PHZ3113: Solution for Homework 11.
1. For small oscillation we have derived the Euler-Lagrange equations, which read
in matrix notation
2 1
2g/l 0
= 0.
+
1 1
0
g/l
This is solved by the exponential ansatz (physical is the real part of the solution):
0
0

Mathematical Physics PHZ 3113
Classwork 12 (April 3, 2013)
Solution Jacobi Determinant
1. Calculate the Jacobi determinant for the
transformation from Cartesian to cylindrical
coordinates.
Solution:
x = cos ,
y = sin ,
z = z
yields the Jacobi determinant

1. Calculate
+
+
I =
(x+y)2(xy)2 .
dx dy e
Solution:
= xy,
1
y = ( ) ,
2
x x
1 1
1
2 2
y y = 1 1 = 2 .
2
2
The minus comes because the transformation switches the righ-handed system
= x+y,
1
x = ( + ) ,
2
dx = x dx ,
dy = y dy
into the left-handed s

Mathematical Physics PHZ 3113
Homework 10 (April 10, 2013)
Pauli Matrices
1. Find three 2 2 matrices i, i = 1, 2, 3,
which fulll the relations
i j = i ijk k for i = j , (1)
i j + j i = 2 ij 12 ,
(2)
where 12 is the 2 2 unit matrix. Solution:
Simple matric

Mathematical Physics PHZ 3113
Classwork 8 (February 27, 2013)
Cylindrical Coordinates
1. Use cylindrical coordinates to calculate
the area of a circle of radius R.
Solution:
d2x =
A =
Scircle
2
=
0
x2+y 2R2
dx dy
R
1 2
d
d = 2 R = R2 .
2
0
2. Calculate

Mathematical Physics PHZ 3113
Midterm 1 (February 18, 2013)
1. Calculate the gradient of the 3D potential
(in arbitrary units) %r2.
2. A force in 3D is (in arbitrary units) given
by F = «erF. Use (with Einstein con-
vention)
V X 7'? :- eijkiiéljrsck
to ca

Mathematical Physics PHZ 3113
Solutions Midterm 2 (March 18, 2013)
1. Use the denition A = ijk xij Ak (with Einstein convention) and properties of the Levi-Civita tensor ijk to
transform
A
into applications of the
involve the curl.
operator, which do no l

Mathematical Physics PHZ 3113
Midterm 2 (March 18, 2013)
1. Use the denition A = ijk xij Ak (with Einstein convention) and properties of the Levi-Civita tensor ijk to
transform
A
into applications of the
involve the curl.
operator, which do no longer
2. C

Mathematical Physics PHZ 3113
Solutions Midterm 1 (February 18, 2013)
1. Calculate the gradient of the 3D potential
(in arbitrary units) 1 r2.
2
Solution (with Einstein convention):
xi
1 2
r = r r = r xi i r = r xi
2
r
r
= r = rr = r.
r
2. A force in 3D i

Solution Paul Trap
In the quasi-static approximation the eld is the electrostatic eld with the given
boundary conditions at the time in question. Due to the cylindrical symmetry we
have = (, z; t). To get the boundary conditions on the end electrodes righ

Mathematical Physics PHZ 3113
Vectors 1 (Classwork January 7, 2013)
Group #
Participating students (print):
In the following i = 1, . . . , n, j = 1, . . . , n.
1. Let xi and xj be Cartesian unit vectors.
It holds the relation
xi xj = ij .
(1)
Name the r.

Mathematical Physics PHZ 3113
Sailboat (Homework January 16, 2013)
One of the major inventions of mankind (not known in the antique) was the keel
(or skeg in dinghies), which allows sailing boats to zigzag against the wind. Below
you nd a sketch for the b

Mathematical Physics PHZ 3113
Solutions Midterm 2 (March 18, 2013)
1. Use the denition A = ijk xij Ak (with Einstein convention) and properties of the Levi-Civita tensor ijk to
transform
A
into applications of the
involve the curl.
operator, which do no l

Mathematical Physics PHZ 3113
Solutions Midterm 1 (February 18, 2013)
1. Calculate the gradient of the 3D potential
(in arbitrary units) 1 r2.
2
Solution (with Einstein convention):
xi
12
r = r r = r xi i r = r xi
2
r
r
= r = rr = r.
r
2. A force in 3D is

Mathematical Physics PHZ 3113
Levi-Cevita Tensor 2 Applications
(January 14, 2013)
Group #
Participating students (print):
1. Write down the values of the cyclic permutations of 123 and then of 213. Do
you get all 3D values this way? Which
are positive an

Mathematical Physics PHZ 3113
Levi-Cevita Tensor 1
(January 11, 2013)
Group #
Participating students (print):
1. Use binary numbers 0, 1 and write down
the numbers 0 to 3. Add one more column in which you substitute 0 1, 1
2 and one last column in which

Mathematical Physics PHZ 3113
Vectors 2 (Classwork January 9, 2013)
Group #
Participating students (print):
1. Write down the commutative law of
vector addition
a + b = b + a.
(1)
2. Write down the associative law of vector addition
a + b + c = a + b + c

Mathematical Physics PHZ 3113
Vectors 1 (Classwork January 7, 2013)
Group #
Participating students (print):
In the following i = 1, . . . , n, j = 1, . . . , n.
1. Let xi and xj be Cartesian unit vectors.
It holds the relation
xi xj = ij .
(1)
Name the r.

Mathematical Physics PHZ 3113
Sailboat (Homework January 16, 2013)
One of the major inventions of mankind (not known in the antique) was the keel
(or skeg in dinghies), which allows sailing boats to zigzag against the wind. Below
you nd a sketch for the b

Mathematical Physics PHZ 3113
Levi-Cevita Tensor Homework 2
(January 25, 2013)
1. Use the identity
3
ijk ilm = jl km jmkl
i=1
(1)
to eliminate the vector products from the
expression
a b c
(2)
Solution (compare book p.33):
a b c
xa
bc
m ijk i j k lm l m
i

Mathematical Physics PHZ 3113
Levi-Cevita Tensor Homework 1
(January 23, 2013)
1. Use the 3D identity
3
i=1
ijk ilm = jl km jmkl
(1)
to calculate
( ) ( )
abab
(2)
where a and are unit vectors. Eliminate
b
b
all a in favor of cos .
Solution: Let
1
b
a

Mathematical Physics PHZ 3113
Levi-Cevita Tensor 2 Applications
(January 14, 2013)
Group #
Participating students (print):
1. Write down the values of the cyclic permutations of 123 and then of 213. Do
you get all 3D values this way? Which
are positive an

Mathematical Physics PHZ 3113
Levi-Cevita Tensor 1
(January 11, 2013)
Group #
Participating students (print):
1. Use binary numbers 0, 1 and write down
the numbers 0 to 3. Add one more column in which you substitute 0 1, 1
2 and one last column in which

Mathematical Physics PHZ 3113
Gradient, (January 23, 2013)
Group #
Participating students (print):
1. Calculate
xj .
xi
(1)
xj = ij .
xi
(2)
n2
xj .
xi j =1
(3)
It holds
2. Calculate
Solution:
2
n2
n dxj xj
=
xj =
j =1 d xj xi
xi j =1
n
2 xj ij = 2 xi .
j