Universitat Internacional de Catalunya
Calculus
Other languages of instruction: Catalan, Spanish
Teaching staff
Office hours will be agreed at ccardo@uic.es
Introduction
Objectives
This course aims to familiarize the student with the concepts of differential and integral calculus in multiple dimensions, as well as provide an introduction to differential equations.
Competences/Learning outcomes of the degree programme
- CP03 - Apply the techniques of graphical representation, both by traditional methods of metric geometry and descriptive geometry, as well as by computer-aided design applications.
- HB09 - Solve problems that may arise in the field of Bioengineering by applying mathematical knowledge (geometry, integral calculation, numerical methods and optimisation) and the general laws of mechanics and biomechanics.
Learning outcomes of the subject
After passing the subject, students will have acquired the following skills:
- A good command of formal mathematical language
- A good domain of differential calculation in multiple variables
- A good domain of multiple integrals
- Ability to analyze and synthesize the information obtained in the course
- Ability to interpret and solve basic differential equations
- Ability to formulate and solve optimization problems
- Ability to use computer calculation programs
Syllabus
Tema 1. UNIVALUATED CALCULUS
1.1 Introduction
1.2 Univaluated calculus. Derivatives. Series. Taylor theorem.
1.3 Calculus with several variables. Directional derivative. Gradient.
Tema 2. OPTIMIZACION
2.1 Optimization with one variable.
2.2 Optimization with several variables. Hessian Matrix. Matrix sign.
2.3 Constrained optimization. Method of Lagrange multiplier.
Tema 3. SIMPLE INTEGRATION
3.1 Riemann integration definition. Integrals immediatas
3.2 Integration methods. By parts. Change of variables. Rational and irrational integration.
Tema 4. MULTIPLE INTEGRATION
4.1 Multiple integration definition. Rectangular domain. Fubini theorem.
4.2 Integration with non rectangular domains.
4.3 Change of variables. Jacobian matrix.
Tema 5. DIFFERENTIAL EQUATIONS
5.1 Differential equations, definition, examples and applications. Picard theorem.
5.2 Types and methods of resolution. Separated variables, homogeneous, linear and Bernoulli equations. Laplace transform.
5.3 Applications of differential equations.
Teaching and learning activities
In person
Methodology
The subject will be taught face-to-face through theoretical classes and problem solving sessions. The theory of the subject will be exposed in a rigorous way avoiding, however, an excess of formalization, which could mask the true purpose of the subject: teach the basics of infinitesimal calculus to bioengineers. For this reason, the conceptual clarity and resolution of multiple examples will be emphasized using the R software. In addition, applications of calculation tools will be shown to problems of engineering interest such as calculation of the center of gravity, or moments of inertia.
Learning activities
Master Class and resolution of exercises and problems: 60 h (Presence: 100%)
Preparation and realization of evaluable activities: 30 h (Presence: 0%)
Autonomous work of study and performance of exercises: 60 h (Presence: 0%)
Evaluation systems and criteria
In person
During the course it will be necessary to present and carry out the following evaluation tasks:
E: submissions of exercises, tests or works.
P: partial exam of half of the syllabus and which will take place in the middle of the course.
F: final exam of the whole syllabus.
Evaluation:
C: continuous evaluation is the weighted average consisting of C=20%E + 20%P* + 60%F.
FINAL GRADE: the final grade in the first call will result from calculating the maximum between the continuous evaluation mark C and the mark of the final exam F. However, when F was less than 3.5, F will be the final grade.
In the second call will remain without altering the mark of the deliveries and the mark of the partial exam. Only final exam will be repeated and the same system of evaluation as in the first call will be applied. .
Re-sit:
Students who register for the second time or more may present the work upon express request and under the conditions specified and in no case will the grade of the work or any exam grade from the previous year count. If the formal request is not made by mail during the first two weeks of the course, the continuous assessment grade will be C=40%P + 60%F. The final grade will be calculated in the same way as mentioned above.
Important considerations:
- Plagiarism, copying or any other action that may be considered cheating will be zero in that evaluation section. Besides, plagiarism during an exam will mean the automatic failing of the whole subject.
- In the second-sitting exams, the maximum grade students will be able to obtain is "Excellent" (grade with honors distinction will not be possible).
- Changes of the calendar, exam dates or the evaluation system will not be accepted.
- Exchange students (Erasmus and others) or repeaters will be subjected to the same conditions as the rest of the students.
Bibliography and resources
Main references
- Cálculus (2012). Robert Adams: A complete Course, Addison Wesley.
- Algebra lineal y Cálculo para estudiantes de química (2015) Jesús Medina Moreno. Paranimfo.
- Cálculo diferencial e integral (2015). Nikolai Piskunov. Limusa
- Calculus (1988) Michael Spivak. Reverté.
Evaluation period
- E1 28/05/2026 A12 10:00h