Computating the components of a tensor

[Rasalhague]

In the first problem here http://elmer.tapir.caltech.edu/ph237/assignments/assignment2.pdf , we're asked to show from the duality relation

$$\mathbf{e}^{\mu} \cdot \mathbf{e}_{\nu} = \delta^{\mu}_{\nu}$$

and the expansion of a tensor

$$\mathbf{T}\left(\underline \quad, \underline \quad, \underline \quad \right)$$

in terms of basis vectors:

$$\mathbf{T} = T^{\alpha \beta}_{\mu} \mathbf{e}_{\alpha} \otimes \mathbf{e}_{\beta} \otimes \mathbf{e}^{\mu}$$

that the components of a tensor can be computed by inserting basis vectors into its slots and lining up the indices, e.g.

$$T^{\alpha \beta}_{\mu} = \mathbf{T}\left(\mathbf{e}^{\alpha},\mathbf{e}^{\beta}, \mathbf{e}_{\mu} \right).$$

The solution ( http://elmer.tapir.caltech.edu/ph237/assignments/solutions/week2/page1.jpg ) does just that.

(Note: in these lectures and the accompanying material, Kip Thorne avoids making an explicit distinction between one-forms and vectors, by associating one-forms with vectors via the metric tensor.)

What puzzles me is that I read (in Schutz: Geometrical Methods of Mathematical Physics) that a tensor isn't always simply a tensor product; it might be a sum of tensor products. So does this proof only apply to simple tensors (those which are tensor products), and if so, what would a general proof look like? Also, is there an easy way to tell whether a given tensor is simple? And is there a way to derive a basis for the vector space a tensor belongs to from the basis of the vector space V in terms of which the tensors are defined?

 

[dx]

The tensor T here is a sum of tensor products. The summation sign is omitted by convention, but there is a sum over alpha, beta and mu. This is called the Einstein summation convention.

 

[Rasalhague]

Aha, yes, thanks, dx, for once again coming to my rescue! I see it now: T is a sum of all those permutations of tensor products of the chosen basis vectors of V and its dual space V* (or using this convention whereby vectors are associated with 1-forms via the inner product, the chosen basis of V and the reciprocal basis), each permutation scalar-multiplied by the appropriate component.