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The purpose of =
these notes
I have an interest in string theory.
The reason I am fascinated with string theory comes fr=
om the
fact that, when modeling waves, I was stunned to find out the
implications of . We do not really convert matter into en=
ergy. We are converting energy in a standing =
wave
(the illusion we call matter) into free energy.
The equation could be written
.
Happily the amount of force required to change the standing energy wave into
free energy is enormous in most cases – less stable atoms such a uranium do=
not
require a large amount of energy to change their state. If that force was m=
uch,
much lower, each step you take would cause an atomic explosion!
I remember my awe when doing 3D modeling of the Schrod=
inger
Wave equation:
which lead me to the previous conclusion, and I would like to model strings and writ=
e a
program that shows 3D or possibly ND strings.
I would like to fully understand the
implications of strings and the multidimensional quality of them and possib=
ly
gain a new stunning view. It’s like climbing a mountain just to l=
ook
out from the top. Currently that view eludes me.
Of course I realize that string theory
might be all wrong, but still it is a fun mental exercise.
The math of string theory is over my head right now. I have never taken a class in tensor ca=
lculus
and am attempting to understand it so that I can model strings. Tensor calc=
ulus
is the calculus of N dimensions. It has uses in stress analysis, fluid
dynamics, electromagnetic and much more. http://en.wikipedia.org/wiki/T=
ensor
I have a feeling for the possible
multidimensional nature of gravity in that it should be much stronger than =
it
is (we should be stuck like pancakes to the earth) and that the reason we a=
re
not is that much of the force of gravity is in another dimension(s) but I w=
ould
like a deeper understanding.
As I study tensor calculus I need to =
take
extensive notes and thought that I would share them with students on my
website.
Note: Since I am doing this as a hobb=
y,
it may take years. J Also, these notes are not intended to t=
each
tensor calculus but, rather, are for my own reference. The order may be somewhat random. Also,=
I may
be wrong in spots!
Fundamentals of notation
Einstein nota=
tion for
summation:
Any expression having a twice repeated
index as a sub or super script or once as a sub and once as a super script
indicates a sum.
Indicates summation because of the re=
peated
subscript i
Indicates summation because of the
repeated subscript j
Does not indicate summation
Definition
of dummy and free indices:
In the equation j can be replaced with any variable name=
(for
example) and is called the dummy index. i is cal=
led
the free index. It is assumed (unless noted) that the dummy and free indices
have the same range. In the previo=
us
equation if n =3D 2:
Multiple
summation indices:
No index may appear more than twice <=
/span> is undefined.
If an expression has more than one
summation index, there are terms. (n =3D number of summation terms)=
For
n =3D 2 becomes 2 equations with 4 ter=
ms
each (order of iteration r,s does not matter):
First Order Tensors
Contravariant tensor
For vector field where =
and can be expressed as real valued function=
s
=
If there is a contravariant vector of order one such t= hat
Then is a contravariant tensor of order one.<= o:p>
Covariant tensor
If there is a covariant vector of ord=
er
one such that
Then the vector field is a covariant tensor of order one.
The
relationship between the two is that there is a change of basis such as
Cartesian coordinates to polar coordinates.
Summary
The tangent vector
of a smooth curve transforms as a contravariant vector of order one.
The gradient of a differentiable function is a covari=
ant
vector of order one.
Mnemonic: the notation for co=
variant
and contravariant tensors can be remembered by looking at the 3rd
letter of covariant and contravariant. The “v” points down in covariant and th=
e “n”
points up in contravariant. then indicate=
s a
contravariant tensor while indicates a covariant tensor.
Substitution
Rules:
If is
to be substituted into you must change the indices for the first
equation to a new notation such as so that the second b=
ecomes .
Kronecker
Delta:
Non-identities:
Identities:
Partial Differentiation of Sums
Note that C does not depend on k and =
is
therefore constant.
Now differentiating with respect to k=
we
get:
Note that when the restriction is removed, each term has an=
.
Using the Einstein notation we get:
can be written as because:
In my opinion this derivation is a th=
ing
of beauty! Very clever. I understand it but prob=
ably
would not have come up with it.
Summary:
Basic Linear Algebr=
a For Tensors
For tensors, rather than notation, since we need the upper and lo=
wer
indices of summation, use where
m is the row and n is the column. This can also be .
For example:
Specific example:
Vector
Notation:
If I have a real n-dimensional column
matrix: it is written and
v is a vector. The collection of a=
ll
n-dimensional vectors is
Vector
Addition:
given
then
Scalar
Multiplication:
Matrix
Multiplication:
Identity
Matrix:
Where =
is the Kronecker del=
ta
Matrix
Inversion:
Useful
information for the construction of this document:
Microsoft Word 2010 supports LaTex ma=
th
symbols in its equation generator.
see: http://mirror.hmc.edu/ctan/info/symbols/comprehensive/symbols=
-a4.pdf
for a comprehensive list.
LaTex
(starts with \):
\equiv \bar
T =
=
x^ab
=
x_ab
\delta
=
\Delta
\neq
=
Non-LaTex:
ħ Unicode 0127 hex