# 406-0253-AAL Mathematics I & II

Semester | Autumn Semester 2016 |

Lecturers | A. Cannas da Silva |

Periodicity | every semester recurring course |

Language of instruction | English |

Comment | Enrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. |

### Catalogue data

Abstract | Mathematics I covers mathematical concepts and techniques necessary to model, solve and discuss scientific problems - notably through ordinary differential equations. Main focus of Mathematics II: multivariable calculus and partial differential equations. |

Objective | Mathematics is of ever increasing importance to the Natural Sciences and Engineering. The key is the so-called mathematical modelling cycle, i.e. the translation of problems from outside of mathematics into mathematics, the study of the mathematical problems (often with the help of high level mathematical software packages) and the interpretation of the results in the original environment. The goal of Mathematics I and II is to provide the mathematical foundations relevant for this paradigm. Differential equations are by far the most important tool for modelling and are therefore a main focus of both of these courses. |

Content | 1. Linear Algebra and Complex Numbers: systems of linear equations, Gauss-Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, cartesian and polar forms for complex numbers, complex powers, complex roots, fundamental theorem of algebra. 2. Single-Variable Calculus: review of differentiation, linearisation, Taylor polynomials, maxima and minima, antiderivative, fundamental theorem of calculus, integration methods, improper integrals. 3. Ordinary Differential Equations: separable ordinary differential equations (ODEs), integration by substitution, 1st and 2nd order linear ODEs, homogeneous systems of linear ODEs with constant coefficients, introduction to 2-dimensional dynamical systems. 4. Multivariable Differential Calculus: functions of several variables, partial differentiation, curves and surfaces in space, scalar and vector fields, gradient, curl and divergence. 5. Multivariable Integral Calculus: multiple integrals, line and surface integrals, work and flow, Green, Gauss and Stokes theorems, applications. 6. Partial Differential Equations: separation of variables, Fourier series, heat equation, wave equation, Laplace equation, Fourier transform. |

Literature | - Bretscher, O.: Linear Algebra with Applications (Pearson Prentice Hall). - Thomas, G. B.: Thomas' Calculus, Part 1 - Early Transcendentals (Pearson Addison-Wesley). - Thomas, G. B.: Thomas' Calculus, Parts 2 (Pearson Addison-Wesley). - Kreyszig, E.: Advanced Engineering Mathematics (John Wiley & Sons). |

Prerequisites / Notice | Prerequisites: familiarity with the basic notions from Calculus, in particular those of function and derivative. Assistance: Tuesdays and Wednesdays 17-19h, in Room HG E 41. |

### Performance assessment

Performance assessment information (valid until the course unit is held again) | |

Performance assessment as a semester course | |

ECTS credits | 13 credits |

Examiners | A. Cannas da Silva |

Type | session examination |

Language of examination | English |

Repetition | The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit. |

Mode of examination | written 180 minutes |

Written aids | 40 pages A4 (= 20 sheets of paper - front & back sides) summary. 1 English dictionary. No (pocket) calculators, no cellular phones. |

This information can be updated until the beginning of the semester; information on the examination timetable is binding. |

### Learning materials

Main link | Mathematics I: course syllabus and suggested work plan in English |

Additional links | Mathematics II: course syllabus and suggested work plan in English |

Only public learning materials are listed. |

### Courses

Number | Title | Hours | Lecturers | |
---|---|---|---|---|

406-0253-AA R | Mathematics I & II Self-study course. No presence required. | 390s hrs | A. Cannas da Silva |

### Groups

No information on groups available. |

### Restrictions

There are no additional restrictions for the registration. |

### Offered in

Programme | Section | Type | |
---|---|---|---|

Health Sciences and Technology Master | Course Units for Additional Admission Requirements | E- | |

Environmental Sciences Master | Course Units for Additional Admission Requirements | E- |