# 並矢張量

${\displaystyle {\boldsymbol {v}}=v_{1}{\boldsymbol {i}}+v_{2}{\boldsymbol {j}}+v_{3}{\boldsymbol {k}}\,}$
${\displaystyle {\boldsymbol {w}}=w_{1}{\boldsymbol {i}}+w_{2}{\boldsymbol {j}}+w_{3}{\boldsymbol {k}}\,}$

${\displaystyle {\boldsymbol {vw}}=v_{1}w_{1}{\boldsymbol {ii}}+v_{1}w_{2}{\boldsymbol {ij}}+v_{1}w_{3}{\boldsymbol {ik}}+v_{2}w_{1}{\boldsymbol {ji}}+v_{2}w_{2}{\boldsymbol {jj}}+v_{2}w_{3}{\boldsymbol {jk}}+v_{3}w_{1}{\boldsymbol {ki}}+v_{3}w_{2}{\boldsymbol {kj}}+v_{3}w_{3}{\boldsymbol {kk}}\,}$

${\displaystyle {\boldsymbol {vw}}={\begin{pmatrix}v_{1}w_{1}&v_{1}w_{2}&v_{1}w_{3}\\v_{2}w_{1}&v_{2}w_{2}&v_{2}w_{3}\\v_{3}w_{1}&v_{3}w_{2}&v_{3}w_{3}\end{pmatrix}}\,}$

## 定義

${\displaystyle A_{ij}'=\sum _{m,n}{\frac {\partial x_{m}}{\partial x_{i}'}}{\frac {\partial x_{n}}{\partial x_{j}'}}A_{mn}\,}$

${\displaystyle \mathbf {A} =A_{11}{\boldsymbol {ii}}+A_{12}{\boldsymbol {ij}}+A_{13}{\boldsymbol {ik}}+A_{21}{\boldsymbol {ji}}+A_{22}{\boldsymbol {jj}}+A_{23}{\boldsymbol {jk}}+A_{31}{\boldsymbol {ki}}+A_{32}{\boldsymbol {kj}}+A_{33}{\boldsymbol {kk}}\,}$

## 並矢張量運算

${\displaystyle \mathbf {A} \cdot {\boldsymbol {v}}=\sum _{m,n}(A_{mn}{\boldsymbol {e}}_{m}{\boldsymbol {e}}_{n})\cdot \sum _{\ell }(v_{\ell }{\boldsymbol {e}}_{\ell })\,}$

${\displaystyle \mathbf {A} \cdot {\boldsymbol {v}}=\sum _{m,n}A_{mn}v_{n}{\boldsymbol {e}}_{m}\,}$

${\displaystyle |\mathbf {A} |=\sum _{m}A_{m}^{m}\,}$

${\displaystyle \langle \mathbf {A} \rangle ={\boldsymbol {e}}_{1}(A_{23}-A_{32})+{\boldsymbol {e}}_{2}(A_{31}-A_{13})+{\boldsymbol {e}}_{3}(A_{12}-A_{21})\,}$

${\displaystyle \langle \mathbf {A} \rangle =\sum _{mn}\epsilon _{imn}A_{mn}\,}$

## 進階理論

${\displaystyle ({\boldsymbol {vw}})\cdot {\boldsymbol {u}}:={\boldsymbol {v}}\,({\boldsymbol {w}}\cdot {\boldsymbol {u}})\,,\qquad {\boldsymbol {u}}\cdot ({\boldsymbol {vw}}):=({\boldsymbol {u}}\cdot {\boldsymbol {v}})\,{\boldsymbol {w}}\,}$

## 进阶定义

${\displaystyle V\,}$  是域 ${\displaystyle F\,}$  上的一个线性空间，则下述定义是等价的。

(1) ${\displaystyle \forall \mathbf {T} \in W\,}$ , ${\displaystyle \exists k\in \mathbb {N} \,}$  以及 ${\displaystyle {\boldsymbol {u}}_{1},{\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {u}}_{k},{\boldsymbol {v}}_{k}\in V\,}$  使得
${\displaystyle \mathbf {T} =\sum _{i=1}^{k}\phi ({\boldsymbol {u}}_{i},{\boldsymbol {v}}_{i})\,}$
(2) 当 ${\displaystyle {\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {v}}_{k}\in V\,}$  线性无关时，${\displaystyle \{\phi ({\boldsymbol {v}}_{i},{\boldsymbol {v}}_{j})\,|\,i,j=1,\ldots ,k\}\,}$ ${\displaystyle W\,}$  中的线性无关向量组，

