变换方式
向量：反变变换
标记法说明：向量
v
{\displaystyle \mathbf {v} \,\!}
向量空间
V
{\displaystyle V\,\!}
e
1
,
e
2
,
.
.
.
,
e
n
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},...,\mathbf {e} _{n}\,\!}
x
1
,
x
2
,
.
.
.
,
x
n
{\displaystyle x^{1},x^{2},...,x^{n}\,\!}
v
{\displaystyle \mathbf {v} \,\!}
v
1
,
v
2
,
.
.
.
,
v
n
{\displaystyle v^{1},v^{2},...,v^{n}\,\!}
v
=
∑
i
v
i
e
i
{\displaystyle \textstyle \mathbf {v} =\sum _{i}v^{i}\mathbf {e} _{i))
（注：
v
2
{\displaystyle v^{2}\,\!}
2
{\displaystyle 2}
平方 ，而是代表第二个坐标，在较基础的数学上，常写作
v
2
{\displaystyle v_{2}\,\!}
爱因斯坦求和约定 。）
向量空间
V
{\displaystyle V}
e
¯
1
,
.
.
.
,
e
¯
n
{\displaystyle {\bar {\mathbf {e} ))_{1},...,{\bar {\mathbf {e} ))_{n}\,\!}
x
¯
1
,
.
.
.
,
x
¯
n
{\displaystyle {\bar {x))^{1},...,{\bar {x))^{n}\,\!}
v
{\displaystyle \mathbf {v} \,\!}
v
¯
1
,
v
¯
2
,
.
.
.
,
v
¯
n
{\displaystyle {\bar {v))^{1},{\bar {v))^{2},...,{\bar {v))^{n}\,\!}
v
=
∑
i
v
¯
i
e
¯
i
{\displaystyle \textstyle \mathbf {v} =\sum _{i}{\bar {v))^{i}{\bar {\mathbf {e} ))_{i))
对于1...n之间任意整数
μ
{\displaystyle \mu \,\!}
v
¯
μ
{\displaystyle {\bar {v))^{\mu }\,\!}
v
1
,
v
2
,
.
.
.
,
v
n
{\displaystyle v^{1},v^{2},...,v^{n}\,\!}
v
¯
μ
=
∂
x
¯
μ
∂
x
1
v
1
+
∂
x
¯
μ
∂
x
2
v
2
+
.
.
.
+
∂
x
¯
μ
∂
x
n
v
n
{\displaystyle {\bar {v))^{\mu }={\frac {\partial {\bar {x))^{\mu )){\partial x^{1))}v^{1}+{\frac {\partial {\bar {x))^{\mu )){\partial x^{2))}v^{2}+...+{\frac {\partial {\bar {x))^{\mu )){\partial x^{n))}v^{n}\,\!}
使用爱因斯坦求和约定 可写成：
v
¯
μ
=
∂
x
¯
μ
∂
x
i
v
i
{\displaystyle {\bar {v))^{\mu }={\frac {\partial {\bar {x))^{\mu )){\partial x^{i))}v^{i}\,\!}
余向量：共变变换
假设对偶空间
V
∗
{\displaystyle V^{*))
d
x
1
,
d
x
2
,
.
.
.
,
d
x
n
{\displaystyle {\mathbf {dx} ^{1},\mathbf {dx} ^{2},...,\mathbf {dx} ^{n))\,\!}
d
x
¯
1
,
d
x
¯
2
,
.
.
.
,
d
x
¯
n
{\displaystyle \mathbf {d{\bar {x))} ^{1},\mathbf {d{\bar {x))} ^{2},...,\mathbf {d{\bar {x))} ^{n}\,\!}
[1] :289-297
假设
ω
∈
V
∗
,
ω
=
∑
i
η
i
d
x
i
=
∑
j
η
¯
j
d
x
¯
j
{\displaystyle \textstyle {\boldsymbol {\omega ))\in V^{*},{\boldsymbol {\omega ))=\sum _{i}\mathbf {\eta } _{i}\mathbf {dx} ^{i}=\sum _{j}{\bar {\mathbf {\eta } ))_{j}d{\bar {\mathbf {x} ))^{j))
1
{\displaystyle 1}
n
{\displaystyle n}
μ
{\displaystyle \mu \,\!}
η
¯
μ
{\displaystyle {\bar {\mathbf {\eta } ))_{\mu }\,\!}
η
1
,
η
2
,
.
