. ca
Coordinate changes. The length of a curve. Definition of regular surface ([G], Def. 15.4).
Tangent plan and normal unit. Area of ​​a surface ([G], Def. 15.6).
Surface integrals. Flow of a field
vector across a surface. Examples. Statement of the divergence theorem.
Proof of the divergence theorem (for normal domains in R ^ 2).
The Rotor theorem (shown for normal domains in R ^ 2).

3. Differential forms and work ([G])
1-Differential forms Integral of a differential 1-form (work
of a vector field), closed and exact forms. An exact form if and only if the integral
on any closed curve zero. Example of a closed form which is not exact.
Simply connected sets. A closed form on a simply connected set isd exact.
Starred sets.
closed forms on a star domain.


4. Series and sequences of functions ([G])
Series and sequence of functions: punctual, uniform and total convergence.
Continuity of the limit, integration and derivation of sequences of functions uniformly
converging. Power series: convergence radius. Examples of
Taylor series of elementary functions.

5. Fourier series ([G])
Fourier series, Fourier coefficients. Properties of Fourier coefficients, inequality
by Bessel, Lemma by Riemann Lebesgue Pointwise convergence
of the Fourier series. Uniform convergence in the case of C1 functions.
Parseval equality. The Fourier series of a piecewise C1 function converges to the jump average
discontinuity points. Linearity of the Fourier series.

6. Complements
Convolution and regularization (par. 3.2). Ascoli's theorem. Stirling formula. The real analytic functions.

Analisi Matematica II, Giusti- Analisi Matematica II, Chierchia