Wave equation in Cartesian coordinates
The wave equation is given in one spatial dimension
2u
1 2u
=2 2
x 2
v t
We again use separation of variables u (x , t ) = X (x )T (t ), and
then we can write the wave equation as,
X
1T
=2
= k 2
X
vT
Hence we wind

Periodic functions: simple harmonic oscillator
Recall the simple harmonic oscillator (e.g. mass-spring system) d 2y 2 + 0 y = 0 dt 2 Solution can be written in various ways: y (t ) = Ae i 0 t y (t ) = A cos 0 t + B sin 0 t The constants of integration A a

Physical application of divergence and divergence theorem
Consider a vector eld J = v , where is mass density and v is a ow velocity The net change of mass in a time dt in a volume element is determined only by J on the surface of dM = J nd dt By the div

Divergence theorem in two dimensions
j i Consider a vector eld V = Vx + Vy (Notice here Vz = 0) If we take Q = Vx and P = Vy , then Vy P Vx Q = + = div V x y x y
idy Consider the outward normal n = jdx 2 = 2 dx +dy idx jdy , ds
and then
Pdx + Qdy = Vy dx

Greens theorem in the plane
We have seen how to do double integrals, now lets do them in the xy-plane over an area A
b yu
dx
a yl
P (x , y ) dy = y
b
[P (x , yu ) P (x , yl )] dx =
a C
Pdx
The integral C Pdx means a counterclockwise integral around th

Gradient operator
In our calculation of d along the vector ds , we see that it can be described as the scalar product d = i+ j+ k ux ds + uy ds + uz ds k i j x y z
We take d = ds = ds u and hence dene the gradient operator (in a Cartesian system) = grad

Dierentiation of vectors
In a Cartesian system, , , and k are xed unit vectors ij + Ay + Az k , where the components If we have a vector A = Ax i j are functions of t , then we can take a derivative dAx dAy dAz dA = i+ j+ k dt dt dt dt For example, A cou

Midterm Review
Wednesday, February 17 Open book No notes Chapters 1-5, focus on lectures and homeworks
Patrick K. Schelling
Introduction to Theoretical Methods
Eigenvalues and eigenvectors; diagonalization
We have described linear operators acting on v

Chapter 6: Vector Analysis
We use derivatives and various products of vectors in all areas of 2r physics. For example, Newtons 2nd law is F = m d 2 . In electricity dt and magnetism, we need surface and volume integrals of various elds. Fields can be scal

Change of variables in the integral; Jacobian
Element of area in Cartesian system, dA = dxdy We can see in polar coordinates, with x = r cos , y = r sin , r 2 = x 2 + y 2 , and tan = y /x , that dA = rdrd In three dimensions, we have a volume dV = dxdydz

Dierentiation of integrals and Leibniz rule
We are interested in solving problems of the type d dx
v (x )
f (x , t )dt
u (x )
Notice in addition to the limits depending on x , the function f (x , t ) also depends on x First lets take the case where the

Complex Fourier series
Recalling the Euler formula, we can write cos nx = sin nx = e inx + e inx 2
e inx e inx 2i This suggest that instead of using cos nx and sin nx , we might use e inx
f (x ) =
n=
cn e inx
To obtain the coecients, multiply left and r

Even and odd functions
We dene an even function such that f (x ) = f (x ) We dene an odd function such that f (x ) = f (x ) Example, sin x is an odd function because sin x = sin x Example, cos x is an even function because cos x = cos x Now consider a Fo

Fourier transforms
We can imagine our periodic function having periodicity taken to
the limits
In this case, the function f (x ) is not necessarily periodic, but we
can still use Fourier transforms (related to Fourier series)
Consider the complex Four

Laplace equation in Cartesian coordinates
The Laplace equation is written
2
=0
For example, let us work in two dimensions so we have to nd
(x , y ) from,
2 2
+
=0
x 2
y 2
We use the method of separation of variables and write
(x , y ) = X (x )Y (y )
X

