# Mathematical Analysis II B

## Objectives

At the end of this curricular unit, students should have acquired knowledge, skills and competences that allow them to:

- Work with basic notions of topology in R^{n}.

- Understand the rigorous notion of limit, continuity and differentiability of vector functions of a real variable.

- Apply vector functions of a real variable in the parameterization and study of curves.

- Understand the rigorous notion of limit and continuity of real and vector functions of several real variables and compute limits.

- Know the notion of partial derivative and differentiability for functions of several real variables.

-Understand and apply the implicit function theorem and the inverse function theorem.

- Know Taylor''s formula and applications to the study of functions and its extremums.

- Know the notion of double and triple integral and how to compute these integrals using appropriate coordinates.

- Know some applications of double and triple integrals.

- Know the notion of line integral, its applications, and fundamental results.

- Know the notion of surface integral, its application and fundamental results.

## General characterization

##### Code

10476

##### Credits

6.0

##### Responsible teacher

Ana Margarida Fernandes Ribeiro

##### Hours

Weekly - 4

Total - 48

##### Teaching language

Português

### Prerequisites

Differencial and integral calculus on R. Basic knowledge of matricial calculus.

### Bibliography

Any multivariate analysis book can be helpful. Some examples:

Calculus; Anton, Bivens and Davis, Wiley (8th edition).

Cálculo, vol 2; Tom M. Apostol, Ed. Reverté.

Curso de Análise, vol 2; Elon L. Lima, Ed IMPA (projecto Euclides).

Calculus III, Jerrold Marsden and Alen Weinstein, Springer.

### Teaching method

The problem-solving sessions consist of a presentation of the subject, along with illustrative examples.

Practical classes consist of problem solving and analysis. Students will be required to previously prepare exercises that will be presented on the board for the class, with subsequent group discussion. These exercises will be chosen from a list provided by the teachers.

Any doubts are clarified during classes or in sessions designed to assist students or even in sessions arranged directly between student and teacher.

### Evaluation method

The Continuous Assessment of the curricular unit comprises:

Theoretical-Practical Assessment: two written tests, each lasting 1h30 minutes, to be carried out during the semester. Each test will be classified between 0 and 8 values.

Summative Assessment: delivery of solved exercises. In each class, two exercises will be choosen to be delivered the resolution in the next class. Among all the students who deliver the resolution, two will be chosen to present their resolution on the board. The final grade for this component, between 0 and 4, will be assigned by the teacher based on the quality and quantity of resolutions delivered. In this evaluation component, more than a correct resolution, the student''s work will be valued.

Frequency: Frequencyis obtained by delivering solutions for the exercises in more than 5 classes. Students with student-worker status and students who attended the previous edition of the curricular unit are exempt from obtaining it. Only students with Frequency will have final classification in the curricular unit.

A student who meets the Frequency criterion will have a final classification by Continuous Assessment equal to T1 + T2 + AS, rounded to the nearest integer. Where T1 and T2 are the final grades of the first and second test, respectively, and AS is the Summative Assessment grade. The student will obtain approval in the curricular unit, by Continuous Assessment, if this classification is equal to or greater than 10 values.

Final Exam: Students who have not been approved by Continuous Assessment and who have obtained Frequency of the curricular unit, can take an Final Exam. This is a written exam, lasting 3 hours, which evaluates all the contents taught in the curricular unit. The exam is divided into two parts, each one classified from 0 to 8 values, whose evaluated material corresponds, respectively, to the first and second test. The final grade will be T1 + T2 + AS, where T1 and T2 are the final grades of the first and second part. The student is approved if it is greater than or equal to 10 values.

Classification Improvement: Students approved in the curricular unit may request, upon compliance with all the conditions imposed by NOVA FCT, Classification Improvement by taking the Final Exam. The final grade will be T1 + T2 + AS.

On the day of the test, students must present themselves with a blank test booklet, writing material and official identification document. All tests and examinations must be carried out without consultation and without the use of any computational calculation material.

Working students, or other students who for some reason cannot attend practical classes, can ask the teacher to be evaluated without the Summative Assessment component. In that case the final Continuous Assessment or Exam grades will be calculated according to the formula (T1 + T2)5/4. In this situation the Frequency is given automatically. Attention: this request must be made by the end of the second week of classes, otherwise the student will be assessed with the Continuous Assessment component.

In any omitted situation, the NOVA FCT Assessment Regulation, of 31 July, 2020, applies.

## Subject matter

Revision of some analytical geometry concepts. Conics. Quadrics.

Topological notions in *R ^{n}*. Vector functions of a real variable and functions of several variables. Domain, graph, level curves and level surfaces. Limits and continuity of functions of several variables.

Partial derivatives and Schwarz theorem. Derivative according to a vector. Jacobian matrix. Gradient vector and the notion of differentiability. Differentiability of composite function. Taylor''s formula.

Implicit function theorem and inverse function theorem. Local and global extrema. Constrained extrema and Lagrange multipliers.

Double and triple integrals. Iterated integrals and Fubini''s theorem. Change of variables in integrals. Double integrals in polar coordinates. Triple integrals in cylindrical and spherical Coordinates. Applications.

Vector fields. Gradient, divergence and rotational. Closed fields. Conservative fields. Applications.

Line integrals of scalar and vector fields. Fundamental theorem for line integrals. Green''s theorem. Applications.

Surface integral. Flow of a vector field through a surface. Stokes'' Theorem of and Gauss'' theorem. Applications.

## Programs

Programs where the course is taught: