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It's easy to find "intuitive" examples of $(0, r)$ tensors or even $(k, r)$ tensors $( k, r > 0)$. For the purposes of this question, I am considering a tensor in the "classical" sense as being represented by a multilinear form. For instance, every inner product space has an associated inner product which is just a $(0, 2)$-tensor that satisfies certain properties. Also, if $V$ is a vector space over a field $\mathbb{F}$ and $V^{*}$ denotes its dual, then the evaluation map $E:V^{*} \times V \rightarrow \mathbb{F}$ given by $E(f,v) = f(v)$ is an example of a $(1, 1)$ tensor. What are some elementary/intuitive examples of $(k,0)$ tensors?

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Examples of completely contravariant tensors:

  • The stress-energy tensor.
  • The inverse metric.
  • Blades, multivectors, or pseudovectors.

Intuitive examples of completely contravariant tensors:

  • The angular momentum bivector.
  • The torque bivector.
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It all depends. Unfortunately, because dimension is the only invariant for vectors spaces there are literally more than five equivalent, common, definitions of $V\otimes V$ I can think of. Probably the one that best fits your needs is to think about a $(m,n)$-tensor as just being a multililinear map $T:\text{Hom}(V,F)^m\times V^n\to F$. So, for example, taking $m=0$ gives that a $(0,n)$-tensor is nothing more than a member of $\text{Mult}_m(V,F)$ ($m$-linear maps on $V^m$) and taking $n=0$ gives that $(m,0)$-tensors are just $m$-linear maps on $\text{Hom}(V,F)^m$.

Does that help?

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I have updated the question to be explicit about the type of tensor I am considering. – ItsNotObvious Nov 2 '11 at 18:21
Well, I think that I went with that definition, so I have no more to add. I hope what I said was helpful. – Alex Youcis Nov 2 '11 at 18:24
You helped me clarify my question and be more explicit about what I was asking, but your examples effectively defined the type of tensors I am considering. Thanks. – ItsNotObvious Nov 2 '11 at 18:32

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