Calculus II

Differential and integral calculus of real functions of a real variable.

In classes, the theoretical exposition of the program of the curricular unit and the resolution of some application exercises are made. The theoretical classes have a more expository nature, while the theoretical-practical and practical classes have a more practical nature in which the main objective is the discussion and resolution of exercises by the students, with the support of the teacher.

Understand and apply the knowledge of differential and integral calculus in IRⁿ, unilateral Laplace transforms, as well as numerical series and power series. Solve and interpret problems presented in the real context.

At the end of this unit the student is expected to know: explain the concepts, discuss and present each problem-solving appropriately; solve exercises with increasing autonomy using the matters dealt with in class and related topics; find and select relevant information from different sources, such as monographs and internet.

1. Real functions of several variables and their derivatives – Conics and quadric surfaces. Topology concepts in IRⁿ. Real functions of several real variables: Domain; Graph; Contours; Limits; Continuity; Partial derivatives; Schwarz theorem; Chain rule; Directional derivative; Gradient vector and its applications; Free and conditional extrema.

2. Multiple integrals – Double integrals: Definition and geometric interpretation; Calculus of a double integral in Cartesian and polar coordinates; Applications. Triple integrals: Definition and geometric interpretation; Calculus of a triple integral in Cartesian, cylindrical and spherical coordinates; Applications.

3. Laplace transform – Definition of unilateral Laplace transform. Exponential order function. Existence and properties of the Laplace transform. Laplace transform of derivative of functions. Derivatives of the Laplace transform. Solving Linear Ordinary Differential Equations with the Laplace transform.

4. Series – Numerical series: Definition of partial sum; Definition of numerical series; Examples (geometric series, telescopic series, Dirichlet series); Properties; Necessary condition of convergence; Criteria for convergence of series of non-negative terms; Alternating series and Leibniz criterion; Simple and absolute convergence. Function series: Definition of power series; Convergence; Taylor series.

NAO

Rodrigues, J. A. (2008). Curso de Análise Matemática – Cálculo em IR^n. Princípia (available in the Library of ISEC: 3-2-347).

Larson, R. E., Hostetler, R. P., & Edwards, B. H. (1998). Cálculo com geometria analítica – Volume 2. Rio de Janeiro: Livros Técnicos e Científicos (available in the Library of ISEC: 3-2-245; 3-2-249).

Stewart, J. (2001). Cálculo – Volume 2. São Paulo: Pioneira – Thomson Learning (available in the Library of ISEC: 3-2-119; 3-2-153; 3-2-160; 3-2-278).

Edwards, C. H., & Penney, D. E.  (1993). Elementary differential equations with boundary value problems. Englewood Cliffs, NJ: Prentice-Hall (available in the Library of ISEC: 3-11-14).

Spiegel, M. R. (1965). Theory and problems of Laplace transforms: Schaum’s outline. New York: McGraw-Hill (available in the Library of ISEC: 3-2-72).