Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 2 | 1 |
The main prerequisite is a standard first-year course in Calculus, covering differential and integral calculus in one real variable. If not fulfilled, it is imperative that, at least, the student understands the notion of convergence of functions or sequences, as well as those of continuity, derivability and integrability of functions. It is also very important that the student is familiar with the computation of limits, differentiation rules, the fundamental theorem of calculus, Taylor's development of elementary functions, etc.
The main objective of this course is to know the basic techniques of the differential and integral calculus in several variables and the most important concepts of vector analysis.
In the first part of the course the student must first become acquainted with the Euclidean space and its metric and topological structure. Next, the key concept is that of differential as a linear approximation of the increment and other better approximations, in terms of higher order differentials, and how the behavior of these approximations translates into local properties of the function. In the same way that in the case of one variable, the techniques of the course will be applied to the resolution of different mathematical problems such as geometric problems, optimization or, in general, issues where quantify the variation of a given magnitude based on other variables. The student must also be familiar with the geometric concepts of curves and regular surfaces, tangential planes, local coordinates, etc.
The second part of the course, more instrumental, is dedicated to multiple integrals and to vector analysis, covering techniques such as change of order of integration, change of variables, Green, Gauss or Stokes's theorems.
1. Differential calculus in several variables:
- Basic geometrical and topological notions in the Euclidean space. Limits and continuity. Parameterization. Graphics and level sets
- Differentiability. Basic properties. Partial derivatives and directional derivatives Relative extremes
- Higher order differentials. Taylor's Formula. Analysis of the critical points.
- The inverse function theorem. Changes of coordinates.
- The implicit function theorem. Geometric viewpoint, regular submanifolds.
- Functional dependence and independence.
- Constrained extrema. Lagrange multipliers
2. Integration
- The Riemann integral of bounded functions on rectangles. Basic properties
- Fubini theorem.
- Integration on general sets
- Changes of variables, meaning of the Jacobian.
- Length and area. Integration on curves and surfaces.
3. Vector analysis
- Orientable surfaces.
- Circulation and flow of a vector field.
- Divergence and rotation of a vector field. Conservative and solenoidal fields.
- Theorems of Green, Gauss and Stokes.
There are three type of activities the student is supposed to attend: the lectures (3 hours /week) mainly concerned with the description of the theoretical concepts, problem solving sessions (1 hour/week) and seminars (2 hours on alternate weeks), similar to the problem solving sessions but where students work in groups supervised by a teaching assistant.
The course has a web page in the UAB online campus gathering all information and communications between students and professors, and where all material, including problem sheets, solutions, etc are published regularly.
Students are supposed to submit a couple of exercise sets to be evaluated in a personalized interview.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Lectures | 39 | 1.56 | |
Problem session | 13 | 0.52 | |
Working seminars | 13 | 0.52 | |
Type: Autonomous | |||
Solving problems | 95 | 3.8 | |
Studying theoretical concepts | 53 | 2.12 |
A continuous assessment is done consisting in a mid-term test (P) and two submissions of exercise sets (LL1,LL2), mandatory and evaluated in a personalized interview. At the end there is a final exam (F) and a resit exam (R)
The final score is obtained in two steps. Let P,LL,F,R denote respectively the scores (between 0 and 10) of the mid-term test, the mean of the two exercise submissions, the final and the resit exams.
With P,F we compute NE=max(F; (0,6)F+(0,4)P) and then C1=(0,80)NE+(0,20)LL. If C1 is greater or equal to 5, C1 is the final score. Otherwise, the students that have submitted the two exercise sets may attend the resit exam. Then the final score is C2=(0,80) R+(0,20)LL.
Students with C1 greater or equal to 5 may attend R to improve their grades, in which case the final score is (C1+C2)/2.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Final test | 48% | 4 | 0.16 | 7, 6, 4 |
Mid-term test | 32% | 4 | 0.16 | 7, 6 |
Seminars | 20% | 4 | 0.16 | 2, 3, 7, 1, 6, 5, 4, 8 |