# Mathematical Analysis II C

## Objectives

1- Understand the basic topological concepts in R ( with special emphasis in R R).

2. Understand the concepts of limit, continuity of real / vector functions of various real variables. Know how to calculate limits of the functions considered at indicated points.

3. Understand the concept of partial derivative, differentiability and its applications. Know how to calculate the derivative of the composite function and apply the implicit and inverse function theorems. Know how to determine the Taylor formula of a multiple variable function. Calculation of local, absolute and conditioned extremes.

4. Understand the concept of double and triple integral and know how to calculate these integrals using various types of coordinates.

5. Understand the concept of line and surface integrals and their applications: Green, Stokes and divergence theorems.

## General characterization

##### Code

10347

##### Credits

6.0

##### Responsible teacher

Joaquim Eurico Anes Duarte Nogueira, João de Deus Mota Silva Marques

##### Hours

Weekly - 4

Total - 42

##### Teaching language

Português

### Prerequisites

Students should know and apply the basic concepts of Mathematical Analysis I as well as linear algebra and analytic geometry.

### Bibliography

1- Cálculo vol. 2, Howard Anton, Irl Bivens, Stephen Davis,8ª edição,Bookman/Artmed

2- Calculus III, Jerrold Marsden and Alen Weinstein

3- Vector Calculus, Jerrold Marsden and Anthony Tromba, 5ª edição

### Teaching method

Theoretical classes consist of exposition of the subject, with the demonstration of the most relevant results, followed by illustrative examples of the subjects exposed.

In practical classes are solved exercises of application of the methods and results presented in the lectures. The exercises are preferably solved on the board by students accompanied by the clarification of the doubts that arise during their resolution. The exercises are chosen from a list previously made available in the Clip by the teachers.

There is a fixed time for clarifying doubts. There is also the possibility of clarifying doubts, beyond the time set for this purpose, in sessions previously agreed between teacher and students.

### Evaluation method

**Assessment Method - Mathematical Analysis II-C**

In accordance with the Knowledge Assessment Regulation (RAC) of the Faculty of Science and Technology, Universidade Nova de Lisboa, approved on February 7, 2017, the discipline of Mathematical Analysis II-C has the following method of assessment:

**Attendance**

Attendance will be given to first-time students who have attended at least two-thirds of the number of actual face-to-face practical classes. Students of second and subsequent enrollment or who have a status that exempts them from attendance are exempt from attendance.

Continuous evaluation

The continuous evaluation of the discipline is carried out through two written tests (theoretical and practical evaluation) with a duration of two hours each. The tests will take place on 12/05 and 18/06 respectively. Each written test will be assigned a classification (t1, t2). Regarding t2, there is a minimum grade requirement of 7.5 points.

The final classification of the Continuous Assessment "AC" is calculated by rounding

(t1 + t2) / 2

units, by the usual conventions.

The student is approved by continuous assessment if he obtained frequency according to the rules explained above, if t2 is greater than or equal to 7.5 and if AC = 10.

Exam

Students who fail continuous assessment to whom attendance has been assigned, or have been dismissed from it, may take the exam.

The exam consists of a written test lasting no less than 3 hours covering the entire contents of the course.

The exam will be assigned an entire classification between 0 and 20 values, with the student being approved to the discipline, with that classification, if it is greater than or equal to 10 values.

The use of calculating machines or any other calculation support instruments is prohibited at all times of assessment.

Note Defense

Students with a final classification greater than or equal to 18 values must take a defense test. Failure to take this test leads to a final classification of 17 values in the discipline. The final grade of a student who takes the defense test will never be lower than 17 values.

Rating Improvement

Students approved by continuous assessment may request, upon compliance with all the provisions imposed by FCT-UNL, to improve their classification. In that case, they can take the exam. The final grade will be the maximum between the grades obtained in continuous assessment and in exam.

Conditions for the written tests (tests and exam)

Students who are regularly enrolled in the Clip can attend each of the Continuous Assessment tests and the Exam. Registration must be done at least one week before the test (test or exam). On the day of the test, each student must have a blank exam book which will be handed over at the beginning of the test to the teacher who is conducting the surveillance. During the test, the student must carry an official identification document with a recent photograph.

The General Regulations of FCT-UNL apply in everything that this Regulation does not contain.

## Subject matter

**1. Analytic Geometry review**

1.1. Conics

1.2. Quadric surfaces.

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

2.1. Topological notions in R^{n}

2.2. Vector valued functions and functions of several real variables: domain, graph, level curves and level surfaces.

2.3. Limits and continuity of functions with several real variables.

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

3.1. Partial derivatives and Schwarz''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem.

3.2. Directional derivative along a vector. Jacobian matrix, gradient vector and differentiability.

3.3. Differentiability of the composition of two functions. Taylor''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Implicit and inverse function theorems.

3.4. Local extrema. Conditional extrema and Lagrange multipliers.

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

4.1. Double integrals. Iterated integrals and Fubini''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Change of variable in double integrals. Double integrals in polar coodinates. Applications.

4.2. Triple integrals. Iterated integrals and Fubini''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s theorem. Change of variable in triple integrals. Triple integrals in cylindrical and spherical coordinates. Applications.

**5. Vectorial Analysis**

5.1. Vector fields: Gradient, divergence and curl. Closed fields. Gradient fields. Applications.

5.2. Formalism of differential forms. Line integrals of scalar and vector fields. Fundamental theorem of line integrals. Green''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s Theorem. Applications.

5.3. Surface integrals of scalar fields. Flux of a vector field across a surface. Stokes Theorem and Gauss-Ostrogradsky theorem. Applications.

## Programs

Programs where the course is taught: