Multivariable Calculus

Presentation of the course
This course will be instructed in English. The course includes the study of the concepts, methods and tools of differential and integral calculus, in order to develop an adequate spatial imagination and treatment of models that involve several variables, an indispensable and necessary quality for the resolution of problems and future professional development.

It is intended to develop the technical skills of students in the tools of multivariable calculus, but even more, to achieve a thorough understanding of the concepts. The student is expected to incorporate the geometric, numerical, algebraic and verbal points of view in their approach to mathematical knowledge and to be able to use technology as a powerful tool to discover and understand those concepts. The student is required to develop his inductive and deductive logical thinking as the ultimate goal of his scientific training. The course is of a theoretical nature and provides students with a very important tool of mathematics, its main objective is to make the student learn to use the differential and integral calculus of functions in several variables to solve a wide variety of problems and that will serve of consolidation to study the courses of the degree. The topics that are developed in this course are: Vector functions, curves, functions of several variables, Applications as maximum and minimum of these functions, double and triple integrals with applications to area of flat regions and volume of solids, line integrals and surface, vector fields, Stokes Theorems and Divergence with their applications in flows and circulation of fluids.

General Objectives
Understand, identify and assess the mathematical tools that contribute and support the solution of complex problems in different areas of Engineering.

Specific Objectives

• Know the basic vocabulary and the properties associated with calculation of functions of several variables.
• Understand and assess the usefulness of vector functions and calculate limits, analyze continuity, derivatives and integrals.
• Values its importance in the solution of engineering problems.
• Incorporate the function concept of several variables. Extend the notions of limit and continuity to two and three dimensions.
• Modeling and solving optimization problems with rigor and precision. 
• Demonstrate the validity or falsity of propositions using mathematical reasoning techniques.
• Identify and graph regions in the plane and space.
• Describe double and triple integrals of real variable that allow to determine, areas, volume, centers of mass, etc.
• Define and calculate line integrals on different trajectories. 
• Apply Theorems of Green, Stokes and Divergence to the problems of engineering.
• Calculate flows and circulation of vector fields.

Teaching methodology
The teaching activities are developed in two areas: the Chair and the Assistantship.
In the Chair the classes will be oriented to discuss the different ideas and concepts of the course incorporating as much as possible the participation of the students through questions that allow interactions in class leading to an effective understanding. Illustrative examples of the topics will be used and the ideas involved in the demonstration of the fundamental theorems will be discussed.

The Assistantship focuses on the development of skills for the treatment of the problems of the course and its specific applications. Students should work individually on selected exercises of the guide text with gradual difficulty and emphasis on problem solving techniques, clarifying any doubts that may arise.

Learning evaluation and regulation
The course includes two types of evaluations: Chair Tests and Assistantship controls. There will be three proficiency tests and three assistantship tests. Each chair test (P1, P2, P3) has a weight of 25%, the average of the three assistants controls (C) has a weight of 25%.
This gives rise to a presentation note (NP) for exam that is calculated by:
NP=0.25*P1+0.25*P2+0.25*P3+0.25*C

The team of professors of the subject will determine an exemption note (NE). This grade must be greater than or equal to 5.0 and subject to the condition that only 10% of the students in the course are exempted.

The exam is of a global nature and aims to evaluate a synthesis of the main contents covered by the course.

If the presentation grade (NP) is greater than or equal to the grade of exemption (NE), then the student will be exempted from the exam and his final grade (NF) will be his presentation grade (NP). Otherwise the student must take the exam (EX) with 30% weighting, which will produce the final grade calculated as follows:
NF=0.7*NP+0.3*EX

All students with a final grade (NF) greater than or equal to 4.0 will pass the course.
Exempted students who so wish may take the exam must assume the qualification obtained in it, whatever it may be.

Only the students who took the exam and their final grade (NF) is greater than or equal to 3.5 and less than or equal to 3.9, have the right to take a second exam, with the possibility of passing with a grade of 4.0 or fail with final grade (NF).

If the student did not attend an assistantship control, the grade of the control will be replaced by the corresponding test grade, as long as it has justified before undergraduate area and the justification is accepted. Otherwise, the control grade will be 1.0.

If the student failed a proficiency test, the absence must be justified before the undergraduate area, who will authorize the proof of the test to be replaced by the exam grade (NE).

A student may justify before undergraduate area, at most two evaluations of the proficiency tests and exams. The rest of these evaluations not taken, will be rated with a grade of 1.0.

In case his/her absence is duly justified and accepted by undergraduate area, the student must take the second exam. This exam fulfills the function of the first exam for the student.

Any special situation that is not contained in this regulation must be known and resolved by the Undergraduate Secretariat. If a student exceeds the absences to evaluations (Chair tests, assistantship controls or examination) foreseen in this regulation, he/she must present his/her duly justified situation before the Undergraduate Secretariat, where the extraordinary procedure to be followed will be decided.

Bibliography
Basic Bibliography:

1. Stewart, James., Cálculo Trascendentes Tempranas, Sexta edición, Cencage Learning, 2008.
2. Stewart, James., Calculus Early Trascendentals, Sixth Edition, Thomson Brooks/Cole, 2008.

Complementary Bibliography:

1. Marsden, J.; Tromba, A., Cálculo Vectorial, 4ª edición, Addison- Wesley, 1998.
2. Pita,Claudio, Cálculo Vectorial, 1º edición, Printece-Hall Hispanoamericana S.A., 1995.
3. Thomas, G.; Finney, R., Cálculo varias Variables, 9ª edición. Addison-Wesley Longman, 1999.
4. Stewart, J., Cálculo multivariable, 3ª edición. Internacional Thomson Editores, 1999.
5. Edwards, CH.; Penney, D., Cálculo con Geometría Analítica, 4ª edición, Printece may Hispanoamericana S.A., 1994.
6. Dennis, G., Cálculo con Geometría Analítica, 1ª edición, Grupo editorial Iberoamericano, 1987.
7. Leithold, L., El Cálculo con Geometría Analítica, 1ª edición, Editorial Harla, 1987.
8. Smith, R.; Minton, R., Cálculo. Volumen 2, Segunda Edición, Mc. Graw Hill.