# Mathematical Methods of Physics

## Objectives

The objective of this course is the acquisition of proficiency in the aplication of mathematical methods to the solution of physical problems.

We will also use this course as an opportunity for students to learn how to use modern mathematical packages on computers to do calculations, graphing, and numerical simulations.

## General characterization

##### Code

11678

##### Credits

6.0

##### Responsible teacher

António Carlos Simões Paiva

##### Hours

Weekly - 4

Total - 42

##### Teaching language

Português

### Prerequisites

Previous approval in the following courses: A.L.G.A., Análise Matemática IB, IIB, IIIB and IVB, or Análise Matemática ID, IID e IIID.

### Bibliography

1- Mathematical Methods in the Physical Sciences, Mary L. Boas, Wiley

2- Métodos Matemáticos para Físicos e Engenheiros, José Paulo Santos e Manuel Fernandes Laranjeira, Fundação da FCT

Also of interest:

3- Mathematical Methods for Scientists and Engineers, Donald A. McQuarrie, Univ Science Books

4- Técnicas Matemáticas da Física, Rui Dilão, IST Press

### Teaching method

The course is organized in lectures where the theory is presented and problems are discussed with the instructor.

### Evaluation method

**Evaluation**

There will be two midterm evaluation tests and a final examination.

**Grading**

Midterms 50% each.

Final Examination 100%

Students must score minimum of 10 out of 20 to have success.

## Subject matter

1. | COMPLEX VARIABLES AND CONFORMAL MAPPING |

1. | Analytic functions and Laplace´s Equation |

2. | The Cauchy-Riemann equations |

3. | Conformal mapping |

4. | Conformal mapping and boundary value problems |

2. | COMPLEX INTEGRATION |

1. | Bound for the absolute value of integrals |

2. | Cauchy´s theorem |

3. | Cauchy´s integral formula |

4. | Singularities and residues |

5. | The Residue Theorem |

6. | Evaluation of Deﬁnite Integrals |

7. | Jordan´s lemma |

8. | Singularities and branch points |

3. | DISTRIBUTIONS |

1. | Test functions |

2. | Distributions |

3. | Support of a distribution |

4. | Operations on distributions |

5. | The Fourier Transform of distributions |

6. | The Fourier Series of distributions |

4. | CALCULUS OF VARIATIONS |

1. | Introduction |

2. | The Euler Equation |

3. | Using the Euler Equation |

4. | The Brachistochrone Problem; Cycloids |

5. | Several Dependent Variables; Lagrange’s Equations |

6. | Isoperimetric Problems |

7. | Variational Notation |

8. | Miscellaneous Problems |