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!<= /span>

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 3^rd 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:

<= /u>

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:

<= /u>

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:<= /p>
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