27213 - Mathematical Analysis 2

Learning outcomes

Students will acquire a mathematical knwledge of informative character using an ample basic instrumentation to describe different physical phenomena. In particular, at the end of the course, students will be able to: solve problems of conditioned extremals, compute simple integrals of functions in many variables; to compute simple integrals of functions defined on surfaces.

Course contents

Differential Calculus in more than one variable Introduction to topology in metric spaces.

Functions from R^n to R^m . Limits and continuity. Bolzano and Weierstrass theorems.

Functions from R^n to R:partial derivatives, directional derivatives,properties of the gradient function, higher order derivatives, Hessian, Schwarz lemma, Taylor formula at second order, tangent plane.

Functions with vector values: the Jacobian, the Jacobian of composite functions.

Applications of differential calculus Local minima and maxima. Fermat theorem. Definition and classification of quadratic forms associated to symmetric matrices, Sylvester theorem, classification of critical points: necessary/sufficient conditions for C^2 functions.

Regular varieties in implicit form. Normal and tangent spaces. Dini theorem and the local parametrization of a variety. Conditioned extrema. Fermat theorem in such case. Lagrange multiplier theorem.

Measure and integration Peano-Jordan measure. Riemann integration for function from R^n to R. Properties of integration: additivity, linearity, monotonicity. The integral average theorem. Reduction theorems for double and triple integrals in normal domains. Cavalieri principle and Cavalieri theorem. The change of variable in the integral. Polar, spherical and cylindrical coordinates.

Curves in parametric form and curvilinear integrals Regular curves. Piecewise regular curves. Orientation. Curvilinear integrals on non oriented curves: length, curvilinear integral of a function (mass, barycenter, inertia moments). Vector fields and differential forms. Curvilinear integral of a differential form and work. Exact differential forms and conservative vector fields. Closed differential forms and irrotational vector fields. Potential of a conservative vector field. Poincarè lemma.

Parametric surfaces and integrals on surfaces. Regular surfaces. Tangent plane and normal vector. Orientation. Area of a surface and integration of scalas functions on non-oriented surfaces (mass, barycenter, inertia momenta). Regular surfaces with boundary. Canonical orientation of the boundary. Piecewise regular surfaces. Stokes theorem. Gauss theorem.

 

 

Readings/Bibliography

Simonetta Abenda : Analisi Matematica (Esculapio)
Simonetta Abenda: Esercizi di Analisi Matematica (Esculapio)

Notes for the parts not included in the text will be distributed to students.

 

S. Ahmad, A. Ambrosetti, A textbook for ordinary differential equations, Unitext series, vol. 73, Springer

C. Parenti, A. Parmeggiani, Algebra lineare ed equazioni differenziali ordinarie, 2a edizione, Springer.

E. Giusti: Analisi Matematica 2 (Boringhieri)

C.D. Pagani, S. Salsa: Analisi Matematica 2 (Zanichelli);

W. Rudin: Analisi Reale e Complessa (Boringhieri);

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Teaching methods

Traditional lessons and exercise classes

Assessment methods

Exam is in written form. Exam calls are publshed on Almaesami and are visible on the degree programme website. Students may partecipate in any exam call. They may refuse the mark only once.

It is obligeatory to enrol on Almaesami list for the exam call. The exam is in two parts to be held in the same call.

In the first part students solve 4 multiple choice exercises (volume, maxima and minima, constrained maxima and minima differential calculus on manifolds)  and and 1 exercise (flow).

The test grade is +3 (correct answer), 0 (answer not given). -.25 (wrong answer)

The computation of the flow is marked from 0 (answer not given or completely wrong) to 6 (correct and complete answer). 

The student who gets at least 7/18 is admitted to the second part.

In the second part they fully solve one exercise of the first part and answer to 2 theoretical questions. The grade for each exercise/theoretical question is from -3 (answer not given or completely wrong) to +7 (answer correct and complete).

Exercises at exam difficulty level shall be distributed during lessons and published on Virtuale.

Teaching tools

Screen shots of the lectures, exercises and solutions will be published on Virtuale. These have to be considered as auxiliary teahing tools and they are not intended to substitute the active participation of students at lessons and their autonomous study.

Students are encouraged to take notes during lessons, to check and study them, to compare them to texts and t solve exercises autonomously.

Students are strongly encouraged to ask questions to the professor during lessons and not just at office hours. Indeed the main teaching tool is the active participation of students to lessons both on-line and in presence.

Office hours

See the website of Simonetta Abenda