The following identities are important in vector calculus :
Operator notations
Gradient
In the three-dimensional Cartesian coordinate system , the gradient of some function
f
(
x
,
y
,
z
)
{\displaystyle f(x,y,z)}
grad
(
f
)
=
∇
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
+
∂
f
∂
z
k
{\displaystyle \operatorname {grad} (f)=\nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }
where i , j , k are the standard unit vectors .
The gradient of a tensor field ,
A
{\displaystyle \mathbf {A} }
n , is generally written as
grad
(
A
)
=
∇
A
{\displaystyle \operatorname {grad} (\mathbf {A} )=\nabla \mathbf {A} }
and is a tensor field of order n + 1
ψ
{\displaystyle \psi }
grad
(
ψ
)
=
∇
ψ
{\displaystyle \operatorname {grad} (\psi )=\nabla \psi }
is a vector field .
Divergence
In three-dimensional Cartesian coordinates, the divergence of a continuously differentiable vector field
F
=
F
x
i
+
F
y
j
+
F
z
k
{\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }
scalar -valued function:
div
F
=
∇
⋅
F
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
⋅
(
F
x
,
F
y
,
F
z
)
=
∂
F
x
∂
x
+
∂
F
y
∂
y
+
∂
F
z
∂
z
.
{\displaystyle \operatorname {div} \,\mathbf {F} =\nabla \cdot \mathbf {F} =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot (F_{x},F_{y},F_{z})={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.}
The divergence of a tensor field ,
A
{\displaystyle \mathbf {A} }
n , is generally written as
div
(
A
)
=
∇
⋅
A
{\displaystyle \operatorname {div} (\mathbf {A} )=\nabla \cdot \mathbf {A} }
and is a contraction to a tensor field of order n − 1
∇
⋅
(
B
⊗
A
^
)
=
A
^
(
∇
⋅
B
)
+
(
B
⋅
∇
)
A
^
{\displaystyle \nabla \cdot (\mathbf {B} \otimes {\hat {\mathbf {A} }})={\hat {\mathbf {A} }}(\nabla \cdot \mathbf {B} )+(\mathbf {B} \cdot \nabla ){\hat {\mathbf {A} }}}
where
B
⋅
∇
{\displaystyle \mathbf {B} \cdot \nabla }
directional derivative in the direction of
B
{\displaystyle \mathbf {B} }
∇
⋅
(
a
b
T
)
=
b
(
∇
⋅
a
)
+
(
a
⋅
∇
)
b
.
{\displaystyle \nabla \cdot (\mathbf {a} \mathbf {b} ^{\mathrm {T} })=\mathbf {b} (\nabla \cdot \mathbf {a} )+(\mathbf {a} \cdot \nabla )\mathbf {b} \ .}
Curl
In Cartesian coordinates , for
F
=
F
x
i
+
F
y
j
+
F
z
k
{\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }
curl(
F
{\displaystyle \mathbf {F} }
∇
×
F
=
|
i
j
k
∂
∂
x
∂
∂
y
∂
∂
z
F
x
F
y
F
z
|
{\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}}
∇
×
F
=
(
∂
F
z
∂
y
−
∂
F
y
∂
z
)
i
−
(
∂
F
x
∂
z
−
∂
F
z
∂
x
)
j
+
(
∂
F
y
∂
x
−
∂
F
x
∂
y
)
k
{\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} -\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} }
where i , j , and k are the unit vectors for the x -, y -, and z -axes, respectively.
v
{\displaystyle \mathbf {v} }
∇
×
v
{\displaystyle \nabla \times \mathbf {v} }
or in Einstein notation as:
ε
i
j
k
∂
v
k
∂
x
j
{\displaystyle \varepsilon ^{ijk}{\frac {\partial v_{k}}{\partial x^{j}}}}
where ε is the Levi-Civita symbol .
Laplacian
In Cartesian coordinates
f
(
x
,
y
,
z
)
{\displaystyle f(x,y,z)}
Δ
f
=
∇
2
f
=
(
∇
⋅
∇
)
f
=
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
.
{\displaystyle \Delta f=\nabla ^{2}f=(\nabla \cdot \nabla )f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}
For a tensor field ,
A
{\displaystyle \mathbf {A} }
Δ
A
=
∇
2
A
=
(
∇
⋅
∇
)
A
{\displaystyle \Delta \mathbf {A} =\nabla ^{2}\mathbf {A} =(\nabla \cdot \nabla )\mathbf {A} }
and is a tensor field of the same order.
