Differential and Integral Calculus II

Objectives

Master the differential and integral calculus of scalar and vector valued functions of several real variables and multiple and line integrals, including the fundamental theorems of calculus for line and double integrals, and geometric and physical applications.

Program

Basic topological notions in R^n, sequences.
Scalar and vector fields. Limits and continuity. Differentiability and gradient. Applications.
Intermediate value theorem.
C^k functions, Schwarz lemma. Extremal and sadle points of scalar fields.
Weierstrass theorem, Taylor's formula, Hessian matrix, Lagrange multipliers.
Inverse and inplicit function theorems. Applications.
Multiple integrals and applications.
Curves, paths and line integrals. Applications.
Fundamental theorem of calculus for line integrals and applications.
Greens's theorem and applications.
Gradient vector fields of scalar fields.

Teaching Methodologies

The teaching methodologies aim to promote learning based on problem solving, reinforcing the practical component, active learning, autonomous work and student accountability. The assessment model incorporates exam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).

Bibliography

* Vector Calculus, Marsden and Tromba, 2012, 6th ed, Freeman;
* Calculus II, Apostol, 2016, 2nd ed, Wiley;
* Functions of Several Variables, Fleming, 1977, 2nd ed, Springer;
* Cálculo Diferencial e Integral em ℝ^n,, Gabriel Pires, 2016, 3ª ed, IST Press.;
* Integrais Múltiplos, Luís T. Magalhães, 1996, 3ª ed, Texto Editora;
* Exercícios de Cálculo Integral em ℝ^n, Gabriel Pires, 2018, 2ª ed, IST Press;
* Exercícios de Análise Matemática I e II, DM-IST, 2003, Departamento de Matemática do IST.