Calculus II


At the end of this course the student must have acquired knowledge and skills that will enable him (i) to know the fundamental concepts and applications of vectorial calculus and of functions of several variables, (ii) to have a developed tridimensional visualization aptitude, (iii) to understand the complexity of working with functions of several variables with respect to functions of one variable, (iv) to recognize on the methods studied to functions of one real variable a motivation to develop suitable techniques to study functions of several variables.

General characterization





Responsible teacher

Paula Cristiana Costa Garcia Silva Patrício


Weekly - 4

Total - 56

Teaching language



Knowledge of Calculus I and Linear Algebra and Analytical Geometry I


H. Anton, I. Bivens & S. Davis, Calculus, 8th edition, John Wiley & Sons, 2005.

S. Salas, E. Hille, G. Etgen, Calculus One and Several Variables, 10th edition, John Wiley & Sons, 2007.

G. Simmons, Calculus with Analytic Geometry, 2nd edition, McGraw Hill, 1996.

Teaching method

Classes are theoretical/practical with oral presentation of concepts and results and complemented with examples and applications. Sometimes proofs or hints for the proof are presented. A list of exercises and problems is provided to the students to be solved independently. The student is encouraged to use computational means supporting the topics under study. Specific student difficulties will be addressed during classes or in individual sessions scheduled with the professor.  Continuous evaluation is based on two tests. Part of the evaluation is based on tasks to be carried out throughout the semester in Moodle. If a student does not obtain approval through continuous evaluation, he can try it in an additional assessment.

Evaluation method

To pass the continuous evaluation (épcoa normal) students are required to have a grade equal ou higher than 9.5. This value is the weighted average of

First test: 45%; second test: 45%; Tasks: 10%

Furthermore, students are required to attain a minimum of 6.5 points on the second test.

If a student does not pass in the continuous evaluation, he or she can do an "exame de recurso".

Grades are rounded to the nearest integer (n.5 -> n+1)

Subject matter

1. Conic sections, polar coordinates and parametric equations.

2. Vectorial calculus: vectorial fields, curves and arc length, curvilinear motion.

3. Functions of several variables: quadric surfaces, graphs, level curves and level surfaces, limits and continuity, cylindrical and spherical coordinates.

4. Gradient, directional derivatives, tangent plane and differentiability. Taylor’s theorem.

5. Extreme values of functions of several variables. Lagrange multipliers.

6. Double and triple integrals, Fubini’s theorem and change of variables.

7. Line and surface integrals, work, and flux through a surface.

8. Divergence and curl, Green’s theorem, divergence theorem, and Stokes theorem.