Special notations
In Feynman subscript notation ,
∇
B
(
A
⋅
B
)
=
A
×
(
∇
×
B
)
+
(
A
⋅
∇
)
B
{\displaystyle \nabla _{\mathbf {B} }\left(\mathbf {A\cdot B} \right)=\mathbf {A} \times \left(\nabla \times \mathbf {B} \right)+\left(\mathbf {A} \cdot \nabla \right)\mathbf {B} }
where the notation ∇B B .[ 1] [ 2]
A less general but similar idea is used in geometric algebra overdot notation is employed.[ 3]
∇
˙
(
A
⋅
B
˙
)
=
A
×
(
∇
×
B
)
+
(
A
⋅
∇
)
B
{\displaystyle {\dot {\nabla }}\left(\mathbf {A} \cdot {\dot {\mathbf {B} }}\right)=\mathbf {A} \times \left(\nabla \times \mathbf {B} \right)+\left(\mathbf {A} \cdot \nabla \right)\mathbf {B} }
where overdots define the scope of the vector derivative. The dotted vector, in this case B , is differentiated, while the (undotted) A is held constant.
For the remainder of this article, Feynman subscript notation will be used where appropriate.
Properties
Distributive properties
∇
(
ψ
+
ϕ
)
=
∇
ψ
+
∇
ϕ
{\displaystyle \nabla (\psi +\phi )=\nabla \psi +\nabla \phi }
∇
⋅
(
A
+
B
)
=
∇
⋅
A
+
∇
⋅
B
{\displaystyle \nabla \cdot (\mathbf {A} +\mathbf {B} )=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} }
∇
×
(
A
+
B
)
=
∇
×
A
+
∇
×
B
{\displaystyle \nabla \times (\mathbf {A} +\mathbf {B} )=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} }
Product rule for the gradient
The gradient of the product of two scalar fields
ψ
{\displaystyle \psi }
ϕ
{\displaystyle \phi }
product rule in single variable calculus .
∇
(
ψ
ϕ
)
=
ϕ
∇
ψ
+
ψ
∇
ϕ
{\displaystyle \nabla (\psi \,\phi )=\phi \,\nabla \psi +\psi \,\nabla \phi }
Product of a scalar and a vector
∇
⋅
(
ψ
A
)
=
ψ
(
∇
⋅
A
)
+
A
⋅
(
∇
ψ
)
{\displaystyle \nabla \cdot (\psi \mathbf {A} )=\psi (\nabla \cdot \mathbf {A} )+\mathbf {A} \cdot (\nabla \psi )}
∇
×
(
ψ
A
)
=
ψ
(
∇
×
A
)
+
(
∇
ψ
)
×
A
{\displaystyle \nabla \times (\psi \mathbf {A} )=\psi (\nabla \times \mathbf {A} )+(\nabla \psi )\times \mathbf {A} }
Quotient rule
∇
(
f
g
)
=
g
∇
f
−
(
∇
g
)
f
g
2
{\displaystyle \nabla \left({\frac {f}{g}}\right)={\frac {g\nabla f-(\nabla g)f}{g^{2}}}}
∇
⋅
(
A
g
)
=
g
∇
⋅
A
−
(
∇
g
)
⋅
A
g
2
{\displaystyle \nabla \cdot \left({\frac {\mathbf {A} }{g}}\right)={\frac {g\nabla \cdot \mathbf {A} -(\nabla g)\cdot \mathbf {A} }{g^{2}}}}
∇
×
(
A
g
)
=
g
∇
×
A
−
(
∇
g
)
×
A
g
2
{\displaystyle \nabla \times \left({\frac {\mathbf {A} }{g}}\right)={\frac {g\nabla \times \mathbf {A} -(\nabla g)\times \mathbf {A} }{g^{2}}}}
Chain rule
∇
(
f
∘
g
)
=
(
f
′
∘
g
)
∇
g
{\displaystyle \nabla (f\circ g)=(f'\circ g)\nabla g}
∇
(
f
∘
A
)
=
(
∇
f
∘
A
)
∇
A
{\displaystyle \nabla (f\circ \mathbf {A} )=(\nabla f\circ \mathbf {A} )\nabla \mathbf {A} }
∇
⋅
(
A
∘
f
)
=
(
A
′
∘
f
)
⋅
∇
f
{\displaystyle \nabla \cdot (\mathbf {A} \circ f)=(\mathbf {A} '\circ f)\cdot \nabla f}
∇
×
(
A
∘
f
)
=
−
(
A
′
∘
f
)
×
∇
f
{\displaystyle \nabla \times (\mathbf {A} \circ f)=-(\mathbf {A} '\circ f)\times \nabla f}
Vector dot product
∇
(
A
⋅
B
)
=
J
A
T
B
+
J
B
T
A
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
.
