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向量恒等式列表
维基媒体列表条目
语言
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编辑
(重定向自
向量恆等式
)
在这篇文章内,
向量
与其量值分别用
粗体
与
斜体
表示;例如,
|
r
|
=
r
{\displaystyle \left|\mathbf {r} \right|=r\,\!}
。
这条目陈列一些常用的
向量代数
的
恒等式
。
目录
1
三重积
2
其他乘积
3
乘积定则
4
二次微分
5
积分
5.1
格林恒等式
6
参阅
三重积
编辑
主条目:
三重积
A
×
(
B
×
C
)
=
(
C
×
B
)
×
A
=
B
(
A
⋅
C
)
−
C
(
A
⋅
B
)
{\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {C} \times \mathbf {B} )\times \mathbf {A} =\mathbf {B} (\mathbf {A} \cdot \mathbf {C} )-\mathbf {C} (\mathbf {A} \cdot \mathbf {B} )}
A
⋅
(
B
×
C
)
=
B
⋅
(
C
×
A
)
=
C
⋅
(
A
×
B
)
{\displaystyle \mathbf {A} \cdot (\mathbf {B} \times \mathbf {C} )=\mathbf {B} \cdot (\mathbf {C} \times \mathbf {A} )=\mathbf {C} \cdot (\mathbf {A} \times \mathbf {B} )}
其他乘积
编辑
(
A
×
B
)
⋅
(
A
×
B
)
=
A
2
B
2
−
(
A
⋅
B
)
2
=
B
⋅
(
A
×
(
B
×
A
)
)
{\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot (\mathbf {A} \times \mathbf {B} )=A^{2}B^{2}-(\mathbf {A} \cdot \mathbf {B} )^{2}=\mathbf {B} \cdot (\mathbf {A} \times (\mathbf {B} \times \mathbf {A} ))}
(
A
×
B
)
×
(
C
×
D
)
=
(
A
⋅
(
B
×
D
)
)
C
−
(
A
⋅
(
B
×
C
)
)
D
{\displaystyle \mathbf {\left(A\times B\right)\times } \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot (\mathbf {B\times D} )\right)\mathbf {C} -\left(\mathbf {A} \cdot (\mathbf {B\times C} )\right)\mathbf {D} }
乘积定则
编辑
∇
(
f
g
)
=
f
(
∇
g
)
+
g
(
∇
f
)
{\displaystyle \mathbf {\nabla } (fg)=f(\mathbf {\nabla } g)+g(\mathbf {\nabla } f)}
∇
(
A
⋅
B
)
=
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
+
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
{\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\mathbf {\nabla } \times \mathbf {B} )+\mathbf {B} \times (\mathbf {\nabla } \times \mathbf {A} )+(\mathbf {A} \cdot \mathbf {\nabla } )\mathbf {B} +(\mathbf {B} \cdot \mathbf {\nabla } )\mathbf {A} }
∇
(
A
⋅
B
)
=
(
A
×
∇
)
×
B
+
(
B
×
∇
)
×
A
+
A
(
∇
⋅
B
)
+
B
(
∇
⋅
A
)
{\displaystyle \mathbf {\nabla } (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \times \mathbf {\nabla } )\times \mathbf {B} +(\mathbf {B} \times \mathbf {\nabla } )\times \mathbf {A} +\mathbf {A} (\mathbf {\nabla } \cdot \mathbf {B} )+\mathbf {B} (\mathbf {\nabla } \cdot \mathbf {A} )}
∇
⋅
(
f
A
)
=
f
(
∇
⋅
A
)
+
A
⋅
(
∇
f
)
{\displaystyle \mathbf {\nabla } \cdot (f\mathbf {A} )=f(\mathbf {\nabla } \cdot \mathbf {A} )+\mathbf {A} \cdot (\mathbf {\nabla } f)}
∇
⋅
(
A
×
B
)
=
B
⋅
(
∇
×
A
)
−
A
⋅
(
∇
×
B
)
{\displaystyle \mathbf {\nabla } \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\mathbf {\nabla } \times \mathbf {A} )-\mathbf {A} \cdot (\mathbf {\nabla } \times \mathbf {B} )}
∇
×
(
f
A
)
=
f
(
∇
×
A
)
+
(
∇
f
)
×
A
{\displaystyle \nabla \times (f\mathbf {A} )=f(\nabla \times \mathbf {A} )+(\nabla f)\times \mathbf {A} }
∇
×
(
A
×
B
)
=
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
+
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
{\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )}
∇
×
(
A
⋅
B
)
=
A
×
(
∇
×
B
)
−
B
×
(
∇
×
A
)
−
(
A
×
∇
)
×
B
+
(
B
×
∇
)
×
A
