Revision and Warmup exercises These problems are designed to exercise your basic mathematical skills. They are not designed to be easy! Each one has some twist that is designed to catch you out if you merely manipulate symbols without thinking. Diff
1) Dielectric Sphere: Consider a solid dielectric sphere of radius a and permittivity . The sphere is placed in a electric field which is takes the constant value E = E0 ^ a long distance z from the sphere. Recall that Maxwell's equations require tha
1) Elastic Rods. The elastic energy per unit length of a bent steel rod is given by 1 Y I/R2 . 2 Here R is the radius of curvature due to the bending, Y is the Young's modulus of the steel and I = y 2 dxdy is the moment of inertia of the rod's cross
1) Test functions and distributions: Read the sections on distributions in chapter two of the lecture notes, then do the following problems: a) Let f (x) be a smooth function. i) Show that f (x)(x) = f (0)(x). Deduce that d [f (x)(x)] = f (0) (x). dx
Solutions to Homework Set 8 1) Pantograph Drag: The cable supports waves with dispersion equation 2 = 2 + c2 k 2 . Here c2 = T /. The phase velocity /k is always > c, while the group velocity /k = c2 k/ is always < c. a) We seek a solution of the for
Here are some optional problems on integral equations. They are taken verbatim from Paul Goldbart's homework sets. 1) Integral equations: a) Solve the inhomogeneous type II Fredholm integral equation
1
u(x) = ex +
0
b) Solve the homogeneous type I
1) Fermat's principle: According to Fermat's principle, the path taken by a ray of light between any two points makes stationary the travel time between those points. A medium is characterised optically by its refractive index n, such that the speed
1) Linear differential operators: ^ a) Consider the differential operator L = id/dx. Find the formal adjoint of L with respect to the inner product uv = wu v dx, and find the corresponding surface term Q[u, v]. b) Now do the same for the operator M
Solutions to Homework Set 0 Differential calculus: The point of the exercise was to make sure that you know how to differentiate integrals with respect to their limits: d da and in general d dt
b(t) b
f (x) dx = f (a),
a
d db
b
f (x) dx = f (b),
1) Flexible rod again: A flexible rod is supported near its ends by means of knife edges that constrain its position, but not its slope or curvature. It is acted on by by a force F (x).
y
x x=0 F(x)
Simply supported rod. The deflection of the rod i
Solutions to Homework Set 3 Test functions and distributions: For part a) we take any test function (x) and look at
(, f + f )

(x) {f (x) (x) + f (x)(x)} dx {[ (x)f (x)  (x)f (x)](x) + (x)f (x)(x)} dx

=
=  (0)f (0), and compare it with
Solutions to Homework Set 6 1) Flexible rod again: This is a somewhat tedious, but ultimately rewarding exercise. a) Look back at your solution for homework set 4, where you showed that
1 0
u v (4) dx = [u v (3)  (u ) v + (u ) v  (u(3) ) v]1 + 0
1) Missing State: In Homework Set 4 you found that the Schrdinger equation o  has eigensolutions k (x) = eikx (ik + tanh x) with eigenvalue E = k 2 . For x large and positive k (x) A eikx ei(k) , while for x large and negative k (x) A eikx ei(k
Solutions to Homework Set 9 1) Conducting strip:
+ a
A(k) =

V (x)eikx dx = V0
a
eikx dx =
2V0 sin ka k
From this we have V (x, 0, y) = 2V0

dk sin(ka) ikx ky e e . 2 k
0 (Ey y=+
Taking the y gradients to get Ey , and then getting
1) Pantograph Drag: A highspeed train picks up its electrical power via a pantograph from an overhead line. The locomotive travels at speed U and the pantograph exerts a constant vertical force F on the power line.
A highspeed train. We make the u
1) Critical Mass: An infinite slab of fissile material has thickness L. The neutron density n(r) in the material obeys the equation n = D 2 n + n + , t where n is zero at the surface of the slab at x = 0, L. Here D is the neutron diffusion constant,
Solutions to Homework Set 4 1) Linear Differential operators: a) Integrating by parts gives us
b
uLv
w
=
a
wu i
d v dx
b
dx w i d wu v dx w dx L uv w .
= [iwu v]b + a
a
Therefore the formal adjoint is L = and the boundary term is
[Q]b a
Physics 498/MMA Handout 9 Oct 10th 2002
Mathematical Methods in Physics I http:/w3.physics.uiuc.edu/mstone5
Homework 9
Prof. M. Stone 2117 ESB University of Illinois
1) Conducting Strip: A thin insulated conducting strip of width 2a extends infi
Do question one, and then any two of the other three questions. Try to answer entire questions. Little, if any, credit will be given for fragmentary answers. 1) Green Function: Consider the homogeneous boundary value problem y = f (x), y(0) = y (1)
Solutions to Homework Set 1 1) Snellius' law: Parts a) and b) are trivial, so I will not write out the solutions. Part c) has two subparts: i) Setting the variation of F1 [y] to zero gives d y n(x) dx 1+y2 = 0.
Now elementary calculus tells us tha
Do question one, and then as many other questions as you can. Try to answer entire questions. Little, if any, credit will be given for fragmentary answers. 1) Green Function: Consider the boundary value problem y = f (x), y (0) = y(1) = 0.
a) Const
Solutions to Homework Set 2 1) Bent bars: First some elementary calculus: the curvature , and the radius R of the osculating circle, at a point (x, y) on a curve y(x) are given by 1 d tan1 y d 1 y = = = = . 2 R ds dx (1 + y 2 )3/2 1+y Here the arc
Solutions to Homework Set 7 1) Critical mass: We expand n(x, t) = and also =
m=1
am (t) sin
mx , L
mx 4 sin , m L m,odd
0 < x < L.
Substituting in the given equation, and using the linear independence of the sine functions, then gives Dm2 2 4
Solutions to Homework Set 10 Dielectric Sphere: The solution outside will be of the form out (r, ) = E0 rP1 (cos ) + = while, inside in (r, ) = A1 rP1 (cos ). These terms have been selected to match the asymptotically uniform electric field at infin
Solutions to Homework Set 5 1) Missing state: The continuum eigenfunctions are are k = (ik + tanh x)eikx , so k (x) = (k  i)eikx /i x (k + i)eikx /i x 0, 0.
It now helps to draw a phasor diagram
Im
i k Re
from which we see that (k) = tan1
This exam has four pages and six problems. Answer question one, and then any other three questions. Do not hand in solutions to more than this number of problems! Try to answer entire questions. Little, if any, credit will be given for fragmentary an
Mathematics for Physics I
A set of lecture notes by
Michael Stone
PIMANDERCASAUBON Alexandria Florence London
ii Copyright c 2001,2002 M. Stone. All rights reserved. No part of this material can be reproduced, stored or transmitted without the