{\displaystyle {\begin{aligned}\nabla (\mathbf {A} \cdot \mathbf {B} )&=\mathbf {J} _{\mathbf {A} }^{\mathrm {T} }\mathbf {B} +\mathbf {J} _{\mathbf {B} }^{\mathrm {T} }\mathbf {A} \\&=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} )\ .\end{aligned}}}
where J A Jacobian of A
Alternatively, using Feynman subscript notation,
∇
(
A
⋅
B
)
=
∇
A
(
A
⋅
B
)
+
∇
B
(
A
⋅
B
)
.
{\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=\nabla _{\mathbf {A} }(\mathbf {A} \cdot \mathbf {B} )+\nabla _{\mathbf {B} }(\mathbf {A} \cdot \mathbf {B} )\ .}
As a special case, when A = B
1
2
∇
(
A
⋅
A
)
=
J
A
T
A
=
(
A
⋅
∇
)
A
+
A
×
(
∇
×
A
)
.
{\displaystyle {\begin{aligned}{\frac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)&=\mathbf {J} _{\mathbf {A} }^{\mathrm {T} }\mathbf {A} \\&=(\mathbf {A} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {A} )\ .\end{aligned}}}
Vector cross product
∇
⋅
(
A
×
B
)
=
(
∇
×
A
)
⋅
B
−
A
⋅
(
∇
×
B
)
{\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )}
∇
×
(
A
×
B
)
=
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
=
(
∇
⋅
B
+
B
⋅
∇
)
A
−
(
∇
⋅
A
+
A
⋅
∇
)
B
=
∇
⋅
(
B
A
T
)
−
∇
⋅
(
A
B
T
)
=
∇
⋅
(
B
A
T
−
A
B
T
)
{\displaystyle {\begin{aligned}\nabla \times (\mathbf {A} \times \mathbf {B} )&=\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+(\mathbf {B} \cdot \nabla )\mathbf {A} -(\mathbf {A} \cdot \nabla )\mathbf {B} \\&=(\nabla \cdot \mathbf {B} +\mathbf {B} \cdot \nabla )\mathbf {A} -(\nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla )\mathbf {B} \\&=\nabla \cdot (\mathbf {B} \mathbf {A} ^{\mathrm {T} })-\nabla \cdot (\mathbf {A} \mathbf {B} ^{\mathrm {T} })\\&=\nabla \cdot (\mathbf {B} \mathbf {A} ^{\mathrm {T} }-\mathbf {A} \mathbf {B} ^{\mathrm {T} })\end{aligned}}}
Second derivatives
Curl of the gradient
The curl of the gradient of any twice-differentiable scalar field
ϕ
{\displaystyle \ \phi }
zero vector :
∇
×
(
∇
ϕ
)
=
0
{\displaystyle \nabla \times (\nabla \phi )=\mathbf {0} }
Divergence of the curl
The divergence of the curl of any vector field A is always zero:
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
Divergence of the gradient
The Laplacian of a scalar field is defined as the divergence of the gradient:
∇
2
ψ
=
∇
⋅
(
∇
ψ
)
{\displaystyle \nabla ^{2}\psi =\nabla \cdot (\nabla \psi )}
Note that the result is a scalar quantity.
Curl of the curl
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \nabla \times \left(\nabla \times \mathbf {A} \right)=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
Here,∇2 is the vector Laplacian operating on the vector field A .