{\displaystyle \nabla \times (\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-(\mathbf {A} \times \nabla )\times \mathbf {B} +(\mathbf {B} \times \nabla )\times \mathbf {A} }
∇
(
1
|
r
−
r
′
|
)
=
−
∇
′
(
1
|
r
−
r
′
|
)
=
−
r
−
r
′
|
r
−
r
′
|
3
{\displaystyle \nabla \left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\nabla '\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-\ {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\,\!}
∇
2
(
1
|
r
−
r
′
|
)
=
−
4
π
δ
(
r
−
r
′
)
{\displaystyle \nabla ^{2}\left({\frac {1}{|\mathbf {r} -\mathbf {r} '|}}\right)=-4\pi \delta (\mathbf {r} -\mathbf {r} ')}
二次微分
编辑
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
∇
×
(
∇
f
)
=
0
{\displaystyle \nabla \times (\nabla f)=\mathbf {0} }
∇
2
(
∇
⋅
A
)
=
∇
⋅
(
∇
2
A
)
{\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla ^{2}\mathbf {A} )}
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
这里,
∇
2
A
{\displaystyle \nabla ^{2}\mathbf {A} }
应被理解为对
A
{\displaystyle \mathbf {A} }
的每个分量取
拉普拉斯算子
,即
向量值函数的拉普拉斯算子
。
积分
编辑
∮
S
A
⋅
d
S
=
∫
V
(
∇
⋅
A
)
d
V
{\displaystyle \oint _{\mathbb {S} }\mathbf {A} \cdot \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\left(\nabla \cdot \mathbf {A} \right)\mathrm {d} V}
(
散度定理
)
∮
S
ψ
d
S
=
∫
V
∇
ψ
d
V
{\displaystyle \oint _{\mathbb {S} }\psi \mathrm {d} \mathbf {S} =\int _{\mathbb {V} }\nabla \psi \,\mathrm {d} V}
∮
S
(
n
^
×
A
)
⋅
d
S
=
∫
V
(
∇
×
A
)
d
V
{\displaystyle \oint _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \mathbf {A} \right)\cdot \mathrm {d} S=\int _{\mathbb {V} }\left(\nabla \times \mathbf {A} \right)\mathrm {d} V}
∮
C
A
⋅
d
l
=
∫
S
(
∇
×
A
)
⋅
d
S
{\displaystyle \oint _{\mathbb {C} }\mathbf {A} \cdot d\mathbf {l} =\int _{\mathbb {S} }\left(\nabla \times \mathbf {A} \right)\cdot \mathrm {d} \mathbf {S} }
(
斯托克斯定理
)
∮
C
ψ
d
l
=
∫
S
(
n
^
×
∇
ψ
)
d
S
{\displaystyle \oint _{\mathbb {C} }\psi d\mathbf {l} =\int _{\mathbb {S} }\left({\hat {\mathbf {n} }}\times \nabla \psi \right)\mathrm {d} S}
格林恒等式
编辑
格林第一恒等式:
∫
U
(
ψ
∇
2
ϕ
+
∇
ϕ
⋅
∇
ψ
)
d
V
=
∮
∂
U
ψ
∂
ϕ
∂
n
d
S
{\displaystyle \int _{\mathbb {U} }(\psi \nabla ^{2}\phi +\nabla \phi \cdot \nabla \psi )\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\psi {\partial \phi \over \partial n}\,\mathrm {d} S}
格林第二恒等式:
∫
U
(
ψ
∇
2
ϕ
−
ϕ
∇
2
ψ
)
d
V
=
∮
∂
U
(
ψ
∂
ϕ
∂
n
−
ϕ
∂
ψ
∂
n
)
d
S
{\displaystyle \int _{\mathbb {U} }\left(\psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi \right)\,\mathrm {d} V=\oint _{\partial \mathbb {U} }\left(\psi {\partial \phi \over \partial n}-\phi {\partial \psi \over \partial n}\right)\,\mathrm {d} S}
格林第三恒等式:
ψ
(
x
)
−
∫
U
[
G
(
x
,
x
′
)
∇
′
2
ψ
(
x
′
)
]
d
V
′
=
∮
∂
U
[
ψ
(
x
′
)
∂
G
(
x
,
x
′
)
∂
n
′
−
G
(
x
,
x
′
)
∂
ψ
(
x
′
)
∂
n
′
]
d
S
′
{\displaystyle \psi (\mathbf {x} )-\int _{\mathbb {U} }\left[G(\mathbf {x} ,\mathbf {x} ')\nabla '^{\,2}\psi (\mathbf {x} ')\right]\,\mathrm {d} V'=\oint _{\partial \mathbb {U} }\left[\psi (\mathbf {x} '){\partial G(\mathbf {x} ,\mathbf {x} ') \over \partial n'}-G(\mathbf {x} ,\mathbf {x} '){\partial \psi (\mathbf {x} ') \over \partial n'}\right]\,\mathrm {d} S'}
参阅
编辑
格林恒等式
数学恒等式列表
(
List of mathematical identities
)
向量微积分恒等式
(
Vector calculus identities
)