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**Showcase_22** A pure contravariant tensor of order (p,0) (the 0 shows that a vector space $\displaystyle V$ involved) has the form:

$\displaystyle \underbrace{ V^* \times \cdots \times V^* }_{p \ times} \rightarrow \mathbb{R}$

where $\displaystyle V^*$ is the dual space of $\displaystyle V$.

I'm having a lot of trouble trying to think of an example! I figure the easiest example is to take a vector space $\displaystyle \mathbb{R}^3$ and any $\displaystyle 3 \times 3$ matrix. These types of matrices will be in the dual space and they will represent a linear map.

To map a linear map to a scalar without using any vectors in $\displaystyle V$, I figure the easiest way is to just take the determinant of the matrix.

I think this example is okay, but I wondered if there was a better one I could use.

Anyway, my question is: Can anyone think of any other good examples of a pure contravariant tensor?