# 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 - 3

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

### 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 homework assignments.

**Grading**

Midterms 35% each and 30% for the assignments.

Final Examination 100%

Students must score minimum of 10 out of 20 to pass.

## Subject matter

1. |
FOURIER SERIES |

1. |
Introduction |

2. |
Simple Harmonic Motion and Wave Motion; Periodic Functions |

3. |
Applications of Fourier Series |

4. |
Fourier Coeﬃcients |

5. |
Dirichlet Conditions |

6. |
Complex Form of Fourier Series |

7. |
Other Intervals |

8. |
Even and Odd Functions |

9. |
An Application to Sound |

10. |
Parseval’s Theorem |

2. |
DISTRIBUTIONS |

1. |
Definitions |

2. |
Operations |

3. |
FOURIER TRANSFORMS |

1. |
Introduction |

2. |
Essential Transforms |

3. |
The Dirac Delta Function |

4. |
Simple Theorems |

5. |
The Convolution Theorem |

6. |
The Parseval’s Formula |

7. |
Applications |

4. |
FUNCTIONS OF A COMPLEX VARIABLE |

1. |
Introduction |

2. |
Updates |

3. |
The Residue Theorem |

4. |
Methods of Finding Residues |

5. |
Evaluation of Deﬁnite Integrals |

6. |
The Point at Inﬁnity; Residues at Inﬁnity |

7. |
Conformal Mapping |

5. |
CALCULUS OF VARIATIONS |

1. |
Introduction |

2. |
The First Variation. Euler’s and Lagrange’s approach |

3. |
Cases and Examples. Minimal surface of revolution; the brachistochrone; Geodesics |

4. |
Generalizations. Lagrangian mechanics; Hamiltonian mechanics |

5. |
Several Dependent Variables |

6. |
Variational Notation |

7. |
Constraints. Isoperimetric constraints |