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
MA/PH 607
Howell
9/14/2011
Homework Handout IV
A. Let
U
"
&
#
'
#3
$!
#
&
"
'
3
$!
!
"
'
&3
$!
.
1. Compute UT and U .
2. Verify that U is unitary.
3. Let U e e" e# e$ f be any orthonormal set of vectors (so e4 le5 $45 ), and let
e b" b# b$ f be given by
MA/PH 607
Howell
9/14/2011
Homework Handout IV
A. Let
U
"
&
#
'
#3
$!
#
&
"
'
3
$!
!
"
'
&3
$!
.
1. Compute UT and U .
2. Verify that U is unitary.
3. Let U e e" e# e$ f be any orthonormal set of vectors (so e4 le5 $45 ), and let
e b" b# b$ f be given by
2
Traditional Vector Theory
The earliest denition of a vector usually encountered is that a vector is a thing possessing
length and direction This is the arrow in space view with length naturally being the
.
length of the arrow and direction being the dir
Three Coordinate Sytems for the Plane
(Each with a Cartesian System)
1. Relative to the e B C f system, the
curves ? ?! and @ @! are
straight lines with slopes
"
and 2,
$
respectively. The ? ?! curve
crosses the \ -axis at B $?! , and
the @ @! curve cross
Preparing for Test II
The test covers everything from Elementary Linear Transform Theory up through
Wednesdays discussion of curves and derivatives (In the online notes: chapters 6, 7
and pages 1 to the top of 19 of chapter 8, excluding section 8.5); (see
MA/PH 607
10/14/2011
Howell
TAKE HOME PORTION OF
TEST II
(40 points)
GENERAL NOTES:
Due at the beginning of class on Friday, October 21.
This is NOT to be a team effort. Do your own work! (You may, of course, consult
with the instructor.)
Hand in the next
MA/PH 607
10/14/2011
Howell
TAKE HOME PORTION OF
TEST II
(40 points)
GENERAL NOTES:
Due at the beginning of class on Friday, October 21.
This is NOT to be a team effort. Do your own work! (You may, of course, consult
with the instructor.)
Hand in the next
Preparing for the First Exam
The test covers everything done up through change of bases" and volumes of hyperparallelepipeds" (Chapter 5 in the online notes). The problems will be more
computational than theoretical, but calculators will be unnecessary.1
The Sheet That Should Be Handed Out
The First Day Of Class
Mathematical Methods for Physicists ~ Fall 2011
(revised 8/18/2011)
General Stuff
Course: Mathematical Methods (for Physicists), MA 607 and PH 607
Partial Prerequisites: A basic course in linear a
Note on Example 1.8.3
(page 46 of A&W)
I've done enough partial derivation of equation 1.80 to believe it is true. However,
their derivation is garbage. In using the BACCAB rule they seem to ignore the fact that
f is an operator, not a vector subject to t
Multidimensional Calculus: Differential Theory
9.5
Chapter & Page: 919
General Formulas for the Divergence and Curl
Later, we will discover the geometric signicance of the divergence and the curl of a vector eld.
Then, using those, we can both redene dive
Multidimensional Calculus: Differential Theory
9.3
Chapter & Page: 913
The Classic Gradient, Divergence and Curl
Basic Denitions in Euclidean Space
For expediency, we will rst dene the classical differential operators for scalar and vector elds
in a Eucli
Multidimensional Calculus: Differential Theory
Chapter & Page: 99
Final Comments on Christoffel Symbols and Acceleration
Using the above, we can nd formulas for acceleration (and Christoffel symbols) in Euclidean
spaces and in nonEuclidean subpaces contai
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
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 607
Howell
11/7/2011
Homework Handout IX
A. For the following problems, we will use Cartesian
coordinates to describe points in the plane, and will
let V" , V# , V$ and V% be the four curves from
! ! to # % indicated in the figure. Note that:
+ V" a
MA/PH 607
Howell
10/31/2011
Homework Handout VIII
A. In a previous problem, you sketched the curves
x>
3> 9>
> #1 >
with
x>
3> 9>
>#
with
x>
3> 9>
% >
1
$
0>
>
with 0 > 41
Now:
.x
in terms of the polar coordinates and the associated tangent vectors e3
MA\PH 607
Howell
10/18/2011
Homework Handout VII
Addendum
(version 1)
A. Let e ? @ f be a basis-based coordinate system for the plane based on a basis e b" b# f
as indicated below. Assume the smallest angle between b" and b# is
lb" l #
#1
and that
$
lb# l
MA\PH 607
Howell
10/14/2011
Homework Handout VII
A. Two e ? @ f coordinate systems for the plane are sketched below. On each, plot the points
! ! , " # and
" # . Also, sketch the curve (well as you can) ? @ .
B. Sketch the curve traced out by each B> give
MA/PH 607
Howell
10/3/2011
Homework Handout VI
A. The following all involve linear operators on a two-dimensional traditional vector space i
with standard basis U e i j f. Try to do all of these problems without refering to the
matrix for the operator.
1.
MA/PH 607
09/23/2011
Homework Handout V
A. Let i be a two-dimensional traditional vector space with standard basis T e i j f .
Using this basis for i , do the following (unless otherwise indicated, view i as both the
input space and the output space).1
1.
MA/PH 607
Howell
9/14/2011
Homework Handout IV
A. Let
U
"
&
#
'
#3
$!
#
&
"
'
3
$!
!
"
'
&3
$!
.
1. Compute UT and U .
2. Verify that U is unitary.
3. Let U e e" e# e$ f be any orthonormal set of vectors (so e4 le5 $45 ), and let
e b" b# b$ f be given by
MA/PH 607
9/9/2011
Homework Handout III
A. Read 4.1 in the online notes, doing any in-notes exercises.
Also skim/read 3.2 of A&W (Arfken & Weber), starting with Basic Definitions at the
bottom of page 177. Skip everything in 3.2 before that (most of pages
MA/PH 607
8/29/11
Homework Handout II
A. For the following, i is a three-dimensional space of traditional vectors with standard basis
f ei jkf
(If you prefer, use e e1 e2 e3 f or e x y z f .)
Also, let
U e b1 b2 b3 f
where
b" i
#j
b2 $ j
k
and b3 # i
$j
MA/PH 607
8/17/2011
Homework Handout I
Note: In skimming/reading the indicated sections of A&W (Arfken and Weber) keep in mind
that their basic approach to the traditional vector theory is quite different from your
instructors. They depend on coordinate f
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
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