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I know the meaning of tensor, but I forgot the meaning of "$(n,m)$-tensor". What do $n$ and $m$ refer to?

Thanks.

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    What do you mean by tensor: an element of a tensor product of vector spaces (if not modules) or a field of tensors on a manifold? Assuming your meaning of tensor is the first one, an $(n,m)$ tensor means an element of $(V^{\otimes n}) \otimes (V^*)^{\otimes m}$ for some vector space $V$.2012-06-27

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An $(n,m)$-tensor on a finite-dimensional real vector space $V$ is (usually) defined to be a multilinear map $\Phi:\underbrace{V^{\ast}\times\cdots \times V^{\ast}}_{n\text{ times}}\times \underbrace{V\times\cdots \times V}_{m\text{ times}}\to \mathbb{R}$; $V^{*}$ denotes the dual space of $V$, i.e., the real vector space of all linear functionals $V\to\mathbb{R}$. The nonnegative integers $n$ and $m$ are referred to as the covariant and contravariant orders of the type $(n,m)$-tensor $\Phi$ on $V$, respectively.

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    I wasn't paying any attention to covariant/contravariant business but rather to your use of $n$ to count the number of components of $\Phi$ equal to $V$ and $m$ to count the number equal to $V^*$, which was backwards. Now you've swapped the roles of $V$ and $V^*$ in the definition of $\Phi$, so that part is okay.2012-06-28