(1) 任意向量 ${\displaystyle {\boldsymbol {v}}\,}$ ${\displaystyle {\boldsymbol {w}}\,}$  并置摆放形成一个并矢积 ${\displaystyle {\boldsymbol {vw}}\,}$
(2) 对于任意的 ${\displaystyle \alpha \in F\,}$  和任意的 ${\displaystyle {\boldsymbol {v}},{\boldsymbol {w}}\in V\,}$ ，规定 ${\displaystyle (\alpha {\boldsymbol {v}}){\boldsymbol {w}}={\boldsymbol {v}}(\alpha {\boldsymbol {w}})=\alpha ({\boldsymbol {vw}})\,}$  ，并把上述结果不加区分地记作 ${\displaystyle \alpha {\boldsymbol {vw}}\,}$
(3) 称有限个并矢积的形式和为一个并矢张量
(4) 对任意正整数 ${\displaystyle k\,}$ ，如果 ${\displaystyle {\boldsymbol {v}}_{1},\ldots ,{\boldsymbol {v}}_{k}\in V\,}$  线性无关，则 ${\displaystyle \{{\boldsymbol {v}}_{i}{\boldsymbol {v}}_{j}\,|\,i,j=1,\ldots ,k\}\,}$  是线性无关向量组——特别是， ${\displaystyle {\boldsymbol {vw}}=0\,}$  的充分必要条件是 ${\displaystyle {\boldsymbol {v}}=0\,}$ ${\displaystyle {\boldsymbol {w}}=0\,}$
(5) 对任意的 ${\displaystyle {\boldsymbol {u}}\,}$ ${\displaystyle {\boldsymbol {v}}\,}$ ${\displaystyle {\boldsymbol {w}}\in V\,}$ ，成立着分配律
${\displaystyle {\boldsymbol {u}}({\boldsymbol {v}}+{\boldsymbol {w}})={\boldsymbol {uv}}+{\boldsymbol {uw}}\,,\qquad ({\boldsymbol {u}}+{\boldsymbol {v}}){\boldsymbol {w}}={\boldsymbol {uw}}+{\boldsymbol {vw}}\,}$

## 并矢张量与向量的缩併

(6) 对于任意的 ${\displaystyle \alpha \in F,\,\mathbf {T} \in V\otimes V\,}$  以及 ${\displaystyle {\boldsymbol {v}}\in V\,}$
${\displaystyle (\alpha \mathbf {T} )\cdot {\boldsymbol {v}}=\mathbf {T} \cdot (\alpha ^{*}{\boldsymbol {v}})=\alpha (\mathbf {T} \cdot {\boldsymbol {v}})\,,\qquad {\boldsymbol {v}}\cdot (\alpha \mathbf {T} )=(\alpha ^{*}{\boldsymbol {v}})\cdot \mathbf {T} =\alpha ({\boldsymbol {v}}\cdot \mathbf {T} )\,}$

(7) 对于任意的 ${\displaystyle \mathbf {S} ,\,\mathbf {T} \in V\otimes V\,}$  以及 ${\displaystyle {\boldsymbol {v}}\in V\,}$ ，总有
${\displaystyle (\mathbf {S} +\mathbf {T} )\cdot {\boldsymbol {v}}=\mathbf {S} \cdot {\boldsymbol {v}}+\mathbf {T} \cdot {\boldsymbol {v}}\,,\qquad {\boldsymbol {v}}\cdot (\mathbf {S} +\mathbf {T} )={\boldsymbol {v}}\cdot \mathbf {S} +{\boldsymbol {v}}\cdot \mathbf {T} \,}$
(8) 对于任意的 ${\displaystyle \mathbf {T} \in V\otimes V\,}$  以及 ${\displaystyle {\boldsymbol {v}},\,{\boldsymbol {w}}\in V\,}$ ，总有
${\displaystyle \mathbf {T} \cdot ({\boldsymbol {v}}+{\boldsymbol {w}})=\mathbf {T} \cdot {\boldsymbol {v}}+\mathbf {T} \cdot {\boldsymbol {w}}\,,\qquad ({\boldsymbol {v}}+{\boldsymbol {w}})\cdot \mathbf {T} ={\boldsymbol {v}}\cdot \mathbf {T} +{\boldsymbol {w}}\cdot \mathbf {T} \,}$
(9) 对任意的 ${\displaystyle {\boldsymbol {u}},\,{\boldsymbol {v}},\,{\boldsymbol {w}}\in V\,}$ ，总有
${\displaystyle ({\boldsymbol {uv}})\cdot {\boldsymbol {w}}={\boldsymbol {u}}\,({\boldsymbol {v}}\cdot {\boldsymbol {w}})\,,\qquad {\boldsymbol {u}}\cdot ({\boldsymbol {vw}})=({\boldsymbol {u}}\cdot {\boldsymbol {v}})\,{\boldsymbol {w}}\,}$

