Mathematical Analysis IV A


The first chapters of the course are devoted to the complementary study of Ordinary Differencial Equations begun in Mathematical analysis III-A. the work of Cauchy, Liouville and others showed the importance of establishing general theorems to guarantee the existence of solutions to certain specific classes of differential equations. One of the chapters of our course is concerned with the proofs of some of these theorems. With respect to multiple integration, our study is intended to familiarize the student with the properties and with the methods of finding the values of doule and triple integrals. Our course also includes line integrals and surface integrals. These kind of integrals are of fundamental importance on both pure and applied mathematics. Finally we present two generalizations of the Green´s theorem, namely the Stokes theorem and the divergence theorem. The divergence theorem is useful in connection with the consideration of solid angles.

General characterization





Responsible teacher

Maria Fernanda de Almeida Cipriano Salvador Marques


Weekly - 6

Total - Available soon

Teaching language



Available soon


Taylor A. E; Man, W.R. - Advanced Calculus - John Wiley and sons

Cálculo vol. 2, Howard Anton, Irl Bivens, Stephen Davis,8ª edição,Bookman/Artmed

Calculus III, Jerrold Marsden and Alen Weinstein

Marsden - Basic Complex Analysis

Teaching method

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Evaluation method

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Subject matter

 1. Double integrals. Definition of a double integral. Some properties. Evaluation of a double integral. Geometric interpretation of a double integral as a volume.  Green´s theorem in the plane. Some applications of  Green´s theorem. Change of variables in a double integral.

2. Triple integrals. Definition of a triple integral. Some properties. Cylindrical co-ordinates. Spherical co-ordinates. Applications.

3. Scalar fields and vector fields. Gradient, curl, divergence and laplacian. Conservative fields.

4. Line integrals. Green theorem.

5. Surface integrals.  Stokes theorem and the divergence theorem.  Applications of the divergence theorem- solid angle.

6. Integrals depending on a parameter. Leibniz´s rule.

7. Complements on ordinary differential equations- Use of power series to obtain solutions of a linear differential equation of order two. The homogeneous linear equation of second order- singular points. The Bessel equation. The Bessel functions. Systems of linear differential equations. Existence and uniqueness theorems for differential equations.


Programs where the course is taught: