MATHEMATICAL ANALYSIS II
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- Versione italiana
- Academic year
- 2015/2016
- Teacher
- DAMIANO FOSCHI
- Credits
- 9
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The course continues the study of calculus tools and mathematical analysis theory started with the course of Analisi Matematica 1.
The main goal is to teach the basic elementary techniques of differential and integral calculus for functions of several real variables and resolution methods for simple ordinary differential equations. The acquisition of these mathematical tools is considered essential in order to face successfully the subsequent technical teachings.
The main acquired knowledge will be:
* introduction to the theory of scalar ordinary differential equations, in particular first order equations with separable variables and linear second-order equations with constant coefficients;
* theory of pointwise and uniform convergence and passage to the limit properties for sequences and serie of functions, in particular power series, Taylor's series, Fourier's series;
* differential calculus for functions of several real variables: partial derivatives and differentiability, search for critical and extremal points;
* elementary differential geometry of regular curves and surfaces in the plane and in space;
* definition and computation of Riemann's integral for functions of several variables;
* definition and computation of curvilinear and surface integrals, applications of the theorems of Gauss-Green and of Stokes.
* introduction to the theory of Lebesgue measure and integral and its passage to the limit properties.
The basic acquired abilities (that are the capacity of applying the acquired knowledge) will be to be able to:
* solve first order ordinary differential equations which can be treated with the method of separable variables;
* solve homogeneous and non homogeneous scalar linear differential equations with constant coefficients;
* study pointwise and uniform convergence for sequences and series of functions, in particular power series;
* compute the Taylor's series of a differentiable function and the Fourier series of a periodic function;
* compute partial derivatives and first order approssimations of differentiable functions of several real variables;
* determine and classify free or constrained critical points of a function of several variables;
* determine extremal values of a function on a given domain;
* determine tangent, normal and binormal versors, and curvature and torsion values for a regular parametrized curve;
* determine the tangent plane and the normal versors for a regular parametrized surface;
* compute double and triple integrals, also exploiting different coordinate systems;
* compute curvilinear integrals of the first and of the second kind, surface integrals, flux integrals. Prerequisites
- All contents of the course of Mathematical Analysis 1 are preliminary. Including in particular:
* Elementary functions.
* Differential and integral calculus in one variable.
* Numerical sequences and series.
Also required is the knowledge of:
* Basic linear algebra: linear maps, matrices, determinants, vector product and scalar product.
* Basic geometry: straight lines, planes, conic sections. Course programme
- - Ordinary differential equations: separation of variables, first order linear equations, constant coefficients secondorder equazions. Cauchy problem and existence and uniqueness theorems.
- Sequences and series of functions: uniform convergence and passage to the limit, power series, Taylor series.
- Multivariable functions: continuity, partial derivatives, gradient, directional derivates, differentiability, critical points, Hessian matrix, Jacobian matrix. Implicit function theorem in R^2 and in R^3, Lagrange multipliers method. Free and constrained maxima and minima.
- Regular curves: length of a curve, arc length parameterization, tangent, normal and binormal versors, curvature and torsion. Regular surfaces in R^3, tangent plane, normal versor, area of a surface, orientable surfaces.
- Riemann integral of multivariable functions: double and triple integrals, change of coordinates formula, integrals using polar, cilindrical, spherical coordinates, generalized integrals. Surfaces and solids of revolution, Guldino's theorems.
- Curvilinear integrals of the first kind. Vectorfields and differential forms, curvilinear integrals of the second kind, work integral of a vector field along a curve. Closed differential forms, exact differential forms, conservative vectorfields, potentials, simply connected sets. Green's formulae, Gauss-Green theorems in the plane. Surface integrals, flux integrals, divergence theorem and Stoke's theorem in R^3.
- Lebesgue's measure and integral, measurable sets, sigma-additivity properties. Zero measure sets and properties verified almost everywhere. Passage to the limit under Lebesgue integral: monotone convergence theorem, Fatou's lemma, dominated convergence theorem. Approximation by the least squares line for square integrable functions.
- Fourier series: periodic function, elementary harmonic functions, trigonometric polynomials, Fourier coefficients, Bessel's inequality, piecewise regular functions, Dirichlet kernels, convolution products, pointwise convergence of Fourier series to the regularized value of a function. Didactic methods
- Classroom lectures with presentation at the blackboard of the theoretical aspects, the applications and exercises.
Periodical tutoring sessions with exercises and review of the course topics. Learning assessment procedures
- The learning assessment of the course content is performed through a written exam and a oral exam.
- During the written exam the student shall solve problems and exercises about the course topics: differential equations, function sequences and series, differential calculus of multivariable functions, maxima and minima of multivariable functions, double and triple integrals, curvilinear integrals, surface integrals, Fourier series. The time allowed for the written exam is about 3 hours. It is not allowed to make useof any textbook, PC, tablet or smartphone. The student may consult a handwritten sheet (A4) made by himself/herself, on which he/she can annotate whatever he/she wants. The student may use pocket scientific calculators, with no graphic capabilities. A score up to 30 points is assigned to the written exam, the exam is passed with a score of at least 15 points. To be admitted to the oral exam the student must first pass the written exam.
- During the oral exam the student will be asked to present aspects of the course topics, by illustrating some definitions, examples, properties, formulae, theorems, proofs, or applications. More than a mnemonic knowledge, we want to evaluate the logic understanding of the various concepts, the rigor and precision of the mathematical language used, and the ability to connect abstrat theory with applications. The time allowed for the oral exam is about 30 minutes. If the student doenot passtheoral exam, he/she may try it again at a later date and is not required to repeat the written exam.
The final grade is proposed at the end of the oral exam and takes into account the active participation of the student during lectures and/or tutoring sessions, the correctness and completeness of the solution of the written exam, the quality of the exposition during the oral exam. Reference texts
- Reference text:
* M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica (Mc Graw Hill)
Recommended texts for study:
* V. Barutello, M. Conti, D.L. Ferrario, S. Terracini, * G. Verzini: Analisi Matematica. Con elementi di geometria e calcolo vettoriale: 2 (Apogeo)
* E. Giusti: Analisi Matematica II (Boringhieri)
* E. Giusti: Esercizi e complementi di Analisi Matematica II (Boringhieri)
* W. Rudin: Principi di Analisi Matematica (Mc Graw Hill)
* G. De Marco: Analisi Matematica II (Zanichelli )
* G. De Marco: Esercizi di analisi Matematica II (Zanichelli)
* S. Salsa, A. Squellati: Esercizi di Analisi Matematica II (Zanichelli)
* E.H. Lieb, M. Loss ; Analysis (American Mathematical Society)
* E. Stein, R. Shakarchi: Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton University Press)