Steady-state temperature in a sphere
Consider a sphere of radius r = 1, with the temperature T = 100
on the top half (z > 0 or 0 < < /2) and T = 0 on the bottom
half (z < 0 or /2 < < )
We know that our solution is a solution to Laplace equation
2 T = 0

Expansion of 1/r potential in Legendre polynomials
In electrostatics and gravitation, we see scalar potentials of the
form V = K
d
Take
r
r
d = |R r | = R 2 2Rr cos + r 2 = R 1 2 R cos + ( R )2
r
Use h = R and x = cos , and then we see we have the
gene

Legendre series
The orthogonality over the interval 1 < x < 1 can be used to
make a series expansion of a function f (x ) over the same interval
f (x ) =
c l Pl ( x )
l =0
We use the orthogonality of the Legendre functions to nd
integrals that determine

The factorial function
We can easily nd for > 0 the integral,
1
1
e x dx = e x | =
0
0
We recall that if we take derivatives with respect to , we can
interchange the order of dierentiation and integration, and we nd
n!
n+1
0
For = 1, this gives us anot

Transformation of tensors
Rather than simply a matrix of numbers, tensors depend on the
denition of a coordinate system
The physics does not depend on the coordinate system, so we
need well-dened rules to describe how tensors transform under
coordinate

Tensors in continuum mechanics
When we apply forces on a deformable body (stress) we get a
deformation (strain)
If the stresses are fairly small, the strains will be small
For small stress/strain, the relationship between stress and strain
is linear (J

Vector operators in curvilinear coordinate systems
In a Cartesian system, take x1 = x , x2 = y , and x3 = z , then an
element of arc length ds 2 is,
2
2
2
ds 2 = dx1 + dx2 + dx3
In a general system of coordinates, we still have x1 , x2 , and x3
For exa

Solving dierential equations with Fourier transforms
Consider a damped simple harmonic oscillator with damping
and natural frequency 0 and driving force f (t )
d 2y
dy
2
+ 0 y = f ( t )
+ 2b
dt 2
dt
At t = 0 the system is at equilibrium y = 0 and at re

Linear second-order dierential equations with constant
coecients and nonzero right-hand side
We return to the damped, driven simple harmonic oscillator
dy
d 2y
2
+ 2b
+ 0 y = F sin t
2
dt
dt
We note that this dierential equation is linear
We call yc th

Maxima/minima with constraints
Very often we want to nd maxima/minima but subject to some constraint Example: A wire is bent to a shape y = 1 x 2 . If a string is stretched from the origin to the wire, at what point along the wire is the length of the st

Total dierential for y = f (x )
df For y=f(x), we have y = dy = dx dx We can treat dx = x as an independent variable In the limit x 0, then
y dy = lim x 0 x dx If x nite, then dy is not exactly y
Patrick K. Schelling
Introduction to Theoretical Methods
To

Chapter 4: Partial dierentiation
It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x ), then dy df dx = dx = f . However, most of the time we are dealing with quantities that are funct

Homework 4
PHZ 3113
Due Monday, February 8, 2010
Chapter 3
1. The Pauli matrices below are related to the spin of a spin 1/2 particle
01
10
x =
y =
0 i
i0
z =
10
0 1
a) Show that the Pauli matrices are unitary and Hermitian
b) Find the eigenvectors and ei

Homework 3
PHZ 3113
Due Friday, January 29, 2010
Chapter 2-3
1. Find z = cos1 (i 8) in the form x + iy (Some of this problem was done in
class)
2. We showed in class that for the simple harmonic oscillator, described the equation
of motion,
d2 y
2
+ 0 y =

Homework 2
PHZ 3113
Due Friday, January 22, 2010
Chapter 2
1. Evaluate the integrals
eax cos bxdx
and
eax sin bxdx
To do this, rst nd the integral
parts of the result.
e(a+ib)x dx and then take the real and imaginary
2. Write (8i)1/3 in the form x + iy .