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

Paulo José Fernandes Louro Ribeiro Doutor


Weekly - 4

Total - 50

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 assessment is based on questions in class, 2 tests, 3 assignments and a video project. Part of these assessment elements will be carried out on Moodle. If a student does not obtain approval through continuous evaluation, he can try it in an additional assessment.

Evaluation method

1 - Attendance

There will be no attendance control.

2 - Continuous assessment

Continuous assessment has two components: theoretical-practical assessment (TP) and summative assessment (AS).

2.1 Theoretical-practical assessment

The theoretical-practical assessment is carried out through two tests T1 and T2. Each test lasts 1 hour and 30 minutes and will be graded from 0 to 5. The second test will have a minimum mark of 1.5.

2.2 Summative assessment

Throughout the semester, questions will be given in moodle which will be assessed and graded between 0 and 2. The average of the 5 best assignments will correspond to the final mark of the summative assessment (SA), graded from 0 to 2.

2.3 Project assessment

Over the course of the semester, students will develop 3 computational assignments (TC1, TC2, TC3), graded out of 1, 2 and 3 respectively, and 1 video project (PV), graded out of 2.

2.4 Final mark for continuous assessment

The continuous assessment mark will be calculated from the elements of the theoretical-practical assessment, the summative assessment and the project assessment, as follows:


If the mark for the 2nd test (T2) is greater than or equal to 1.5 (minimum mark) and the continuous assessment mark (CA) is greater than or equal to 9.5, the student passes with this mark rounded up to the nearest integer. If the mark for the 2nd test (T2) is less than 1.5 (minimum mark) or the continuous assessment mark (AC), rounded to the nearest integer, is less than 10, the student must take an Appeal Exam.

3 - Appeal exam

The exam will be marked out of 20 and will determine the final grade (NF).

If the final mark (NF) is greater than or equal to 9.5, the student will pass with that mark rounded up to the nearest integer. If the final mark (NF), rounded to the nearest integer, is less than 10, the student will fail.

For the purposes of the appeal exam, the grade for the summative assessment and project is not counted.

4 - Improvement

All students wishing to sit an improvement exam must register for this purpose at CLIP.

If the mark for the improvement exam is higher than the mark obtained previously in the subject, it will be considered as the final mark. Otherwise, there is no grade improvement.

For the purposes of the appeal exam, the marks for the summative assessment and the project are not taken into account.

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.