# Mathematical Analysis II B

## Objectives

At the end of this course the student must have acquired knowledge, skills and powers to:

- Work with elementary notions of topology in R^{n} (neighborhood, open, closed, etc.).

- Understand the concept and definition of limit, continuity and differentiability of vectorial functions of real variable.

- Apply vectorial functions of real variable to parameterize space curves and to study the properties of the curves.

- Understand the definition of limit and continuity of real and vectorial functions of several variables and calculate limits.

- Understand the notion of parcial derivative, differentiability, directional derivative and understand the implicit and inverse function theorems.

- Understand the Taylor development and its applications to the study of functions and calculus of extreme values.

- Understand the notion of double and triple integral and perform calculations with the adequate coordinates.

- Understand some applications of the double and triple integral.

- Understand the notion of line integral, some applications and fundamentals results.

- Understand the notion of surface integral, some applications and fundamentals results.

## General characterization

##### Code

10476

##### Credits

6.0

##### Responsible teacher

Marta Cristina Vieira Faias Mateus

##### Hours

Weekly - 8

Total - 70

##### Teaching language

Português

### Prerequisites

Diferencial and integral calculus in R. Basic knowledge of matricial calculus.

### Bibliography

H. Anton, I. Bivens, S. Davis, Calculus, volume 2, 8th edition, John Wiley and Sons, 2005.

### Teaching method

Theoretical classes consist on a theoretical exposition illustrated by examples of applications.

Practical classes consist on the solving of exercises of application of the methods and results presented in the theoretical classes. These exercises are chosen from a list provided by the teachers.

Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it or even at special sessions previously arranjed between professors and students.

### Evaluation method

Please, contact Professor Marta Faias (mcm@fct.unl.pt).

## Subject matter

1. **Revision of concepts of analytical geometry**

1.1.Conics.

1.2.Quadrics.

2. **Limits and Continuity in R^{n}**

2.1.Topological notions in *R ^{n}*.

2.2.Vector functions and functions of several variables: Domain, Graph, curves and level surfaces.

2.3.Limit and Multiple Variables Real functions Continuity.

3. **Differential calculus in R ^{n}**

3.1.Partial derivatives and Schwarz theorem.

3.2.Derivative according to a vector. Jacobian matrix. Gradient Vector and notion of Differentiability.

3.3.Composite Function Differentiability. Taylor''''''''s theorem. Implicit Function Theorem and Inverse Function Theorem.

3.4.Local extremes. Conditioned Extremes and Lagrange multipliers.

4. **Integral Calculus in R^{n}**

4.1.Double integrals. Iterated integrals and Fubini Theorem. Change of variables in double integrals. Double integrals in Polar Coordinates. Applications.

4.2.Triple integrals. Iterated integrals and Fubini Theorem. Change of variables in triple integrals. Triple integrals in Cylindrical and Spherical Coordinates. Applications.

5. **Vector analysis**

5.1.Vector fields: gradient, divergence and rotational. Closed fields. Conservative fields. Applications.

5.2.Formalism of Differential Forms. Line Integrals of scalar fields and vector fields. Fundamental Theorem of Line integrals. Green''''''''s Theorem. Applications.

5.3.Integrals of scalar fields surface. Flow of a Vector Field through a surface. Stokes Theorem of and Gauss-Ostrogradsky Theorem. Applications.

## Programs

Programs where the course is taught: