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what is a tensor ?

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Ozone Radical

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Jul 20, 2002, 2:05:02 AM7/20/02
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Hi!

Could you please tell me what exactly is a tensor ? What kind of a quantity is it ?

Larry Hammick

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Jul 21, 2002, 2:24:35 AM7/21/02
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Ozone Radical wrote ...

> Could you please tell me what exactly is a
> tensor ? What kind of a quantity is it ?

Hello O3,
A vector is an element of a vector space, or more generally an element of a
module. A tensor is an element of a "tensor product" of vector spaces or
modules. Given two vector spaces V and W, their tensor product is (up to a
canonical isomorphism) just the set of bilinear mappings of V* x W* into the
scalars. For modules which aren't vector spaces, it is a little more
complicated.

Tensor products are part of the subject known as "multilinear algebra". See
e.g. Serge Lang's _Algebra_.
Larry


George Jones

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Jul 21, 2002, 3:02:32 AM7/21/02
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This definition is only true for finite-dimensional vector spaces,
which is probably the situation that the original poster had in mind.

For infinite-dimensional vector spaces (such as function spaces), the
tensor product of vector spaces V and W is (naturally isomorphic to) a
particular proper subspace of the space of bilinear mappings of V* x W*
into the scalars.

Regards,
George

Shmuel (Seymour J.) Metz

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Jul 27, 2002, 11:03:14 PM7/27/02
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In <8e0a2732.02071...@posting.google.com>, on 07/19/2002

at 11:05 PM, ozoner...@yahoo.com (Ozone Radical) said:

>Could you please tell me what exactly is a tensor ?

There are several differwnt definitions. Very loosely, a tensor is a
sum of products of vectors. I could express that more precisely, but
I'd have to explain the vocabulary.

>What kind of a quantity is it ?

It isn't a quantitiy; it's more general than that. But physicists use
vectors and tensors[1] to represent various physical phenomena. From
their perspective, a tensor is a set of numbers that transforms in a
certain way when you change coordinate systems.

Read a good book on Differential Geometry and it will all become
clear.

[1] Actually, vector and tensor fields.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Atid/2, Team OS/2, Team PL/I

Any unsolicited commercial junk E-mail will be subject to legal
action. I reserve the right to publicly post or ridicule any
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I mangled my E-mail address to foil automated spammers; reply to
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reply to spam...@library.lspace.org

Lee Rudolph

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Jul 28, 2002, 7:27:03 AM7/28/02
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A better question would be, "how does a tensor...?".

Lee Rudolph

Kevin Foltinek

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Jul 29, 2002, 3:40:45 PM7/29/02
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lrud...@panix.com (Lee Rudolph) writes:

> A better question would be, "how does a tensor...?".

Here's an even better question: How does a flurble...?

(What's that? You don't know what a flurble is? And you don't know
what the "..." stands for? Well! Perhaps that wasn't a better
question after all.)

Kevin.

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