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Chap 12. And then there are tensors.
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Joined: 07 May 2009, 16:45
Posts: 62
Chap 12. And then there are tensors.
We have vectors that have components in the form

that are scalars for basis of the form

Then we have simple q-vectors that have components

that are scalars for basis of the form

Then we have covectors, or 1-forms

with basis

and simple p-forms

with basis

Now we get these tensor things, which we are only given the component form, something like

Am I correct in assuming the basis of such an animal is something like

?

Or would it be something like

?

Or some other permutation of vectors and 1-forms?

16 Aug 2009, 05:06

Joined: 26 Mar 2010, 04:39
Posts: 109
Re: Chap 12. And then there are tensors.
No.

Tensors live in a space like this

where is a vector space and is its dual. Note that while Grassmann algebras deal with antisymmetric multilinear forms, tensors are just multilinear and in general will split into a symmetric and antisymmetric part.

Once we get to differentiable manifolds then will be the tangent space at a given point on the manifold and will be the corresponding cotangent space. A typical (p,q) basis element will be

16 Apr 2010, 03:55
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