Summary of important identities
Addition and multiplication
A
+
B
=
B
+
A
{\displaystyle \mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }
A
⋅
B
=
B
⋅
A
{\displaystyle \mathbf {A} \cdot \mathbf {B} =\mathbf {B} \cdot \mathbf {A} }
A
×
B
=
−
B
×
A
{\displaystyle \mathbf {A} \times \mathbf {B} =\mathbf {-B} \times \mathbf {A} }
(
A
+
B
)
⋅
C
=
A
⋅
C
+
B
⋅
C
{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\cdot \mathbf {C} =\mathbf {A} \cdot \mathbf {C} +\mathbf {B} \cdot \mathbf {C} }
(
A
+
B
)
×
C
=
A
×
C
+
B
×
C
{\displaystyle \left(\mathbf {A} +\mathbf {B} \right)\times \mathbf {C} =\mathbf {A} \times \mathbf {C} +\mathbf {B} \times \mathbf {C} }
A
⋅
(
B
×
C
)
=
B
⋅
(
C
×
A
)
=
C
⋅
(
A
×
B
)
{\displaystyle \mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)=\mathbf {B} \cdot \left(\mathbf {C} \times \mathbf {A} \right)=\mathbf {C} \cdot \left(\mathbf {A} \times \mathbf {B} \right)}
scalar triple product )
A
×
(
B
×
C
)
=
(
A
⋅
C
)
B
−
(
A
⋅
B
)
C
{\displaystyle \mathbf {A} \times \left(\mathbf {B} \times \mathbf {C} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\mathbf {B} -\left(\mathbf {A} \cdot \mathbf {B} \right)\mathbf {C} }
vector triple product )
(
A
×
B
)
×
C
=
(
A
⋅
C
)
B
−
(
B
⋅
C
)
A
{\displaystyle \left(\mathbf {A} \times \mathbf {B} \right)\times \mathbf {C} =\left(\mathbf {A} \cdot \mathbf {C} \right)\mathbf {B} -\left(\mathbf {B} \cdot \mathbf {C} \right)\mathbf {A} }
vector triple product )
(
A
×
B
)
⋅
(
C
×
D
)
=
(
A
⋅
C
)
(
B
⋅
D
)
−
(
B
⋅
C
)
(
A
⋅
D
)
{\displaystyle \left(\mathbf {A} \times \mathbf {B} \right)\cdot \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \mathbf {C} \right)\left(\mathbf {B} \cdot \mathbf {D} \right)-\left(\mathbf {B} \cdot \mathbf {C} \right)\left(\mathbf {A} \cdot \mathbf {D} \right)}
(
A
⋅
(
B
×
C
)
)
D
=
(
A
⋅
D
)
(
B
×
C
)
+
(
B
⋅
D
)
(
C
×
A
)
+
(
C
⋅
D
)
(
A
×
B
)
{\displaystyle \left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D} =\left(\mathbf {A} \cdot \mathbf {D} \right)\left(\mathbf {B} \times \mathbf {C} \right)+\left(\mathbf {B} \cdot \mathbf {D} \right)\left(\mathbf {C} \times \mathbf {A} \right)+\left(\mathbf {C} \cdot \mathbf {D} \right)\left(\mathbf {A} \times \mathbf {B} \right)}
(
A
×
B
)
×
(
C
×
D
)
=
(
A
⋅
(
B
×
D
)
)
C
−
(
A
⋅
(
B
×
C
)
)
D
{\displaystyle \left(\mathbf {A} \times \mathbf {B} \right)\times \left(\mathbf {C} \times \mathbf {D} \right)=\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {D} \right)\right)\mathbf {C} -\left(\mathbf {A} \cdot \left(\mathbf {B} \times \mathbf {C} \right)\right)\mathbf {D} }
Differentiation
Gradient
∇
(
ψ
+
ϕ
)
=
∇
ψ
+
∇
ϕ
{\displaystyle \nabla (\psi +\phi )=\nabla \psi +\nabla \phi }
∇
(
ψ
ϕ
)
=
ϕ
∇
ψ
+
ψ
∇
ϕ
{\displaystyle \nabla (\psi \,\phi )=\phi \,\nabla \psi +\psi \,\nabla \phi }
∇
(
A
⋅
B
)
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
{\displaystyle \nabla \left(\mathbf {A} \cdot \mathbf {B} \right)=\left(\mathbf {A} \cdot \nabla \right)\mathbf {B} +\left(\mathbf {B} \cdot \nabla \right)\mathbf {A} +\mathbf {A} \times \left(\nabla \times \mathbf {B} \right)+\mathbf {B} \times \left(\nabla \times \mathbf {A} \right)}
Divergence
∇
⋅
(
A
+
B
)
=
∇
⋅
A
+
∇
⋅
B
{\displaystyle \nabla \cdot (\mathbf {A} +\mathbf {B} )=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} }
∇
⋅
(
ψ
A
)
=
ψ
∇
⋅
A
+
A
⋅
∇
ψ
{\displaystyle \nabla \cdot \left(\psi \mathbf {A} \right)=\psi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla \psi }
∇
⋅
(
A
×
B
)
=
B
⋅
(
∇
×
A
)
−
A
⋅
(
∇
×
B
)
{\displaystyle \nabla \cdot \left(\mathbf {A} \times \mathbf {B} \right)=\mathbf {B} \cdot (\nabla \times \mathbf {A} )-\mathbf {A} \cdot (\nabla \times \mathbf {B} )}
Curl
∇
×
(
A
+
B
)
=
∇
×
A
+
∇
×
B
{\displaystyle \nabla \times (\mathbf {A} +\mathbf {B} )=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} }
∇
×
(
ψ
A
)
=
ψ
∇
×
A
+
∇
ψ
×
A
{\displaystyle \nabla \times \left(\psi \mathbf {A} \right)=\psi \nabla \times \mathbf {A} +\nabla \psi \times \mathbf {A} }
∇
×
(
A
×
B
)
=
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
{\displaystyle \nabla \times \left(\mathbf {A} \times \mathbf {B} \right)=\mathbf {A} \left(\nabla \cdot \mathbf {B} \right)-\mathbf {B} \left(\nabla \cdot \mathbf {A} \right)+\left(\mathbf {B} \cdot \nabla \right)\mathbf {A} -\left(\mathbf {A} \cdot \nabla \right)\mathbf {B} }
Second derivatives
DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles(dashed) mean that DD and GG do not exist.
