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

\alpha^a

that are scalars for basis of the form

\frac{\partial}{\partial x^a}

Then we have simple q-vectors that have components

\alpha^{ab...d}

that are scalars for basis of the form

\frac{\partial}{\partial x^a} \wedge \frac{\partial}{\partial x^b} \wedge ... \wedge  \frac{\partial}{\partial x^d}

Then we have covectors, or 1-forms

\beta_e

with basis

dx^e

and simple p-forms

\beta_{ef...h}

with basis

dx^e \wedge dx^f \wedge ... \wedge dx^h


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

Q^{f...h}_{a...c}

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

\frac{\partial}{\partial x^f} \wedge ... \wedge  \frac{\partial}{\partial x^h} \wedge dx^a \wedge ... \wedge dx^c

?

Or would it be something like

dx^a \wedge ... \wedge dx^c \wedge \frac{\partial}{\partial x^f} \wedge ... \wedge  \frac{\partial}{\partial x^h}

?

Or some other permutation of vectors and 1-forms?


16 Aug 2009, 05:06

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

Tensors live in a space like this
V^*\otimes \dots\otimes V^*\otimes V\otimes \dots\otimes V
where V is a vector space and V^* 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 V will be the tangent space at a given point on the manifold and V^* will be the corresponding cotangent space. A typical (p,q) basis element will be
dx^{i_1}\otimes \dots\otimes dx^{i_p}\otimes{\partial\over \partial x^{j_1}}\otimes\dots\otimes{\partial\over \partial x^{j_q}}


16 Apr 2010, 03:55
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