Faculdade de Ciências e Tecnologia

Mathematical Analysis II C



Academic unit

Faculdade de Ciências e Tecnologia


Departamento de Matemática



Teacher in charge

Nuno Filipe Marcelino Martins

Weekly hours


Teaching language


Subject matter

1. Analytic Geometry review

1.1. Conics

1.2. Quadric surfaces.

2. Limits and Continuity in Rn

2.1. Topological notions in Rn

2.2. Vector valued functions and functions of several real variables: domain, graph, level curves and level surfaces.

2.3. Limits and continuity of  functions with several real variables.

3. Differential Calculus in Rn

3.1. Partial derivatives and Schwarz''s theorem.

3.2. Directional derivative along a vector. Jacobian matrix, gradient vector and differentiability.

3.3. Differentiability of the composition of two functions. Taylor''s theorem. Implicit and inverse function theorems.

3.4. Local extrema. Conditional extrema and Lagrange multipliers.

4. Integral Calculus in Rn

4.1. Double integrals. Iterated integrals and Fubini''s theorem. Change of variable in double integrals. Double integrals in polar coodinates. Applications.

4.2. Triple integrals. Iterated integrals and Fubini''s theorem. Change of variable in triple integrals. Triple integrals in cylindrical and spherical coordinates. Applications.

5. Vectorial Analysis

5.1. Vector fields: Gradient, divergence and curl. Closed fields. Gradient fields. Applications.

5.2. Formalism of differential forms. Line integrals of scalar and vector fields. Fundamental theorem of line integrals. Green''s Theorem. Applications.

5.3. Surface integrals of scalar fields. Flux of a vector field across a surface. Stokes Theorem and Gauss-Ostrogradsky theorem. Applications.  


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

2- Calculus III, Jerrold Marsden and Alen Weinstein

3- Vector Calculus, Jerrold Marsden and Anthony Tromba, 5th edition

Evaluation method

Please contact the course responsible.