# 张量 (内蕴定义)

## 用向量空间的张量积定义

${\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}$

${\displaystyle \left(1,1\right)}$ 型张量

${\displaystyle V\otimes V^{*}}$

${\displaystyle V^{*}\otimes V^{*}}$

## 其它记法

${\displaystyle {\begin{matrix}T_{n}^{m}(V)&=&\underbrace {V\otimes \dots \otimes V} &\otimes &\underbrace {V^{*}\otimes \dots \otimes V^{*}} \\&&m&&n\end{matrix}}.}$

${\displaystyle L(V,W)\ }$

${\displaystyle V^{*}\cong L(V,\mathbb {R} );}$

${\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \dots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \dots \otimes V} _{n},\mathbb {R} )\cong L^{m+n}(V^{*},\dots ,V^{*},V,\dots ,V,\mathbb {R} ).}$

${\displaystyle T_{0}^{1}(V)\cong L(V^{*},\mathbb {R} )\cong V,}$

${\displaystyle T_{1}^{0}(V)\cong L(V,\mathbb {R} )\cong V^{*},}$

${\displaystyle T_{1}^{1}(V)\cong L(V,V).}$

${\displaystyle GL(V,W)\ }$

## 張量在不同座標間的變換公式

${\displaystyle V\otimes V\otimes V^{*}}$

${\displaystyle \mathbf {T} =T^{ij}{}_{k}\,\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \omega ^{k},}$

${\displaystyle T^{ij}{}_{k}\ }$

${\displaystyle \mathbf {\hat {e}} _{i}=\sum _{j}R_{i}^{j}\mathbf {e} _{j}}$

，设 ${\displaystyle (R^{-1})_{k}^{l}}$ ${\displaystyle (R_{i}^{j})}$ 逆矩陣，對同一張量在新基底的張量分量設為${\displaystyle \textstyle {\hat {T}}^{i'j'}\!{}_{k'}}$ ，則兩者之間的變換公式為：

${\displaystyle {\hat {T}}^{i'j'}\!{}_{k'}=\sum _{p,q,r}(R^{-1})_{p}^{i'}\,(R^{-1})_{q}^{j'}\,R_{k'}^{r}\,T^{pq}{}_{r}=(R^{-1})_{p}^{i'}\,(R^{-1})_{q}^{j'}\,R_{k'}^{r}\,T^{pq}{}_{r},}$