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 a
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
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 =
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)
w
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
Mathematical Physics PHZ 3113
Classwork 13 (April 10, 2013)
Solutions Linear Equations
After nding solutions: Check that they
are correct!
1. Find all solutions of the equations
x + 2y = 3 ,
3x + y =
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 (wit
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 applic
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
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
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
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 vect
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 z
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 applic
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
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. D
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
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 as
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 vect
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 z
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)
Solutio
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. Elimina
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. D
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
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
Lecture Notes on General Relativity
arXiv:gr-qc/9712019v1 3 Dec 1997
Sean M. Carroll
Institute for Theoretical Physics
University of California
Santa Barbara, CA 93106
[email protected]
December 19
PHZ-4601/5606 Special and General Theories of Relativity
Final Exam
This is a take-home exam which will be due in my oce no later than Tuesday December 11, 2012 by
12:00 PM noon. You are encouraged to