27213 - Mathematical Analysis 2 (A-L)

Learning outcomes

The student

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. 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

Notes in preparation by the teacher.

Teaching methods

Traditional lecures with blackboard

Assessment methods

Examination made of a written and an oral part, both compulsary. Admission to the oral part subordinated to a minimal score in the written. 

The written part consists in four exercises. In each of them the student must justify the results obtained.

Teaching tools

As already written, notes of the teacher and blackboard.

Office hours

See the website of Davide Guidetti