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
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1. |
FOURIER SERIES |
|
1. |
Introduction |
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2. |
Simple Harmonic Motion and Wave Motion; Periodic Functions |
|
3. |
Applications of Fourier Series |
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4. |
Fourier Coefficients |
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5. |
Dirichlet Conditions |
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6. |
Complex Form of Fourier Series |
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7. |
Other Intervals |
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8. |
Even and Odd Functions |
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9. |
An Application to Sound |
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10. |
Parseval’s Theorem |
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2. |
DISTRIBUTIONS |
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1. |
Definitions |
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2. |
Operations |
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3. |
FOURIER TRANSFORMS |
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1. |
Introduction |
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2. |
Essential Transforms |
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3. |
The Dirac Delta Function |
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4. |
Simple Theorems |
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5. |
The Convolution Theorem |
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6. |
The Parseval’s Formula |
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7. |
Applications |
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4. |
FUNCTIONS OF A COMPLEX VARIABLE |
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1. |
Introduction |
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2. |
Updates |
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3. |
The Residue Theorem |
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4. |
Methods of Finding Residues |
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5. |
Evaluation of Definite Integrals |
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6. |
The Point at Infinity; Residues at Infinity |
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7. |
Conformal Mapping |
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5. |
CALCULUS OF VARIATIONS |
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1. |
Introduction |
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2. |
The First Variation. Euler’s and Lagrange’s approach |
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3. |
Cases and Examples. Minimal surface of revolution; the brachistochrone; Geodesics |
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4. |
Generalizations. Lagrangian mechanics; Hamiltonian mechanics |
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5. |
Several Dependent Variables |
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6. |
Variational Notation |
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7. |
Constraints. Isoperimetric constraints |