# Intuitive Examples of (r,0) Tensors

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?

Examples of completely contravariant tensors:

• The stress-energy tensor.
• The inverse metric.
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 $\text{Hom}(V,F)^m$.