## 範例

### 旋轉

${\displaystyle \mathbf {M} ={\boldsymbol {ji}}-{\boldsymbol {ij}}=\left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)\,}$

${\displaystyle \mathbf {M} \,}$  是一個二維空間的 90° 旋轉算子 (rotation operator) 。它可以從左邊點積一個向量來產生一個旋轉

${\displaystyle \mathbf {M} \cdot (x{\boldsymbol {i}}+y{\boldsymbol {j}})=({\boldsymbol {ji}}-{\boldsymbol {ij}})\cdot (x{\boldsymbol {i}}+y{\boldsymbol {j}})=x{\boldsymbol {ji}}\cdot {\boldsymbol {i}}-x{\boldsymbol {ij}}\cdot {\boldsymbol {i}}+y{\boldsymbol {ji}}\cdot {\boldsymbol {j}}-y{\boldsymbol {ij}}\cdot {\boldsymbol {j}}=-y{\boldsymbol {i}}+x{\boldsymbol {j}}\,}$

${\displaystyle \left({\begin{array}{cc}0&-1\\1&0\end{array}}\right)\left({\begin{array}{c}x\\y\end{array}}\right)=\left({\begin{array}{c}\ -y\\x\end{array}}\right)\,}$

${\displaystyle \cos \theta \mathbf {I} +\sin \theta \mathbf {M} ={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\ \cos \theta \end{pmatrix}}\,}$

### 量子力学

${\displaystyle V\,}$ 量子力学中所有的角动量本征态所张成的希尔伯特空间（囊括了所有可能的总角动量量子数 ${\displaystyle 0\,}$ ${\displaystyle 1/2\,}$ ${\displaystyle 1\,}$ ${\displaystyle 3/2\,}$ ${\displaystyle \ldots \,}$ ），则 ${\displaystyle F=\mathbb {C} \,}$ 。当我们要考虑角动量耦合的时候，就会遇到态矢量的并矢张量 ${\displaystyle |j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \,}$ ，而且时常把它记作 ${\displaystyle |j_{1}m_{1}j_{2}m_{2}\rangle \,}$ ${\displaystyle |j_{1}m_{1},j_{2}m_{2}\rangle \,}$  等等。任取一些复数 ${\displaystyle C_{j_{1}m_{1}j_{2}m_{2}}\,}$ （但是其中只能有有限个非零），则

${\displaystyle \sum _{j_{1}}\sum _{m_{1}}\sum _{j_{2}}\sum _{m_{2}}C_{j_{1}m_{1}j_{2}m_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \,}$

${\displaystyle \mathbf {T} \cdot |jm\rangle =\sum _{j_{1}}\sum _{m_{1}}\sum _{j_{2}}\sum _{m_{2}}C_{j_{1}m_{1}j_{2}m_{2}}\langle jm|j_{2}m_{2}\rangle \,|j_{1}m_{1}\rangle \,}$
${\displaystyle |jm\rangle \cdot \mathbf {T} =\sum _{j_{1}}\sum _{m_{1}}\sum _{j_{2}}\sum _{m_{2}}C_{j_{1}m_{1}j_{2}m_{2}}\langle jm|j_{1}m_{1}\rangle \,|j_{2}m_{2}\rangle \,}$

