The Road to Reality

Chap 12. And then there are tensors.
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Author:  DimBulb [ 16 Aug 2009, 05:06 ]
Post subject:  Chap 12. And then there are tensors.

We have vectors that have components in the form


that are scalars for basis of the form

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

Then we have simple q-vectors that have components


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


with basis


and simple p-forms


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


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?

Author:  robin [ 16 Apr 2010, 03:55 ]
Post subject:  Re: Chap 12. And then there are tensors.


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}}

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