27213 - Mathematical Analysis 2 (A-L)

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

Metric spaces: the Eucledean space R^n. Generalities on metric spaces. Limits and continuous functions between metric spaces. Generalizations of the theorems of Weierstrass an Bolzano. Complete metric spaces. The theorem of contractions. An application to the Cauchy problem for ordinary diffrential equations.

Differential calculus for functions of several real variables. Derivatives with respect to vectors. partial derivatives. Differentiability. Functions of class C^1. Mean value theorem. Higher order derivatives. Functions of class C^k. Theorem of Schwarz. Taylor formula for functions of several real variables. Critical points and their study. Chain rule. Implicit functions. Manifolds. Theorem of the multipliers of Lagrange.

Measure and integration of functions (Introduction to the Lebesgue measure). Measurable sets. Simple functions. Measurable functions. Integration of nonnegative measurable functions. Summable functions. Comparison betweem the integrals of Riemann and Lebesgue in dimension one. Theorems of Fubini and Tonelli. Change of variable. Remarkable changes of variable.

Paths and line integrals. Length of a path. Equivalent paths. Line integrals of first kind. Line inegrals of second kind. Vector fields and potentials. Conservative vector fields. Theorem of Poincaré. Cental vector fileds.

Regular plane domains. Gauss-Green fomulas in the palne. Scalar product in R^3. Regular surfaces. Area of a surface and surface integrals. Tangent space and normal space to a surface in a point. Orientation of a regular surface. Regular domains in R^3. Normal vectors to a surface in a point. Gauss-Green formulas in R^3. Stokes formula

Readings/Bibliography

Lecture notes

Suggested readings:

G. Molteni, M. Vignati; Analisi Matematica 3 (Cortina)

E. Giusti: Analisi Matematica 2 (Boringhieri)

W. Rudin: Principles of Mathematical Analysis (McGraw-Hill) 3a edizione

Davide Guidetti, Analisi Matematica B (2024)

Teaching methods

Blackboard lectures

Assessment methods

Exercise and theory exam, both written, to be passed in the same exam session: first the exercise part, then the theory part. The exercises have to be solved explaining how the solution os found, justifying the steps. The student can take the theory exam if she/he has a grade not inferior to 18/30 in the exercise exam.

Teaching tools

Lecture notes on Virtuale

Office hours

See the website of Nicola Arcozzi