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 Rn.
- 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
Diferencial 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 TP sessions consist of a presentation of the subject, along with illustrative examples and some problem solving. A list of exercises is provided to be solved autonomously by the students and supported by the P (problem-solving) sessions.
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
Frequency: Frequency is obtained by submission of at least 7 lists of exercises on Moodle over a total of 10 lists, that are provided weekly. Students who obtained frequency in the previous edition of the curricular unit are exempt from obtaining it. Only students with Frequency will have final classification in the curricular unit.
Continuous Assessment. The Continuous Assessment is made of three elements of Theoretical-Practical Assessment, namely, a mini-test to be carried out in a class lasting 20 minutes and graded between 0 and 2 values, and two written tests, each lasting 2 hours and graded between 0 and 9 values. A student that obtained frequency gets as final mark in the Continuous Assessment the sum of the evaluation on the two tests and the mini.test. The student will obtain approval in the curricular unit, by Continuous Assessment, if this classification is equal to or greater than 10 values.
Students with student-worker status who, for this reason, cannot attend classes and therefore do not take the mini-test, will have the tests classified between 0 and 10 values and a final classification by Continuous Assessment equal to the sum of the grades obtained in the two tests.
Final Exam: Students who have not been approved by Continuous Assessment and who have obtained Frequency on the curricular unit, can take the Final Exam. This is a written exam, lasting 3 hours, which evaluates all the contents taught in the curricular unit and classified between 0 and 20 values. The student is approved if the evaluations 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 FCT NOVA, Classification Improvement by taking the Final Exam.
On the day of the test or exam, 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.
In any omitted situation, the NOVA FCT Assessment Regulation, applies.
Subject matter
Revision of some analytical geometry concepts. Conics. Quadrics.
Topological notions in Rn. 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.