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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
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
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
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