Module/Subject matter
Basic training / M11 Mathematics.
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Contents
Vector functions of several variables.
Integral Calculus of functions of several variables. Applications.
Ordinary and partial differential equations.
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General proficiencies
General proficiencies that a student should acquire in this course:
- CG3: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations.
- CG4: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in engineering.
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Specific proficiencies
Specific proficiencies that a student should acquire in this course:
- CFB1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, geometry, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization.
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Learning outcomes
At the end of the course, the student is able to:
O1. Apply the basic elements of differential calculus on several variables: gradient, divergence, curl, Stokes theorems.
O2. Apply the basic elements of integral calculus in several variables, e.g., to determine the length of a curve, the area of a surface, the volume of a solid,... using integrals, and use numerical differentiation and integration techniques.
O3. Apply Calculus to Engineering.
O4. Understand the concept of differential equation, and solve basic ordinary differential equations.
O5. Apply partial differential equations: wave equation and heat equation.
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Methodology
Methodology - Activity |
On-site hours |
Off-site hours |
A-1: Theoretical lessons |
45 |
|
A-2: Practical lessons |
15 |
|
A-3: Individual study |
|
75 |
A-4: Tutoring and exams |
15 |
|
TOTAL |
75 |
75 |
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Evaluation
Learning outcome |
Assessment activity |
Weight (%) |
It allows test resit |
Minimum required grade |
All |
Long-answer exam questions |
60 |
Yes (in the final exam) |
5 |
All |
Individual work |
30 |
Yes (in the final exam) |
5 |
All |
Practical exam questions |
10 |
Yes (in the final exam) |
5 |
|
|
|
|
|
For assessment purposes, the course is divided into two parts:
- Part A (related to lessons 1, 2 and 3) is worth 55%.
- Part B (related to lessons 4 and 5) is worth 45%.
In order to pass the subject the average mark of all two parts must be greater or equal than 5. The mark on a final exam covering the whole course (to be scheduled during the resit assessment period) is not less than 5. Only students who did not pass the course by continuous assessment can sit this exam.
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Agenda
- Functions, limits and continuity in Rn. Basic concepts in scalar and vector functions of several variables. Limits. Continuity.
- Differential calculus in Rn. Partial and directional derivatives. Gradient vector and Jacobian matrix. Differentiability. Chain rule. Higher-order derivatives. Hessian matrix. Taylor polynomials. Relative extrema. Constrained optimization: the theorem of Lagrange multipliers. Absolute extrema on closed and bounded regions. Applications.
- Differential equations. Basic notions on differential equations. First-order ordinary differential equations. Some elementary integration methods. Second-order linear differential equations. Introduction to partial differential equations. Applications.
- Integral calculus in Rn. The Riemann integral. Elementary regions. Fubini's theorem. Change of variables. Polar, cylindrical and spherical coordinates. Applications.
- Line and surface integrals. Scalar and vector fields. Conservative fields. Potential function. Line and surface integrals of scalar and vector fields. Circulation and flux. Divergence and curl. Green's, divergence and Stokes' theorems. Applications.
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Experimental practice program
Use of Mathematica and Wolfram Alpha computational intelligence for solving integral and differential problems.
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Bibliography
Access the bibliography that your professor has requested from the Library.
Basic bibliography:
- R.A. Adams. Calculus: a complete course, Ninth edition. Addison Wesley.
- E. Kreyszig. Advanced engineering mathematics. John Wiley & Sons.
- J.E. Marsden, A.J. Tromba. Vector calculus. W.H. Freeman.
- R.K. Nagle, E.B. Saff, Ecuaciones diferenciales y problemas con valores en la frontera. Pearson Educación.
Additional bibliography:
- M. Braun. Differential equations and their applications: an introduction to applied mathematics. Springer-Verlag.
- R.E. Larson, R.P. Hostetler. Cálculo y geometría analítica. McGraw-Hill.
- S.L. Salas, E. Hille, G.J. Etgen. Calculus: una y varias variables. Reverté.
- D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado. Thomson.
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Location
Lecture room building at Arrosadia Campus.
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