MATHEMATICAL ANALYSIS II

The goal is to calculate areas of surfaces and volumes of solids, polynomial expansions of some functions and to solve particular classes of ordinary differential equations.

This course prepares the student to the study of Fourier series and to the Fourier and Laplace transforms.

All the goals are to be considered of knowledge and understanding, applying knowledge and understanding.

Lectures in classroom.

1. Sequences and Series of Functions.
Sequences of real functions of real variable. Pointwise convergence, uniform convergence. Theorems of continuity , the passage to the limit under the integral sign and derivativation. Series of functions. Pointwise, uniform and total convergence. Theorems of continuity, integration by series. Real power series. Pointwise onvergence. Theorem of D'Alembert and Cauchy - Hadamard. Radius of convergence of the derivatives series. Theorems of differentiation and integration of power series. Taylor series. Criterion for the expansion in a Taylor series. Major series.
2. Real functions of two or more real variables.
Elements of topology in R^2 and R^3.
Bounded sets. Open connected sets.
Limits and continuity. Weierstrass theorem.
Partial derivatives. Successive derivatives. Schwartz theorem. Gradient. Differentiability.
Differentiability and continuity. Differential Theorem.
Composition of functions. Theorem of derivation of composite functions. Functions with zero gradient in a connected set.
Extremals. Necessary and sufficient conditions for an extremal.
Conditioned extremals. Lagrange multipliers.

2. Outline of ordinary differential equations and solution methods of some of them.
Position of the problem. Cauchy problem.
Property 'of the general linear equations.
linear differential equations of the first
order. Homogeneous linear differential equations
the second order.
Vectorial space and basis
Non-homogeneous second order linear differential equations. Method of variations of constant. Euler equation. Resolution of some types of differential equations of the first order in normal form. Equation with separable variables, Bernoulli equation. Linear equations of higher order.

3. Curvilinear integrals and differential forms in R^2 and R^3.
Regular curves. tangent vector and the normal vector of a smooth curve at a point.
Rectificability . Length of a smooth curve. Oriented curves. Arc length.
Curvilinear integral of a function. Differential forms.
Curvilinear integral of a differential form.
Exact differential forms. Integration theorem of exact differential forms. Characterization of the exact differential forms. Potential of a differential form. Closed differential forms. Differential forms in a rectangle. Differential forms in a simply connected open set of R2 and R3

4. Riemann integration in R^2 and R^3.
Normal domains in R^2. Integrability in normal domains. Reduction formulas for double integrals. Fubini-Tonelli's theorem. First theorem of Guldinus. Gauss - Green formulas. The divergence theorem. Stokes' formula. Integration by parts formulas. Formulas for the area calculation. Change of variables in double integrals. Triple integrals. Change of variables in triple integrals.

5. Surfaces and surface integrals.
Regular surfaces. Area of ​​a smooth surface. Surface integrals.

6. Basics on Fourier series. Trigonometric polynomials. Trigonometric series. Convergence in L^2 of Fourier series.

[1] Bramanti, C. Pagani, S. Salsa, Analisi Matematica due, Zanichelli.

[2] N. Fusco, P. Marcellini, C. Sbordone, Analisi Matematica due, Liguori Editore.