Mathematical Analysis II B
Code
10476
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
João Pedro Bizarro Cabral, Oleksiy Karlovych
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
At the end of this course the student must have acquired knowledge, skills and powers to:
- Work with elementary notions of topology in Rn (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.
Prerequisites
Diferencial and integral calculus in R. Basic knowledge of matricial calculus.
Subject matter
1. Topological notions in Rn
1.1 Norms and metrics
1.2 Topological notions in Rn
2. Functions of several variables
2.1 Real functions of several variables
2.2 Vector valued functions
2.3 Limits and continuity
3. Differential calculus in Rn
3.1 Partial derivatives. The Schwarz theorem
3.2 Differential of a function
3.3 Directional derivatives
3.4 The chain rule
3.5 Taylor''s formula
3.6 The implicit function theorem
3.7 The inverse function theorem
3.8 Relative extrema
3.9 Conditional extrema. Lagrange multipliers
4. Multiple integrals. Vector Calculus
4.1 Double integrals
4.2 Iterated integrals: Fubini''s Theorem
4.3 Change of variables in double integrals
4.4 Double integrals in polar coordinates
4.5 Applications of double integrals
4.6 Triple integrals
4.7 Change of variables in multiple integrals
4.8 Triple integrals in cylindrical and spherical coordinates
4.9 Vector fields
4.10 Line integrals
4.11 The fundamental theorem of calculus for line integrals
4.12 The Green theorem
4.13 Curl and divergence
4.14 Parametric surfaces and their areas
4.15 Surface integrals
4.16 The Stokes theorem
4.17 The divergence theorem
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.