# Mathematical Analysis II D

## Objectives

The aproved student should be abble to study regularity of functions of several variables in their domains, determine maximums and minimums values by identifying their extremals and to compute line, area, surface and volume integrals.

Also de student should know and apply the classical results on the convergence of numerical series.

## General characterization

##### Code

10572

##### Credits

6.0

##### Responsible teacher

Ana Maria de Sousa Alves de Sá, Cláudio António Raínha Aires Fernandes

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

The student should have the required skills of aproval to basic Analytical Geometry, Analysis 1 and Linear Algebra.

### Bibliography

Adams, Robert A.; Essex, Christopher, Calculus - A Complete Course, 9th edition, Pearson, 2018.

Marsden, Jerrold.; Tromba, Anthony, Vector Calculus, 6th edition,

W. H. Freeman and Company Publishers, 2012.

Marsden, Jerrold.; Weinstein, Alan, Calculus III, 2nd edition, Springer, 1985.

Thomas, George B;, Weir, Maurice D.; Hass, Joel et al. - Thomas Calculus in SI Units, Pearson, 2016

### Teaching method

Our teaching methos is based upon a dual approach, theoretical and pratical. In the theoretical sessions, concepts are introduced providing their motivation as well as its applications in the context of engineering. Problem solving sessions will allow the revision of the fundamental theoretical notions, as well as develop student''''''''s operational skills in view of their utility in a professional context.

Besides bibliography, students will have the support of a theoretical-pratical text (available on CLIP) as well as of an attending schedule where individual questions and dificulties can be treated.

### Evaluation method

Continuous evaluation.

In order to be eligible for continuous evaluation, the student may attend 2/3 of pratical sessions. The continuous evaluation method is based upon three tests. The student is aproved if the grade average is greater or equal to 9.5 and in the third the grade is greater or equal to 7.0. In the final exam date, the student may optionally improve one of the tests. If an approved student desires to improve the grade by exam he must express his intention to the responsible teacher so that the aproval grade is previously registered.

Exam evaluation

Any student inscribed in the discpline may attend to the exam, being aaproved if the final grade is greater or equal than 9.5.

If necessary, the responsible teacher may individually require an extraordinary evaluation.

## Subject matter

1. Topological notions in R^{n}

1.1 Norms and metrics

1.2 Topological notions in R^{n}

2. Functions of several variables

2.1 Real Functions of several variables

2.2 Vector valued functions

2.3 Limits and continuity

3. Differential calculus in R^{n}

3.1 Partial derivatives. Schwarz theorem

3.2 Diferential of a function

3.3 Directional derivatives

3.4 The Chain rule

3.5 Taylor''''s Formula

3.6 Implicit differentiation

3.7 Inverse functions

3.8 Maximum and minimum values

3.9 Lagrange Multipliers

4. Multiple integrals. Vector Calculus

4.1 Double integrals

4.2 Iterated integrals: Fubini''''s Theorem

4.3 Double integrals in polar coordinates

4.4 Applications of double integrals

4.5 Surface area

4.6 Triple integrals

4.7 Triple integrals in cylindrical and spherical coordinates

4.8 Change of variables in multiple integrals

4.9 Vector fields

4.10 Line integrals

4.11 The fundamental theorem for line integrals

4.12 Green''''s Theorem

4.13 Curl and divergence

4.14 Parametric surfaces and their areas

4.15 Surface integrals

4.16 Stokes''''s Theorem

4.17 The divergence theorem

5. Series

## Programs

Programs where the course is taught: