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### Abstract Index Notation and Tensors

Date: 07/15/2002 at 18:11:54
From: Thomas Mainiero
Subject: Abstract Index Notation and Tensors

Hello,

I am currently studying the mathematical side of General Relativity.
The problem is that I do not entirely understand abstract index
notation for tensors. I comprehend what they are, what they are used
for, tensor ranks, and representation of tensors by matrices. I know
the notation cannot be typed out in ASCII, so just use ^ for
superscript and _ for subscript. I tried searching for an introduction
to the notation, but the very few sites I have found are not that
clearly typed. Also can you provide an example of how tensors are used
in coordinate transformations?

Date: 07/18/2002 at 09:08:58
From: Doctor Nitrogen
Subject: Re: Abstract Index Notation and Tensors

Hi, Thomas:

Here is an example of the index notation:

If there is no index, that is, if the tensor has a form, say, like

v,

then it is a tensor of rank zero, meaning no index, and v is a scalar.
If it looks like

v_i, or v^i, i = 1, 2, ...n,

then it is a tensor of rank one because it has one index. If it has a
form like

v^i_j,

or covariant in j, contravariant in i, it is a tensor of rank two, and
so on. Summation for the indices is implied, so there is no summation
symbol.

This is called Einstein summation. The rank is equal to the number of
indices. What the indices sum to is determined by the space you are
in. For Euclidean 3D space it would be just i = 1, 2, 3, j = 1, 2, 3.
Since you say you are studying General Relativity, you must know that
for the Riemannian Geometry used in it, the indices are summed from
1 to 4.

Covariant tensors transform thusly:

A_ij = (&x_i/&x'_p)(&x_j/&x'_q)A'_pq.

(Here "&" denotes the partial derivative symbol.)

Again, that transformation is for a covariant transformation. There is
also a similar one for a contravariant transformation.

The modern terminology for tensor calculus  has become pretty
abstract, using terms like "charts," "atlases," "manifolds,"
"diffeomorphisms," "p-forms," etc., but the basic idea of tensors is
that one can quickly transform from one coordinate system (chart) to
another in a few simple steps.

Suppose you know what the metric tensor g_ij is for a General
Relativity problem, maybe spherical coordinates. Then using tensor
methods, you can compute the square of the arc length using the
coordinates involved.

ds^2 = g_mn(dx^mdx^n)

There are also things called Christoffel symbols, which help you
compute the "covariant derivative" and the "intrinsic derivative" of a
tensor.

Be innovative when you try to find information on tensors and their
indices online. If you enter as keywords

tensors AND indices,

or

(tensors and their indices)

you will find links that include both those expressions and you will
probably find a tutorial somewhere on the topic.

There are math/physics professors who might scoff at their use, but
McGraw Hill publishes excellent books in a book series on a variety of
pure/applied math topics. Their Schaum Outline Series has an excellent
book called _Vector and Tensor Analysis_.

I hope this was helpful.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/

Date: 07/18/2002 at 16:24:52
From: Thomas Mainiero
Subject: Thank you (Abstract Index Notation and Tensors)

Thank you a lot, it was really helpful and clear! You also answered a
question about how to find the squared distance between two points
with the metric tensor that I was about to ask. You have been really

I have two other questions. Is the matrix that represents a tensor
always a square matrix (cube matrix, hypercube matrix, etc.) except in
the case of vectors? Also, what is it meant by the "contraction" of a
tensor?

Thomas

Date: 07/21/2002 at 19:43:21
From: Doctor Nitrogen
Subject: Re: Abstract Index Notation and Tensors

Hello again, Thomas:

To my knowledge, the matrix representations for tensors generally
involve square matrices. For example, mechanical engineers use special
tensors called the stress and strain tensors, which are represented by
3X3 square matrices, and physicists in General Relativity use a
tensor, the metric tensor g^ik or g_ik, which is represented by a 4X4
matrix.

The following illustrates the "contraction" of the indices in a
tensor. Let:

R^i_jkl

be a tensor of rank four, contravariant index i, covariant indices
j, k, and l. You can lower the number of indices if, say, i = j:

R^i_ikl.

If you multiply a tensor  A^i_jklm  by g^jk, you can also get
contraction of the indices:

g^jk(A^i_jklm) = A^i_lm.

Here, j and k were contracted.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/

Date: 07/26/2002 at 22:28:37
From: Thomas Mainiero
Subject: Thank you (Abstract Index Notation and Tensors)

Thank you once again. You have been a great help in answering my
questions. I also recently purchased that Schaum Outline book on
Vector and Tensor analysis and it is a great resource; thank you for
recommending it.

Thomas
Associated Topics:
College Physics

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