Subject: Calculus II

Scientific Area:



96 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

1 - Stimulate and develop thinking skills, rigor, deduction and abstraction. Provide the students with theoretical and practical knowledge of basic Mathematics to be used as fundamental mathematical toolsin the most advanced courses.
2 - Study the fundamentals of the theory of Mathematical Analysis to several real variables, with corresponding applications to real world problems appropriate to the various areas of knowledge.
3 - Mastering the generalization to space IR^n of the concepts of limit, continuity, differentiability and integrability introduced in the curricular unit of Calculus I.


1 - Improper integrals: Definition and classification.
1.1 - Concept of improper integral convergent and divergent.
1.2 - Properties of improper integrals of 1st and 2nd kind.
1.3 - Discussion of the nature of improper integrals by comparison. Criteria for convergence of improper integrals.
1.4 - Study of the nature of the integral of 3rd kind.
2 - Real functions of vector variable: Definition. Domain.
2.1 - Notions of norm and distance in IR^n, with particular emphasis to the Euclidean norm and Eucliden distance. Balls in IR^n.
2.2 - Limits and continuity.
2.3 - Coordinate systems including: Cartesian, polar, cylindrical and spherical.
2.4 - Partial derivatives. Schwartz's theorem. Tangents and tangent planes.
2.5 - Differential of a function, the total differential.
2.6 - Differentiable functions, functions of class C^p and infinitely differentiable functions.
2.7 - Gradient of a scalar function.
2.8 - Composite function theorem, chain rule.
2.9 - Implicit function. Implicit function theorem.
2.10 - Derivative according to a vector, directional derivative and their relationships.
2.11 - Taylor's formula for functions of two variables.
2.12 - Extrema: free and constrained.
3 - Integral Calculus in IR^n: Vector functions of a real variable.
3.1 - Concept of line in IR^n and parameterizations. Definition of a curve.
3.2 - Line integrals of a scalar function and of a vector field. Properties of line integral. Application to Physics.
3.3 - Multiple integrals: definition, geometric interpretation and properties.
3.4 - Calculation of double and triple integrals.
3.5 - Change of variables in double and triple integrals.
3.6 - Areas and volumes.
3.7 - Green's theorem.
3.8 - Parametrizations of surfaces; vector normal to a surface at a point P, orientable surfaces and not globally orientable.
3.9 - Surface integral. Notion of flow of a vector across a surface.
3.10 - Divergence and Rotational (curl).
3.11 - Gauss' theorem.
3.12 - Stokes' theorem.


Agudo, D. , 1969 , Lições de Análise Infinitesimal, Vol I e II, , Escolar Editora
Apostol, T. , 1983 , Cálculo, Vol II, 2ª Edição , Ed. Reverté Lda.
Ávila, G. S. S. , 1979 , Cálculo III-Diferencial e Integral , Livros Técnicos e Científicos Editora S.A.
Campos Ferreira, J. , 2002 , Introdução à Análise em IR^n , IST
Kaplan, W. , 1972 , Cálculo Avançado, Vol II , Editora da Universidade de S. Paulo
Lang, S. , 1980 , Cálculo, Vol II , Livros Técnicos e Científicos Editora
Piskounov N. , 1986 , Cálculo Diferencial e Integral, Vol. I e II , Lopes da Silva Editora
Swokowski, E. W. , 1983 , Cálculo com Geometria Analítica, Vol II , McGraw-Hill
Pires, Gabriel E. , 2012 , Cálculo Diferencial e Integral em Rn , IST Press

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Oral and written presentation of the syllabus contents of the curricular unit. Discussion and resolution of exercises and application problems in small groups or individually. Realization of two tests (with weight of 50% each) to be solved individually during the normal season. In this way, the student can, during the semester, evaluate their performance and change strategies if necessary. In the resit period of exams, the student can perform only one of the tests or, alternatively, a complete exam corresponding to 100% of the final grade. The importance of this examination, in addition to the assessment objectives, focuses on the student's ability to relate different parts of the subject.

Subject Leader:

Nelli Aleksandrova