Subject: Calculus III

Scientific Area:



80 Hours

Number of ECTS:

7,5 ECTS



Overall objectives:

1 - Stimulate and develop thinking skills, rigor, deduction and abstraction. Provide students with the basic calculation techniques to be used as mathematical tools in the most advanced courses.
2 - Study the fundamentals of the theory of series to several real variables, with corresponding applications to real world problems appropriate to the various areas of knowledge.
3 - Studying the foundations of the theory of differential equations and the corresponding applications to real world problems appropriate to the various areas of knowledge.
4 - Generalize the results of real analysis to complex space, in particular the concepts of limit, continuity, differentiability and integrability.


1 - Series: Definition of numerical series. General term of a series. Sequence of partial sums. Notion of convergence and divergence of a series. Geometric series: theorem on the convergence of a geometric series. Harmonic series. Corollary of the general criterion of Cauchy. Fundamental theorems on convergence of series. Dirichlet series: convergence of a Dirichlet series. Series of nonnegative terms. Convergence criteria: For comparison, the ratio criterion, criterion of the root, the threshold criterion for comparison. Series of fixed terms with no sign: Leibnitz criterion. Absolute convergence. Power series: domain of convergence, radius of convergence and interval of convergence. Theorem on the convergence of power series.
2 - Ordinary Differential Equations: Definitions: differential equation (DE), ordinary differential equation (ODE), partial differential equation or equations in partial derivatives (PDE). Notion of order and degree of an ODE. General solution and a particular ED. Geometric interpretation of an ED: orthogonal and oblique trajectories. Ordinary differential equations of 1st order: equations of separable variables, exact differential equations, integrating factor, homogeneous equations, linear equations: method of variation of constants. Linear differential equations of higher orders to one.
3 - Linear Systems of Ordinary Differential Equations: Concept of system of differential equations of first order and linear system of differential equations of first order. Transformation of an ED in a linear system of linear first-order and vice versa. Homogeneous linear differential systems of 1st order with constant coefficients: algebraic resolution. Non homogeneous linear differential systems of 1st order with constant coefficients: the constant variation method.
4 - Partial Differential Equations: Partial Differential Equations of first order: solution of a PDE of 1st order with constant coefficients. Partial differential equations of 2nd order: solution of a PDE of 2nd order linear constant coefficients with only second-order partial derivatives. Method of separation of variables. Principle of superposition of solutions
5 - Complex Analysis: Complex numbers and operations: representation of a complex number, taking roots. Functions of complex variable: notions of limit, continuity and differentiability. Cauchy-Riemann conditions. Harmonic function. Regular point and singular point. Line integrals, Cauchy's Theorem. Integral representation of a function and its derivatives: application of the Cauchy integral representation of the Taylor series. Laurent series. Concept and calculation of residual waste. Residue theorem. Application to the calculation of real integrals.


Barreira, L. , 2009 , Análise Complexa e Equações Diferenciais , IST Press
Boyce, W. E. and DiPrima, R. C. , 1992 , Elementary Differential Equations and Boundary Value Problems, 5th Edition , Jonh Wiley & Sons
Campos Ferreira, J. , 1993 , Introdução à Análise Matemática, 4ª Edição , Ed. Fundação Calouste Gulbenkian
Braun, M. , 1993 , Differential Equations and Their Applications , Springer
Hoffman, M. J. and Marsden, J. E. , 1987 , Basic Complex Analysis , W. H. Freeman and Company
Rudin, W. , 1976 , Principles of Mathematical Analysis , McGraw-Hill
Tiernet, J. A. , 1989 , Differential Equations , Wm. C. Brown Publishers
Wylie, I. C. R. and Barrett, L. C. , 1982 , Advanced Engineering Mathematics, 5th Ed. , McGraw-Hill International Editions
G. F. Simmos, S. G. Krantz , 2007 , Equações Diferenciais, Teoria, Técnica e Prática , McGraw-Hill
Rudin, W. , 1987 , Real and Complex Analysis, 3rd Ed. , McGraw-Hill Book Company

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