## 并矢张量的展开

${\displaystyle {\boldsymbol {vw}}={\Big (}\sum _{i}v^{i}{\boldsymbol {e}}_{i}{\Big )}{\Big (}\sum _{j}w^{j}{\boldsymbol {e}}_{j}{\Big )}{\stackrel {(5)}{=}}\sum _{i}{\Big [}(v^{i}{\boldsymbol {e}}_{i})\sum _{j}w^{j}{\boldsymbol {e}}_{j}{\Big ]}{\stackrel {(5)}{=}}\sum _{i}\sum _{j}(v^{i}{\boldsymbol {e}}_{i})(w^{j}{\boldsymbol {e}}_{j})\,}$

${\displaystyle {\boldsymbol {vw}}{\stackrel {(2)}{=}}\sum _{i}\sum _{j}{\boldsymbol {e}}_{i}{\Big (}v^{i}(w^{j}{\boldsymbol {e}}_{j}){\Big )}=\sum _{i}\sum _{j}{\boldsymbol {e}}_{i}{\Big (}(v^{i}w^{j})\,{\boldsymbol {e}}_{j}){\Big )}{\stackrel {(2)}{=}}\sum _{i}\sum _{j}(v^{i}w^{j})\,{\boldsymbol {e}}_{i}{\boldsymbol {e}}_{j}\,}$

## 实线性空间上的并矢张量和线性变换互相等同（爱因斯坦指标升降）

${\displaystyle g_{ij}={\boldsymbol {e}}_{i}\cdot {\boldsymbol {e}}_{j}\,}$

${\displaystyle {\hat {T}}{\boldsymbol {e}}_{j}=T_{\ j}^{i}{\boldsymbol {e}}_{i}\,}$

${\displaystyle G^{-1}=(g^{ij})\,,\qquad g_{ij}g^{jk}=\delta _{i}^{k}\,}$

${\displaystyle \mathbf {T} =T_{\ j}^{i}g^{jk}\,{\boldsymbol {e}}_{i}{\boldsymbol {e}}_{k}\,}$

${\displaystyle (R_{\flat }\mathbf {T} ){\boldsymbol {e}}_{j}=\mathbf {T} \cdot {\boldsymbol {e}}_{j}=T_{\ l}^{i}g^{lk}\,{\boldsymbol {e}}_{i}{\boldsymbol {e}}_{k}\cdot {\boldsymbol {e}}_{j}=T_{\ l}^{i}g^{lk}\,{\boldsymbol {e}}_{i}({\boldsymbol {e}}_{k}\cdot {\boldsymbol {e}}_{j})=T_{\ l}^{i}g^{lk}\,{\boldsymbol {e}}_{i}\,g_{kj}=T_{\ l}^{i}\delta _{j}^{l}\,{\boldsymbol {e}}_{i}=T_{\ j}^{i}\,{\boldsymbol {e}}_{i}={\hat {T}}{\boldsymbol {e}}_{j}\,}$

${\displaystyle \mathbf {T} '=T_{\ j}^{i}g^{jk}\,{\boldsymbol {e}}_{k}{\boldsymbol {e}}_{i}\,}$

${\displaystyle n\,}$  维欧几里得空间 ${\displaystyle V\,}$  上的所有的线性映射所构成的线性空间记为 ${\displaystyle {\mathfrak {gl}}(V)\,}$ ，则后者的维数为 ${\displaystyle n^{2}\,}$ . 由并矢张量和向量的缩併中的规则 (6) 和 (7) 不难得到

${\displaystyle \dim(V\otimes V)=n^{2}\,}$

## 參考文獻

1. ^ Papanastasiou, Tasos C.; Georgios C. Georgiou, Andreas N. Alexandrou. Viscous Fluid Flow. CRC Press. 2000: pp. 26–27. ISBN 9780849316067.
2. ^ Spencer, Anthony James Merrill. Continuum mechanics. Courier Dover Publications. 2004: pp. 19–20. ISBN 9780486435947.
3. ^ Morse, Philip; Feshbach, Herman, Methods of theoretical physics, Part 2, McGraw-Hill: pp. 54–92, 1953, ISBN 978-0070433175
1. H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Massachusetts 1980, p.194.
2. 吳望一，《流體力學》上册，北京：北京大学出版社，1982：1.13节，1.14节。