Randall D. Kamien
Fall 2014
Physics 500: Mathematical Methods of Physics
Math 594: Advanced Methods in Applied Mathematics
Topics to be covered
1. Vector Spaces
2. Linear Algebra
3. Computation of Eigenvalues and Eigenvectors
4. Partial Differential Equat
Physics 500 Mathematical Methods of Physics
Problem Set 10: Due Friday, December 10
Bringing it all together.
1. Laplaces Equation
Here you will solve for the Greens function solution of Laplaces equation in 3D (aka
Coulombs Law in electromagnetism). The
Physics 500 Mathematical Methods of Physics
Problem Set 6: Due Friday, October 31
(But will be accepted until Monday, November 3 due to Halloween)
Bessel and Legendre Equations
1. Solve:
with
u(r, , t)
r
r=
2 u(r, , t)
= 2 u(r, , t)
t2
u(r, , t)
= 0,
Physics 500 Mathematical Methods of Physics
Problem Set 8: Due Friday, November 14
Fourier Transforms
The Fourier transform and its inverse are defined, respectively, as
Fcfw_f (x) = f(k) =
Z
dxf (x)eikx
F
1
1
cfw_f(k) = f (x) =
2
Z
dk f(k)eikx
1. Compute
Physics 500 Mathematical Methods of Physics
Problem Set 3: Due Friday, September 26
Sturm-Liouville Theorem
1. Find the Sturm-Liouville functions p(x), q(x), r(x), and the eigenvalue for the associated
Legendre equation
2
m2
2
u = l(l + 1)u
(1 x ) 2 2x
x
Physics 500 Mathematical Methods of Physics
Problem Set 2: Due Friday, September 19
More Matrix and Linear Algebra
1. For the following matrix
0 12
A= 0 1 0
1
0 12
2
1
2
(a) Find the matrix log(1 + A), where 1 is the identity matrix.
(b) Find the matrix
Physics 500 Mathematical Methods of Physics
Problem Set 5: Due Friday, October 10
Bessels Equation
1. Tolstov Chapter 8, Problem 1 (p.243)
Write the general solution of the differential equation
5
y 00 + y 0 + y = 0
x
2. Each of the following equations ha
Physics 500 Mathematical Methods of Physics
Problem Set 7: Due Friday, November 7
Greens Functions
1. In this problem you will solve
d2 u
= sin(x)
dx2
subject to the boundary conditions u(1) = u(1) = 0, and x [1, 1], by Greens function
methods.
(a) Start
Physics 500 Mathematical Methods of Physics
Problem Set 4: Due Friday, October 3
Orthogonal Polynomials
1. Tolstov Chapter 2, Problem 9 (p. 64)
A system of functions 0 (x), 1 (x), . . . , n (x), . . . is said to be linearly independent if given
any n, the