PHZ 3113 PROBLEM SET 3
1) An upper triangular matrix is a matrix for which all elements that have a row
number exceeding the column number are zero, i.e., Aij=0 if i>j. Prove that the
product of two upper triangular matrices is another upper triangular ma

PAST SET
PHZ 3113 PROBLEM SET 8
1) Using Cauchy's integral theorems, including the extension of the second theorem
to derivatives of arbitrary order, show that
(z-zo)n dz = 0 if n-1,
(z-zo)-1 dz = 2i
where in both cases, the integral is around a closed

PAST SET
PHZ 3113 PROBLEM SET 7
1) Consider the complex quantity z=x+iy, where x and y are real. In polar form, z is
given by z=R exp(i).
a) Express the inverse of z, 1/z, in terms of x and y and also in polar form.
b) Suppose z=i. Find the real and imagi

PHZ 3113 PROBLEM SET 5
1) Text Problem 4.4 in part a, note that after moving the origin, the function is
symmetric.
2) Consider the function f(x) = x, defined over the interval -<x.
a) Find the Fourier series for this function, noting that it is symmetric

PHZ 3113 PROBLEM SET 1
1) A particle moves so that its position vector is given by the equation
r(t) = r cos(t) x-hat + r sin(t) y-hat
where x-hat and y-hat are the unit vectors along the x and y-axes, and is a
constant.
a) Evaluate the vector dr/dt assum

PHZ 3113 PROBLEM SET 8
1) Using Cauchy's integral theorems, including the extension of the second theorem
to derivatives of arbitrary order, show that
(z-zo)n dz = 0 if n-1,
(z-zo)-1 dz = 2i
where in both cases, the integral is around a closed contour i

PHZ 3113 PROBLEM SET 7
1) Consider the complex quantity z=x+iy, where x and y are real. In polar form, z is
given by z=R exp(i).
a) Express the inverse of z, 1/z, in terms of x and y and also in polar form.
b) Suppose z=i. Find the real and imaginary part

PHZ 3113 PROBLEM SET 2
1) In cylindrical coordinates, the position vector of a moving particle is given by
r = -hat + z z-hat
a) Derive an expression for the velocity vector of the particle, v = dr/dt, in
cylindrical coordinates. Note that the unit vector

PHZ 3113 PROBLEM SET 6
1) Text Problem 5.1 in part a, note that the function is symmetric. In part b, the
trig substitution suggested should enable you to perform the integral of the
square of the Fourier transform.
2) Text Problem 5.2 in part a, note tha

PHZ 3113 PROBLEM SET 4
1) Suppose that two different matrices are diagonalized by the same linear
transformation. Prove that the two matrices must commute, i.e., BA=AB.
Hint: examine the matrix elements of the transformed product matrices using the
fact t

PHZ 3113 PROBLEM SET 5
1) Text Problem 4.4 in part a, note that after moving the origin, the function is
symmetric.
2) Text Problem 4.5 in part a, note that the function is antisymmetric. In part b,
note that the given series corresponds to the Fourier se

PHZ3113 homework 1
x 2 y 2 z 2
1) Make a 3D plot of ellipsoid
1 using elabs Mathematica Plot3D
9
4
2) Make a 2D plot of intersecting curves x 2 and Tanh(x) find solution of
x 2 Tanh( x) .
3) Calculate the area of an ellipse x a cos(t ), y b sin(t ) (its

PHZ3113 homework 8
(1) Given a damped harmonic oscillator with a linear driving force, it satisfies the
2
2
2
&
following equation: & + x + 0 x = 0 (t + / 0 ) . 0 t 1 , find the particular
x
integral by first extend the range of time to 1 t 1 as an even f

PHZ3113 homework 4
1) For multi-electron atoms, the low-lying excitations are often the re-arrangement of last-shell
electrons. Carbon has two electrons in its 2p shell, which has three sub-levels ( m z = 1, 0, 1 ).
With electron spin, there are 6 sub-lev

PHZ3113 homework 8
(1) Elliptical orbit of planetary motion is determined by two constant p and e , as:
p
d
r=
; r2
= pK , 0 e < 1; K = GM s , where G is the
1 + e Cos ( )
dt
gravitational constant and M s is the solar mass. The semi-major axis of the
p
.

3113-example-multi-variable.nb
1
example of prove product of cyclic partial derivatives equals -1
Solve@P V = R T Exp@ al HT V RLD, PD
Solve@P V = R T Exp@ al HT V RLD, VD
Solve@P V = R T Exp@ al HT V RLD, TD
99P
al
E R T V R T
=
V
99V
99T
al
R T Produ

PHZ3113 homework 1.2
1) Fit a second-degree polynomial to the following three (x,y) points: (-1,3), (0,1),
and (1,4). And find the interpolation value for x 0.5 .
2) Using cubic spline method to fit the three points in (1), and find interpolation for
x 0.

PHZ3113 homework 3
3
1
8
1) Given Van der Waals equation (of reduced form): ( P 2 )(V ) T ,
3
3
V
1 V
calculate the expansivity under constant pressure,
. (Hint: Use
V T P const
the tumbling relationship and express expansivity in terms of volume and
te

PHZ3113 homework 7
1 2 1 x1 1
(1) Use LU decomposition method to solve 2 7 4 x 2 = 2 by first
1 8 7 x 2
3
0 0
1
U 11 U 12 U 13
decomposing the matrix into L = L21 1 0 and U = 0 U 22 U 23 .
L
0
0 U 33
31 L32 1
1 0
1
(2) Given three degener

PHZ3113 homework 6
1) Find the Fourier series of the function f ( x) = x in a range < x < . Check the
1 1 1
value at x = / 2 and show that = 1 + + . . Consider x = and show
4
3 5 7
that the Fourier series does not converge to the function, and explain why

PHZ 3113 PROBLEM SET 3
1) Text Problem 1.9 note that two matrices anticommute if BA=-AB. Evaluation
of the last part is facilitated by first demonstrating that the matrices A and C
anticommute.
2) An upper triangular matrix has zeros for all elements for