Calculus
Module: FUNDAMENTALS
Matter: MATHEMATICS I
Main language of instruction: Spanish
Other languages of instruction: Catalan, English
Head instructor
Dr. Carles CARDÓ - ccardo@uic.es
Office hours
Office hours will be agreed at ccardo@uic.es
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.
After passing the subject, students will have acquired the following skills:
Topic 0. Preliminaries
0.1 Review of derivatives and integrals in 1 dimension.
0.2 Series.
Topic 1. Differential calculus in several variables
1.1 Functions of several variables.
1.2 Partial derivatives of the function of several variables.
1.3 Derivatives of compound functions and derivatives of implicit functions.
1.4 Directional derivatives. Gradient.
1.5 Extremes of a function of several variables.
1.6 Advanced topics of practical interest: a) Lagrange multipliers b) least squares method.
Topic 2. Comprehensive calculus in multiple dimensions
2.1 Preliminaries: integrals in a variable.
a) integration methods b) engineering and geometric applications.
2.2 Introduction to double integrals.
2.3 Calculation of areas and volumes by double integrals.
2.4 Change of variables in a double integral.
2.5 Triple integrals.
2.6 Change of variables in a triple integral.
2.7 Engineering applications of multiple integrals (center of gravity, moments of inertia, density of matter).
2.8 Numerical integration.
Topic 3. Differential equations
3.1 Preliminary: approach to the problem. Definitions.
3.2 Differential equations of the first order.
3.3 Equations of separated and separable variables.
3.4 Homogeneous first order equations.
3.5 First order linear equations.
3.6 Differential equations of second order that are reduced to the first order.
3.7 Some differential equations of engineering importance.
3.8 Laplace transform.
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 computerization, 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%)
The subject is approved making 4 deliveries and 2 written exams.
The 4 deliveries (homework) will involve problems to solve. Homework delivered with a delay of up to 3 days after the deadline will be evaluated with half the maximum grade. Otherwise (more than 3 days late) will not be evaluated. It will be obligatory to deliver the "homeworks" on paper or scanned and uploaded in the corresponding space of the intranet of the subject. The homeworks sent by email will not be evaluated.
The written exam will consist of two sessions held on different dates. The first exam will be done towards the middle of the course. The second exam will be held at the end of the course and will include the syllabus of the whole subject. The exams will not allow the consultation of books or notes.
The following weighting will be applied to evaluable activities:
1. Problem solving/deliveries (weighting: 20%)
2. Partial examination (weighting: 20%)
3. Final exam (weighting: 60%)
Second call
Students who fail the subject will have the opportunity to repeat the final exam. The notes of the first partial and of the deliveries will remain unchanged.
Re-sit
Students who repeat the subject will have to deliver the 4 "homeworks" again and perform both exams.
Important considerations:
Main references
- "Cálculo diferencial e integral" (2015). Nikolai Piskunov. Limusa
Supplement references
- "Cálculo" (2009). Robert Adams (trad. esp. de Inés Portillo García), Addison Wesley, 6ª edición. Sistemas y criterios de evaluación.
- "Start R in Calculus" (2013). Daniel Kaplan. Project Mosaic.
- "Solving Differential Equations in R" (2012). Karline Soetaert, Jeff Cash, y Francesca Mazzia. Springer Science & Business Media.
E: exam date | R: revision date | 1: first session | 2: second session: