Mathematical Analysis III A


The essential of the program is devoted to the study of the real functions of several variables paying special attention to the questions related to limits, continuity, differentiation and the fundamental theorems of differential calculus, including chain rules and differentiation of functions defined implicitly. In one of the sections, we propose to make a study of curves as preparation for the line integrals ( in Mathematical Analysis IV-A) and we also includes the problem of finding the envelope of plane curves. The second part of the program deals with the study of ordinary differential equations. We hope that the students be able to determine the general solution and particular solutions of various types of ordinary differential equations.

General characterization





Responsible teacher

Elvira Júlia Conceição Matias Coimbra


Weekly - 6

Total - 84

Teaching language



The fundamental theory of differential caculus of functions of a single real variable ( presented in Mathematical analysis II-A)


1. Apostol, T. M. - Volume I e Volume II - Blaidsell Publishing Company
2. Braun, Martin - Differential Equations and their Applications, Springer-Verlag
3. Freitas, A.C. - Linhas e Superfícies - Aplicações; Equações diferenciais Ordinárias - Notas de lições para o 2º ano das Licenciaturas da FCT.
4. Kreysig - Advanced Engineering Mathmatics
5. Taylor, A. E.; Man, W. R. - Advanced Calculus

Teaching method

Theoretical issues are presented and explained  in the first part of each lecture. These issues are immediatly applied  by solving problems, where the application of the concepts is necessary. The students also solve a few exercices as homework.

Evaluation method

Evaluation is made by two tests along the semester or a final exam. The final classification is the weighted mean of the classification of the tests or, in alternative, the mark obtained in the final exam.

Subject matter

Real Functions of Several Variable ­ The euclidian space n-dimensional. Euclideam metric. Examples of real functions of several variables. Limits. Continuity. Partial derivatives. Schwarz Theorem. Differentials. Composite functions and the chain rule. Directional derivatives. The law of the mean. Taylor´s formula. Sufficient conditions for a relative extreme. The implicit function theorem. A generalization of the implicit function theorem-simultaneous equations.

 2. Curves . Representation of curves. Smooth and sectionally smooth curves. The tangent vector. Envelope of plane curves. Arc length.

 3. Ordinary differential equations

 First-order differential equations. Exact equations. The integrating factor method. Separable equations. Homogeneous equations. The linear equation of first order. First-order equations solvable by special methods. Second-order and higher order equations. Equations where does not explicitly appear the independent variable or the unknown function. Homogeneous linear equation of order n. Nonhomogeneous linear equations of order n.­ Variation of parameters. Homogeneous linear equation of order n with constant coefficients. Special methods for determining particular solutions of the nonhomogeneous linear equation of order n, with constant coeficients. Euler´s equation.


Programs where the course is taught: