# Mathematical Analysis II C

## Objectives

1- Understand the basic topological concepts in R ( with special emphasis in R^2 and R^3).

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

Maria Fernanda de Almeida Cipriano Salvador Marques

##### Hours

Weekly - 4

Total - 56

##### 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

The theoretical-practical classes consist of exposition of the subject, with the demonstration of the most relevant results, followed by illustrative examples and exercises 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

**Frequency**

Students who have any of the special statutes provided for by law, and students with Frequency obtained in the 2022-2023 academic year are exempt from Frequency in this semester.

Frequency will be granted to any student enrolled in the course that does not unjustifiably miss more than 1/3 of the face-to-face practical classes taught, corresponding to the practical class in which they are enrolled in CLIP.

The justifications must be delivered to the teacher of the pratical course in which the student is enrolled in CLIP, within a maximum period of 14 days from the first presence (in the pratical course in which the student is enrolled in CLIP) after the absences to which the justification refers to. If the consecutive period of absences extends after the end of classes, this must be delivered until the end of the day of the assessment test that the student presents (test or exam).

**Evaluation**

Knowledge assessment is carried out through Continuous Evaluation or Exam Evaluation Examination, presential. The Continuous Assessment consists of two tests.

** ****Continuous evaluation**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

During the semester two tests will be carried out. Each test is rated up to a maximum of 20 values.

1st Test: all students enrolled in the course may present themselves to the 1st test.

2nd Test: all students enrolled in the discipline who have obtained frequency or are exempt from it may appear at the 2nd test.

The classification of the Continuous Evaluation is obtained by making the arithmetic mean of the classifications obtained in the 2 tests.

If the classification of Continuous Evaluation falls within the range [9.4, 9.5[, the Professor of the theoretical-practical classes may give the student a classification of 9.5, taking into account the participation and commitment demonstrated by the student during the theoretical-practical classes, as well as attendance at theoretical-practical classes.

If the Continuous evaluation classification is greater than, or equal to, 9.5 and less than 17.5, the student is approved with that classification rounded up to the units. If the classification of the tests is greater than, or equal to, 17.5, the student can choose between keeping the final classification of 17 or taking a complementary test to defend the grade.

If the classification of the Continuous Assessment is less than 9.5, the student may take the Exam.

**Exam**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

All students enrolled in the discipline who have obtained Frequency or are exempted from it, and who have not passed the continuous assessment, may take the Exam.

If the exam classification is greater than or equal to 9.5 values and less, or 17.5 values, the student is approved with this classification rounded up to the units.

If the Exam classification is greater than or equal to 17.5 values, the student can choose between maintaining the final classification of 17 values or taking a complementary test to defend the grade. If the classification obtained in the exam is less than 9.5 points, the student is not is approved.

**Grade improvement**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

Students have the right to improve their grades, upon enrollment within the deadlines, at the time of exam.

**Logistics**

Only students who, at the time of the exam, carry an official identification document, containing a photograph (for example, Citizen Card, Identity Card, Passport, some versions of Student Card) and blank exam notebook, are able to do the exam.

**Final considerations**

In any omitted situation, the Knowledge Assessment Regulation of the FCT-UNL applies

## 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: