MA/PH 609
4/9/2012
Howell
Name:
TEST III
(100 points)
Hand in the following few pages with the final answers neatly and clearly
written in the space provided for each, along with additional sheets
clearly showing how you got your answers. I will be lookin
Preparing for Test II
The test covers the material on complex analysis, chapters 14 through 17 in the online
notes. See also the homework in handouts III, IV and V. As usual, the problems will be
more computaional than theoretical, but calculators will be
Preparing for Test I
The test covers everything done this term up through the material covered on Wednesday
February 1. In the notes this is everything up to the top of page 14-8. The problems will
be more computaional than theoretical, but calculators wi
11
Tensors
Actually, weve been doing tensor analysis all along. All we will do now is to add a few elements
which will make things look even more like classical tensor analysis One element of this
.
look is a slight extension of the Kronecker delta notati
The Sheet That Should Be Handed Out
The First Day Of Class
Mathematical Methods for Physicists ~ Spring 2012
General Stuff
Course: Mathematical Methods (for Physicists), MA 609 and PH 609
Basic Prerequisite: MA/PH 607
The Text: Mathematical Methods for Ph
12
Innite Series
Many of the functions we will be dealing with later are described using innite series often either
,
power series or series of eigenfunctions for certain self-adjoint operators (Fourier series are the
best-known examples of eigenfunction
10
Multidimensional Integration
These sections continue the development of multidimensional calculus. Here we concentrate on
integration.
The main goals in this chapter are to develop the basics of potential theory as well as
,
the classical theorems of G
9
Multidimensional Calculus:
Mainly Differential Theory
In the following, we will attempt to quickly develop the basic differential theory of calculus in
multidimensional spaces. Youve probably already seen much of this theory. Hopefully, we will
develop
8
Multidimensional Calculus: Basics
Weve nished the basic linear algebra part of the course; now we start a major part on
multidimensional calculus This will included discussions of eld theory, differential geometry
.
and a little tensor analysis. Here is
1
Intro
1.1
Some of What We Will Cover
Here is a very rough idea of what the rst part of the Math Physics course will cover:
1.
Linear Algebra
(a) We will start with a fundamental development of the theory for traditional vectors
(developed in a manner th
MA/PH 609
Howell
3/30/2012
Homework Handout IX
Note: Youve already done much of the work for the problems in this set in earlier homework
sets. Feel free to make use of what youve already done!
A 1
Find the solution to the following heat flow problem
?>
$
MA/PH 609
Howell
3/16/2012
Homework Handout VIII
Note: P always denotes some finite distance in the following problems.
A. Compute the following, assuming the interval is ! $ and the weight function is
A B ".
1. B l sin#1B
3. *
2. B# l *
3)B l B#
3)B
4
MA/PH 609
Howell
3/12/2012
Homework Handout VII
A. For each of the following homogeneous boundary-value problems:
i Separate the problem into the corresponding eigenvalue/eigenfunction problem and
the other" problem.
ii Solve the eigenvalue/eigenfunction
MA/PH 609
Howell
03/7/2012
Homework Handout VI
A. Find all separable solutions to each of the following pdes. Assume - denotes some
positive constant.
1.
`?
`>
-#
`#?
!
`B#
2.
`?
`>
-#
`#?
!
`B#
3.
`#?
`>#
-#
4.
`#?
`B#
`#?
!
`C #
`#?
!
`B#
B. Separate ea
MA/PH 609
Howell
02/29/2012
Homework Handout V
Note: In the following, assume B and C are real variables while D is a complex varible
with D B 3C
A. For each of the following functions,
1. Identify all the singularities and zeroes of each function in the
MA/PH 609
Howell
02/10/2012
Homework Handout IV
Note: In the following, assume B and C are real variables while D is a complex varible
with D B 3C
A. Let V be the semicircle from D " to
D " sketched to the right.
1. Using the Cauchy integral theorem, conv
MA/PH 609
Howell
01/30/2012
Homework Handout III
In the following, B and C are real variables while D is a complex varible with D B
A. Find the real and imaginary parts of
"
#
$3
3C .
$3 # .
and #
B. Find the real part, ?B C , and the imaginary part, @B C
MA/PH 609
Howell
1/23/2012
Homework Handout II
A. Consider the differential equation
B
.# C
.B#
#
.C
.B
BC ! .
Set B! ! and, using the basic Frobenius method described in the notes, do the following:
1. Find and solve the indicial equation (using B! !
2.
MA/PH 609
Howell
1/9/12
Homework Handout I
A. 5.1 of A&W, page 325: 1, 2 (Ignore the hint for each. Instead, rewrite each term using
partial fractions, then write out the partial sums, see what cancels, and take the limits.)
B. Which of the following geom
3
General Vector Spaces
Much of what we did with traditional vectors can also be done with other sets of things We
.
will develop the appropriate theory here, and extend our notation appropriately.
One change we will make is that we will no longer restric
13
Power Series and Differential Equations:
The Method of Frobenius
Its all well and good to be able to nd power series representations for functions you know
via the standard computations for Taylor series. Even better is to be able to nd power series
re
MA/PH 609 , Take-Home Part of the Final
IV.
page 4
Consider the following nonhomogeneous heat flow problem:
`?
`>
%)
?B ! > !
`#?
0B
` B#
!
B
?B % > !
and
?B ! ?! B
!
%
B
>!
%
In the last test you should have shown that a complete set of eigenfunctions ar
MA/PH 609
4/13/2012
Howell
(Most of the)
Take-Home Part of the Final
(100 points)
Due by noon on Wednesday, April, 25, 2012. This is not to be a group effort. The only
person you may consult regarding these problems is your instructor. You may use the cla
MA/PH 609
4/13/2012
Howell
Take-Home Part of the Final
(100 points)
Due by noon on Wednesday, April, 25, 2012. This is not to be a group effort. The only
person you may consult regarding these problems is your instructor. You may use the class
notes (incl
Preparing for the In-Class Part of the Final
The final covers everything done this term, but is only worth 50 points. Expect one or
two problems on series, one or two problems on complex variables (with a possible
leaning towards problems involving integr
6
Elementary Linear Transform Theory
Whether they are spaces of arrows in space functions or even matrices, vector spaces quickly
,
become boring if we dont do things with their elements move them around, differentiate or
integrate them, whatever. And oft
4
Elementary Matrix Theory
We will be using matrices:
1.
For describing change of basis formulas.
2.
For describing how to compute the result of any given linear operation acting on a vector
(e.g., nding the components with respect to some basis of the fo
7
Eigenvectors and Hermitian Operators
7.1
Eigenvalues and Eigenvectors
Basic Denitions
Let L be a linear operator on some given vector space V . A scalar and a nonzero vector v
are referred to, respectively, as an eigenvalue and corresponding eigenvector
5
Change of Basis
In many applications, we may need to switch between two or more different bases for a vector
space. So it would be helpful to have formulas for converting the components of a vector with
respect to one basis into the corresponding compon