Table of Contents

# PH461/561: Math Methods

## Fall 2010

PH461/561 serves as a capstone undergraduate course and introductory graduate level course. Graduate students taking PH561 should also enroll in the companion one credit class PH505. In Fall 2010 these courses are being taught by Prof. Ethan Minot.

## Calendar

Day | Topic | Reading | Summary | Assignments | |
---|---|---|---|---|---|

1 | M 9/27 | Complex numbers | Ch 3 | HW #1 | |

2 | W 9/29 | Hyperbolic functions | Ch 3 | ||

3 | F 10/1 | Series | Ch 4 | hw 1 soln | |

Day | Topic | Reading | Summary | Assignments | |

4 | M 10/4 | Series convergence | Ch 4 | Including power series | HW #2 |

5 | W 10/6 | Application of Power Series: Taylor series | Ch 4 | ||

6 | F 10/8 | Fourier series | Ch 4 | hw2 soln | |

Day | Topic | Reading | Summary | Assignments | |

7 | M 10/11 | Matrices | Ch 8 | eigenvectors & eigenvalues | HW #3 |

8 | W 10/13 | Matrices | Ch 8 | approaches to solve linear equations | |

9 | F 10/15 | Vector Algebra | - | worksheet_2.doc | hw3 soln |

Day | Topic | Reading | Summary | Assignments | |

10 | M 10/18 | Review of worksheet 2 | Chpt 8 | Rank, orthonormal basis | HW #4 |

11 | W 10/20 | Vector Calculus | Chpt 10 | Cartesian Coordinates | |

12 | F 10/22 | Vector Calculus | Chpt 10 | Numerical method ∇²Φ=0, cylindrical coords | hw4 soln |

Day | Topic | Reading | Summary | Assignments | |

13 | M 10/25 | Vector Calculus | Chpt 10 | Curvilinear coordinates | |

14 | W 10/27 | MIDTERM | - | - | Midterm |

15 | F 10/29 | Ordinary Diff. Eqns. | Chpt 14 | Separable variable, Exact, Inexact | - |

Day | Topic | Reading | Summary | Assignments | |

16 | M 11/1 | Ordinary Diff. Eqns. | Chpt 14 | Homogeneous, Isobaric, Bernoulli's | HW #5 |

17 | W 11/3 | Ordinary Diff. Eqns. | Chpt 15 | worksheet3.doc, Euler's ODE method, Higher Order: n lin. ind. solns. | |

18 | F 11/5 | Ordinary Diff. Eqns. | Chpt 15 | Complementary, Particular, Guessing y_c and y_p | hw5 soln |

Day | Topic | Reading | Summary | Assignments | |

19 | M 11/8 | Ordinary Diff. Eqns. | Chpt 15 | Method of variation of parameters | HW #6 |

20 | W 11/10 | Power series solns of ODEs | Chpt 16 | Special 2nd order Ord. Diff Eqs, Wronskian, Series Solution | |

21 | F 11/12 | Power series solns of ODEs | Chpt 16 | Recursion relation | hw6 soln |

Day | Topic | Reading | Summary | Assignments | |

22 | M 11/15 | Power series solns of ODEs | Chpt 16 & 18 | Legendre Eqn., Series solution about regular singular point, indicial eqn | HW #7 |

23 | W 11/17 | Special functions | Chpt 18 | Generating functions, Bessel functions | |

24 | F 11/19 | hw7 soln | |||

Day | Topic | Reading | Summary | Assignments | |

25 | M 11/22 | - | worksheet_4.doc | HW #8 | |

26 | W 11/24 | Partial Differential Equations | Chpt 20 | ||

27 | F 11/26 | THANKSGIVING HOLIDAY | - | ||

Day | Topic | Reading | Summary | Assignments | |

28 | M 11/29 | Partial Differential Equations | Chpt 20 | - | |

29 | W 12/1 | Partial Differential Equations | Chpt 20 | - | |

30 | F 12/3 | Review for final | - | ph461exercises.pdf | hw8 soln |

## Syllabus

- Complex numbers
- Manipulating complex numbers
- Functions of complex numbers
- Hyperbolic functions

- Series
- Convergence of series
- Taylor series expansions
- Fourier series

- Matrices and vector spaces
- Eigenvalues and eigenvectors
- Vector algebra

- Vector calculus
- in cartesian coordinates,
- in cylindrical coordinates
- in spherical coordinates
- Line integrals
- Conservative fields
- Surface and volume integrals

- Differential Equations
- First-order ordinary differential equations (ODEs)
- Higher-order ODEs
- Series solutions of ODEs
- Eigenfunction method for ODEs

- Special functions
- Legendre
- Hermite
- Gamma
- Bessel
- Beta

- Partial differential equations (PDEs)
- First order PDEs
- Second order PDEs
- Diffusion equation