Course code: 250206 |
Subject title: MATHEMATICS II |
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Credits: 6 |
Type of subject: Basic |
Year: 1 |
Period: 2º S |
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Department: |
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Lecturers: |
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PORTERO EGEA, LAURA (Resp) [Mentoring ] | ARRARAS VENTURA, ANDRÉS [Mentoring ] |

Elements of differential and integral calculus in several variables. Ordinary differential equations.

General proficiencies that a student should acquire in this course:

**G8**Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations**G9**Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in computer engineering**T1**Analysis and synthesis ability**T3**Oral and written communication**T4**Problem solving**T8**Self-learning

Specific proficiencies that a student should acquire in this course:

**FB1**Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential and integral calculus, numerical methods, numerical algorithms, statistics and optimization**FB3**Ability to understand and master the basic concepts of discrete matemathics, logic, algorithmics and computational complexity, and their applications to problem solving in engineering.

At the end of the course, the student is able to:

**O1:**Apply the basic elements of differential calculus in several variables: limits, continuity, differentiability.

**O2:**Formulate and solve unconstrained and constrained optimization problems.

**O3:**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, etc.

**O4:**Understand the basic elements of vector calculus: flux integral, gradient, divergence, curl, integral theorems.

**O5:**Recognize and solve some basic types of ordinary differential equations.

The following table shows the distribution of activities in the course:

Methodology - Activity | On-site hours | Off-site hours |

A-1: Theoretical lectures | 45 | |

A-2: Practical lectures | 15 | |

A-3: Self-study | 80 | |

A-4: Exams and assessment | 5 | |

A-5: Tutoring | 5 | |

TOTAL | 70 | 80 |

Proficiency | Activities |

G8 | A-1, A-2, A-3, A-5 |

G9 | A-1, A-2, A-3, A-5 |

FB1 | A-1, A-2, A-3, A-4, A-5 |

FB3 | A-1, A-2, A-3, A-5 |

T1 | A-1, A-2, A-3, A-5 |

T3 | A-1, A-2, A-4 |

T4 | A-2, A-4, A-5 |

T8 | A-2, A-3 |

Continuous assessment is considered along the semester based on the following activities:

Learning outcome | Assessment activity | Weight (%) | Resit assessment |

O1, O2 | Midterm exam A on lessons 1 and 2 | 35 | Yes (final exam) |

O3, O4 | Midterm exam B on lessons 3 and 4 | 45 | Yes (final exam) |

O5 | Midterm exam C on lesson 5 | 20 (Minimum to be considered in the final mark: 3 out of 10) | Yes (final exam) |

In order to pass the subject, one of the following conditions must be fulfilled:

- the mark of the midterm exam C is not less than 3 (out of 10) and the weighted mark of all three midterm exams is not less than 5 (out of 10);
- the mark of the final exam covering the whole course (to be scheduled during the resit assessment period) is not less than 5 (out of 10). Only students who did not pass the course by continuous assessment can sit this exam.

- Functions of several variables: limits, continuity, differentiation, Taylor series and graphics.
- Multiple integration. Applications.
- Vector calculus.
- Ordinary differential equations.

**Functions, limits and continuity in R**. Definition. Scalar and vector functions. Limits. Continuity.^{n}**Differential calculus in R**. Partial and directional derivatives. Gradient vector and Jacobian matrix. Differentiability. The chain rule. Higher-order derivatives. Hessian matrix. Taylor polynomials. Relative extrema. Constrained optimization: the theorem of Lagrange multipliers. Absolute extrema on closed bounded regions.^{n}**Integral calculus in R**. The Riemann integral. Elementary regions. Fubini's theorem. Change of variables. Polar, cylindrical and spherical coordinates.^{n}**Vector calculus**. Conservative fields. Potential function. Line and surface integrals. Circulation and flux. Divergence and curl. Green's, divergence and Stokes' theorems.**Ordinary differential equations**. Basic notions on differential equations. First-order ordinary differential equations. Existence and uniqueness of solution. Some elementary integration methods. Higher-order linear differential equations. Applications.

**Access the bibliography that your professor has requested from the Library.**

Basic bibliography:

- R.A. Adams.
*Calculus: a complete course*. 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, A.D. Snider.
*Fundamentals of differential equations and boundary value problems*. Addison-Wesley.

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.