.
.
,
η
n
{\displaystyle \mathbf {\eta } _{1},\mathbf {\eta } _{2},...,\mathbf {\eta } _{n}\,\!}
η
¯
μ
=
∂
x
1
∂
x
¯
μ
η
1
+
∂
x
2
∂
x
¯
μ
η
2
+
.
.
.
+
∂
x
n
∂
x
¯
μ
η
n
{\displaystyle {\bar {\mathbf {\eta } ))_{\mu }={\frac {\partial x^{1)){\partial {\bar {x))^{\mu ))}\mathbf {\eta } _{1}+{\frac {\partial x^{2)){\partial {\bar {x))^{\mu ))}\mathbf {\eta } _{2}+...+{\frac {\partial x^{n)){\partial {\bar {x))^{\mu ))}\mathbf {\eta } _{n}\,\!}
或使用爱因斯坦求和约定写成：
η
¯
μ
=
∂
x
i
∂
x
¯
μ
η
i
{\displaystyle {\bar {\mathbf {\eta } ))_{\mu }={\frac {\partial x^{i)){\partial {\bar {x))^{\mu ))}\mathbf {\eta } _{i}\,\!}
向量的共变分量和反变分量
在欧几里得空间
V
{\displaystyle V\,\!}
内积 运算从向量求得余向量；对于所有余向量
w
{\displaystyle \mathbf {w} \,\!}
v
{\displaystyle \mathbf {v} \,\!}
线性泛函
α
(
w
)
{\displaystyle \alpha (\mathbf {w} )\,\!}
w
{\displaystyle \mathbf {w} \,\!}
α
(
w
)
=
v
⋅
w
{\displaystyle \alpha (\mathbf {w} )=\mathbf {v} \cdot \mathbf {w} \,\!}
逆过来，通过上述方程，线性泛函
α
(
w
)
{\displaystyle \alpha (\mathbf {w} )\,\!}
v
{\displaystyle \mathbf {v} \,\!}
给予
V
{\displaystyle V\,\!}
f
=
(
X
1
,
X
2
,
…
,
X
n
)
{\displaystyle {\mathfrak {f))=(X_{1},X_{2},\dots ,X_{n})\,\!}
f
♯
=
(
Y
1
,
Y
2
,
…
,
Y
n
)
{\displaystyle {\mathfrak {f))^{\sharp }=(Y^{1},Y^{2},\dots ,Y^{n})\,\!}
Y
i
⋅
X
j
=
δ
j
i
{\displaystyle Y^{i}\cdot X_{j}=\delta _{j}^{i}\,\!}
其中，
δ
j
i
{\displaystyle \delta _{j}^{i}\,\!}
克罗内克函数 。
以这两种基底，任意向量
v
{\displaystyle \mathbf {v} \,\!}
v
=
∑
i
v
i
[
f
]
X
i
=
f
v
[
f
]
=
∑
i
v
i
[
f
]
Y
i
=
f
♯
v
[
f
♯
]
{\displaystyle {\begin{aligned}v&=\sum _{i}v^{i}[{\mathfrak {f))]X_{i}={\mathfrak {f))\,\mathbf {v} [{\mathfrak {f))]\\&=\sum _{i}v_{i}[{\mathfrak {f))]Y^{i}={\mathfrak {f))^{\sharp }\,\mathbf {v} [{\mathfrak {f))^{\sharp }]\end{aligned))\,\!}
； 其中，
v
i
[
f
]
{\displaystyle v^{i}[{\mathfrak {f))]\,\!}
v
{\displaystyle \mathbf {v} \,\!}
f
{\displaystyle {\mathfrak {f))\,\!}
v
i
[
f
]
{\displaystyle v_{i}[{\mathfrak {f))]\,\!}
v
{\displaystyle \mathbf {v} \,\!}
f
{\displaystyle {\mathfrak {f))\,\!}
欧几里得空间
将向量
a
{\displaystyle \mathbf {a} \,\!}
投影 于坐标轴
e
i
