Vector and Multivariable Calculus
Code: 107598Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Physics | FB | 1 |
Contact lecturer
- Name :
- Alessio Celi
- Email :
- alessio.celi@uab.cat
Teaching staff
- Axel Pérez-Obiol Castañeda
- Marc Miranda Riaza
Group languages
You can consult this information at the end of the document.
Prerequisites
There are no prerequisites for enrolment.
However, for the development of the course, it is assumed that students have assimilated the contents of the Càlcul d'una variable course from the first semester.
Objectives
This course is the natural continuation of the Càlcul d'una variable course from the first semester. It deals with calculus involving functions of several real variables and the study of their differential properties. The main objective is to provide students with the necessary mathematical tools to successfully follow second-year courses such as Classical Mechanics, Electromagnetism, and Optics.
Learning outcomes
- CM09 (Justify the use of calculus in one and several variables and differential equations in the resolution of general problems.) Justify the use of calculus in one and several variables and differential equations in the resolution of general problems.
- CM10 (Adapt the basic mathematical strategy when approaching a given problem from an analytical point of view.) Adapt the basic mathematical strategy when approaching a given problem from an analytical point of view.
- KM09 (Identify the basic concepts of limits, continuity, derivatives and integrals, vector and subspace space, linear and scalar product and the methodology of matrix diagonalisation.) Identify the basic concepts of limits, continuity, derivatives and integrals, vector and subspace space, linear and scalar product and the methodology of matrix diagonalisation.
- KM10 (Describe the basic concepts of the calculation of several variables and the different methods of solving differential equations in their different typologies.) Describe the basic concepts of the calculation of several variables and the different methods of solving differential equations in their different typologies.
- SM07 (Apply the mathematical knowledge acquired to the resolution of mathematical and physical problems with mathematical representation.) Apply the mathematical knowledge acquired to the resolution of mathematical and physical problems with mathematical representation.
Contents
- Functions of several variables (limits, continuity, scalar and vector-valued functions, 𝑅𝑛 spaces, norm, distance)
- Differential calculus (partial derivative, directional derivative, differential, gradient, Hessian, implicit function theorem, inverse function theorem, maxima and minima, constrained extrema, Lagrange multipliers)
- Integration of functions of several variables (parameter-dependent integrals, Leibniz rule, Riemann integral in two variables, double and triple integrals, change of integration order)
- Vector-valued functions (examples, vector fields, divergence and curl, vector differential calculus)
- Line and surface integrals (integration over curves and surfaces, surface area calculation, arc length of a curve)
- Integral theorems of vector calculus (Green’s theorem, Stokes’ theorem, Gauss’ theorem, conservative vector fields)
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Problem solving | 45 | 1.8 | CM10, SM07 |
| Problem classes | 14 | 0.56 | CM10, SM07 |
| Theory lectures | 28 | 1.12 | CM09, KM09, KM10 |
| Study | 44 | 1.76 | CM09, KM09, KM10 |
| Seminars/In-depth study | 8 | 0.32 | CM09, CM10, SM07 |
Theory classes: Presentation of the theoretical content of the course.
Problem-solving classes: Presentation of the solution to some of the problems from the list previously given to the students, along with guidance for solving the remaining ones. In-class problem-solving by the students, working on proposed exercises under the supervision of the instructor.
Seminar/Advanced session: Activity aimed at reviewing and deepening a theoretical topic and/or solving specific problems.
Note: Fifteen minutes of one class session, within the schedule established by the department/degree program, will be reserved for students to complete surveys evaluating the teaching performance of the instructor and the course/module.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Recovery Exam | 40-80% | 3 | 0.12 | CM09, CM10, KM09, KM10, SM07 |
| Partial exam 1 | 40% | 2 | 0.08 | CM09, CM10, KM09, KM10, SM07 |
| Tests (and in-class revision) | 0-40% | 4 | 0.16 | CM10, KM09, KM10, SM07 |
| Partial exam 2 | 40% | 2 | 0.08 | CM09, CM10, KM09, KM10, SM07 |
Continuous Assessment
- Graded tests T1 and T2: These will be taken in class and graded from 0 to 10. They will cover the topics addressed from the beginning of the first and second parts of the course, respectively, up to the time of the test. This grade cannot be improved through the resit exam.
- Midterm exams E1 and E2: These will take place in the middle and at the end of the semester, respectively, and will be graded from 0 to 10.
- Final grade N: The final grade for the course will be calculated according to the following formula:
- N = 0.1*(T1+T2) + (10-0.1*(T1+T2))*(E1+E2)/20
- Resit exam R: This exam allows students to improve the grade obtained in the midterm exams (graded from 0 to 10), and either one or both midterms can be retaken. To be eligible for the resit exam, students must have sat both midterm exams and achieved a final continuous assessment grade of N ≥ 3/10. Students taking the resit exam must mandatory retake any midterm where they scored below 4.9. The resit exam grade will replace the corresponding midterm grade.
- Non-assessable (No avaluable): Students will be graded as "Non-assessable" if they have not completed assessment activities representing at least 50% of the final grade and have a grade of N < 3/10.
Single Assessment
- Final exam: Students enrolled in the single assessment modality will take a single final exam covering the entire syllabus. This exam will take place on the same day, time, and location as the second midterm exam of the continuous assessment modality.
Resit: Single assessment students have the opportunity to pass the course or improve their grade through the same resit exam as continuous assessment students (both exams will be identical and will take place on the same day, time, and location). To be eligible for the resit, students must have sat the final exam and achieved a grade of ≥ 3/10.
Bibliography
Basic Biliografy:
- T.M. Apostol, Calculus (vol.2), Reverté.
Basic more advanced bibliography:
- J.E. Marsden and J. Tromba, Vector Calculus, W.H. Freeman and Co.
- A. Méndez, Càlcul de vàries variables, notes de classe
- J.M. Ortega, Introducció a l'anàlisi matemàtica, Manuals de la UAB.
- J. Rogawski, Càlculo (vol.2), Reverté.
- R. Courant and F. John, Introducción al análisis matemático (vol.2), Limusa.
Software
The use of AI is permitted solely as an individual study tool. All graded activities will take place in class. During these activities, the use of AI is strictly prohibited. Any unauthorized use will result in the invalidation of the assessment and will be reported to the degree coordinator.
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Catalan | second semester | morning-mixed |
| (TE) Theory | 2 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 11 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 12 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 21 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 22 | Catalan | second semester | morning-mixed |