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