{\displaystyle \mathbf {e} ^{i}\,\!}
a
i
{\displaystyle a^{i}\,\!}
a
{\displaystyle \mathbf {a} \,\!}
坐标曲面 的法线
e
i
{\displaystyle \mathbf {e} _{i}\,\!}
a
i
{\displaystyle a_{i}\,\!}
在欧几里得空间 Ｒ3 里，使用内积 运算，能够从向量求得余向量。给予一组可能不是标准正交基 的基底，其基底向量为
e
1
{\displaystyle \mathbf {e} _{1}\,\!}
e
2
{\displaystyle \mathbf {e} _{2}\,\!}
e
3
{\displaystyle \mathbf {e} _{3}\,\!}
e
1
=
e
2
×
e
3
τ
;
e
2
=
e
3
×
e
1
τ
;
e
3
=
e
1
×
e
2
τ
{\displaystyle \mathbf {e} ^{1}={\frac {\mathbf {e} _{2}\times \mathbf {e} _{3)){\tau ));\qquad \mathbf {e} ^{2}={\frac {\mathbf {e} _{3}\times \mathbf {e} _{1)){\tau ));\qquad \mathbf {e} ^{3}={\frac {\mathbf {e} _{1}\times \mathbf {e} _{2)){\tau ))\,\!}
其中，
τ
=
e
1
⋅
(
e
2
×
e
3
)
{\displaystyle \tau =\mathbf {e} _{1}\cdot (\mathbf {e} _{2}\times \mathbf {e} _{3})\,\!}
e
1
{\displaystyle \mathbf {e} _{1}\,\!}
e
2
{\displaystyle \mathbf {e} _{2}\,\!}
e
3
{\displaystyle \mathbf {e} _{3}\,\!}
平行六面体 的体积。
反过来计算，
e
1
=
e
2
×
e
3
τ
′
;
e
2
=
e
3
×
e
1
τ
′
;
e
3
=
e
1
×
e
2
τ
′
{\displaystyle \mathbf {e} _{1}={\frac {\mathbf {e} ^{2}\times \mathbf {e} ^{3)){\tau '));\qquad \mathbf {e} _{2}={\frac {\mathbf {e} ^{3}\times \mathbf {e} ^{1)){\tau '));\qquad \mathbf {e} _{3}={\frac {\mathbf {e} ^{1}\times \mathbf {e} ^{2)){\tau '))\,\!}
其中，
τ
′
=
e
1
⋅
(
e
2
×
e
3
)
=
1
/
τ
{\displaystyle \tau '=\mathbf {e} ^{1}\cdot (\mathbf {e} ^{2}\times \mathbf {e} ^{3})=1/\tau \,\!}
e
1
{\displaystyle \mathbf {e} ^{1}\,\!}
e
2
{\displaystyle \mathbf {e} ^{2}\,\!}
e
3
{\displaystyle \mathbf {e} ^{3}\,\!}
虽然
e
i
{\displaystyle \mathbf {e} _{i}\,\!}
e
j
{\displaystyle \mathbf {e} ^{j}\,\!}
e
i
⋅
e
j
=
δ
i
j
{\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}\,\!}
这样，任意向量
a
{\displaystyle \mathbf {a} \,\!}
a
1
=
a
⋅
e
1
;
a
2
=
a
⋅
e
2
;
a
3
=
a
⋅
e
3
{\displaystyle a^{1}=\mathbf {a} \cdot \mathbf {e} ^{1};\qquad a^{2}=\mathbf {a} \cdot \mathbf {e} ^{2};\qquad a^{3}=\mathbf {a} \cdot \mathbf {e} ^{3}\,\!}
类似地，共变坐标为
a
1
=
a
⋅
e
1
;
a
2
=
a
⋅
e
2
;
a
3
=
a
⋅
e
3
{\displaystyle a_{1}=\mathbf {a} \cdot \mathbf {e} _{1};\qquad a_{2}=\mathbf {a} \cdot \mathbf {e} _{2};\qquad a_{3}=\mathbf {a} \cdot \mathbf {e} _{3}\,\!}
