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Aug12-04, 10:32 AM
Sci Advisor
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mathwonk's Avatar
P: 9,499
Hi Pete, another day another dollar.

Well, let's try again.

I believe that since x is a function, its differential dx is a covariant 1-tensor, i.e. a section of the cotangent bundle, because it acts on a tangent vector, via directional differentiation, and spits out a number.

Similarly, the tensor product of dx^j and dx^k is a covariant tensor of rank 2, because it acts on a pair of tangent vectors and spits out a number, namely the product of their jth and kth coordinates in the x - coordinate system. so dx^j dx^k is a covariant 2 tensor, i.e. a section of (T^) tensor(T^).

Now you are saying that gjk is also a 2 tensor. well, from what you have told me, it is called a 2 tensor as abuse of language. But what is the actual 2 tensor it is shorthand for? I understand it to be shorthand for the covariant 2 tensor:
summation gjk dx^j dx^k.

That would make it also a section of (T^)tensor(T^),

hence not a candidate for contraction against another such bird.

So here is the crucial point. You must contract tensors of opposite variance.

I,e, summation gjk dx^j dx^k is not actually a contraction.

The problem to me seems to be that if one thinks any expression with some indices up and others down is a contraction, one gets in trouble.

I.e. you are mixing two different languages here. the notation dx^j always stands for a section of the cotangent bundle, namely the differential of x^j. Hence dx^j really is a tensor, not just the components of one, i.e. it is not abuse of language to call dx^j a tensor.

if you want to contract two tensors, they have to be written in the same way, not one as an actual tensor and the other as components of a tensor.

If you want to contract gjk with some thing, it has to be with something like h^(jk), which would be the components of a contravariant 2 tensor like

summation h^(jk) ej ek.

The problem is that the components of a tensor transform opposite to the way the basis elements of the tensor transform.

Thus the basic covariant tensors are written with indices up, while their components are written with indices down.

Thus you do contract components having opposite indices, but a sum of components and basis elements with opposite placed indices is not a contraction.

Lets be more concrete.

What bundle do you think dx^j dx^k is a section of? and what bundle do you think gjk represents a section of?

You canno, contract them unless your answer is that they are sections of mutually dual bundles, but to me that would be hard to bring in line with any accepted notation or usage.