这样，
a
{\displaystyle \mathbf {a} \,\!}
a
=
a
i
e
i
=
a
1
e
1
+
a
2
e
2
+
a
3
e
3
{\displaystyle \mathbf {a} =a_{i}\mathbf {e} ^{i}=a_{1}\mathbf {e} ^{1}+a_{2}\mathbf {e} ^{2}+a_{3}\mathbf {e} ^{3}\,\!}
或者，
a
=
a
i
e
i
=
a
1
e
1
+
a
2
e
2
+
a
3
e
3
{\displaystyle \mathbf {a} =a^{i}\mathbf {e} _{i}=a^{1}\mathbf {e} _{1}+a^{2}\mathbf {e} _{2}+a^{3}\mathbf {e} _{3}\,\!}
综合上述关系式，
a
=
(
a
⋅
e
i
)
e
i
=
(
a
⋅
e
i
)
e
i
{\displaystyle \mathbf {a} =(\mathbf {a} \cdot \mathbf {e} _{i})\mathbf {e} ^{i}=(\mathbf {a} \cdot \mathbf {e} ^{i})\mathbf {e} _{i}\,\!}
向量
a
{\displaystyle \mathbf {a} \,\!}
a
i
=
a
⋅
e
i
=
(
a
j
e
j
)
⋅
e
i
=
(
e
j
⋅
e
i
)
a
j
=
g
j
i
a
j
{\displaystyle a_{i}=\mathbf {a} \cdot \mathbf {e} _{i}=(a^{j}\mathbf {e} _{j})\cdot \mathbf {e} _{i}=(\mathbf {e} _{j}\cdot \mathbf {e} _{i})a^{j}=g_{ji}a^{j}\,\!}
其中，
g
j
i
=
e
j
⋅
e
i
{\displaystyle g_{ji}=\mathbf {e} _{j}\cdot \mathbf {e} _{i}\,\!}
度规张量 。
向量
a
{\displaystyle \mathbf {a} \,\!}
a
i
=
a
⋅
e
i
=
(
a
j
e
j
)
⋅
e
i
=
(
e
j
⋅
e
i
)
a
j
=
g
j
i
a
j
{\displaystyle a^{i}=\mathbf {a} \cdot \mathbf {e} ^{i}=(a_{j}\mathbf {e} ^{j})\cdot \mathbf {e} ^{i}=(\mathbf {e} ^{j}\cdot \mathbf {e} ^{i})a_{j}=g^{ji}a_{j}\,\!}
其中，
g
j
i
=
e
j
⋅
e
i
{\displaystyle g^{ji}=\mathbf {e} ^{j}\cdot \mathbf {e} ^{i}\,\!}
共轭度规张量 。
共变坐标的标号是下标，反变坐标的标号是上标。假若共变基底向量组成的基底是标准正交基，或反变基底向量组成的基底是标准正交基，则共变基底与反变基底相互等价。那么，就没有必要分辨共变坐标和反变坐标，所有的标号都可以用下标来标记。
![{\displaystyle {\begin{aligned}v&=\sum _{i}v^{i}[{\mathfrak {f))]X_{i}={\mathfrak {f))\,\mathbf {v} [{\mathfrak {f))]\\&=\sum _{i}v_{i}[{\mathfrak {f))]Y^{i}={\mathfrak {f))^{\sharp }\,\mathbf {v} [{\mathfrak {f))^{\sharp }]\end{aligned))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/706accb597961c64fa1ffe9c4db0bf94314c7488)
![{\displaystyle v^{i}[{\mathfrak {f))]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e403b5fdaee2e8bd8cb405934e5ff4ac90eb12d)
![{\displaystyle v_{i}[{\mathfrak {f))]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fdff97f5ab560841a1baccf4f27b6a756a1dfd7)