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
∇
×
(
∇
ψ
)
=
0
{\displaystyle \nabla \times (\nabla \psi )=\mathbf {0} }
∇
⋅
(
∇
ψ
)
=
∇
2
ψ
{\displaystyle \nabla \cdot (\nabla \psi )=\nabla ^{2}\psi }
scalar Laplacian )
∇
(
∇
⋅
A
)
−
∇
×
(
∇
×
A
)
=
∇
2
A
{\displaystyle \nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla \times \left(\nabla \times \mathbf {A} \right)=\nabla ^{2}\mathbf {A} }
vector Laplacian )
∇
⋅
(
ϕ
∇
ψ
)
=
ϕ
∇
2
ψ
+
∇
ϕ
⋅
∇
ψ
{\displaystyle \nabla \cdot (\phi \nabla \psi )=\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi }
ψ
∇
2
ϕ
−
ϕ
∇
2
ψ
=
∇
⋅
(
ψ
∇
ϕ
−
ϕ
∇
ψ
)
{\displaystyle \psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi =\nabla \cdot \left(\psi \nabla \phi -\phi \nabla \psi \right)}
∇
2
(
ϕ
ψ
)
=
ϕ
∇
2
ψ
+
2
∇
ϕ
⋅
∇
ψ
+
ψ
∇
2
ϕ
{\displaystyle \nabla ^{2}(\phi \psi )=\phi \nabla ^{2}\psi +2\nabla \phi \cdot \nabla \psi +\psi \nabla ^{2}\phi }
∇
2
(
ψ
A
)
=
A
∇
2
ψ
+
2
(
∇
ψ
⋅
∇
)
A
+
ψ
∇
2
A
{\displaystyle \nabla ^{2}(\psi \mathbf {A} )=\mathbf {A} \nabla ^{2}\psi +2(\nabla \psi \cdot \nabla )\mathbf {A} +\psi \nabla ^{2}\mathbf {A} }
∇
2
(
A
⋅
B
)
=
A
⋅
∇
2
B
−
B
⋅
∇
2
A
+
2
∇
⋅
(
(
B
⋅
∇
)
A
+
B
×
∇
×
A
)
{\displaystyle \nabla ^{2}(\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \cdot \nabla ^{2}\mathbf {B} -\mathbf {B} \cdot \nabla ^{2}\mathbf {A} +2\nabla \cdot ((\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {B} \times \nabla \times \mathbf {A} )}
Green's vector identity )
Third derivatives
∇
2
(
∇
ψ
)
=
∇
(
∇
⋅
(
∇
ψ
)
)
=
∇
(
∇
2
ψ
)
{\displaystyle \nabla ^{2}(\nabla \psi )=\nabla (\nabla \cdot (\nabla \psi ))=\nabla (\nabla ^{2}\psi )}
∇
2
(
∇
⋅
A
)
=
∇
⋅
(
∇
(
∇
⋅
A
)
)
=
∇
⋅
(
∇
2
A
)
{\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla (\nabla \cdot \mathbf {A} ))=\nabla \cdot (\nabla ^{2}\mathbf {A} )}
∇
2
(
∇
×
A
)
=
−
∇
×
(
∇
×
(
∇
×
A
)
)
=
∇
×
(
∇
2
A
)
{\displaystyle \nabla ^{2}(\nabla \times \mathbf {A} )=-\nabla \times (\nabla \times (\nabla \times \mathbf {A} ))=\nabla \times (\nabla ^{2}\mathbf {A} )}
Integration
Below, the curly symbol ∂ means "boundary of ".
In the following surface–volume integral theorems, V denotes a 3d volume with a corresponding 2d boundary S = ∂V (a closed surface ):
∂
V
{\displaystyle \scriptstyle \partial V}
A
⋅
d
S
=
∭
V
(
∇
⋅
A
)
d
V
{\displaystyle \mathbf {A} \cdot d\mathbf {S} =\iiint _{V}\left(\nabla \cdot \mathbf {A} \right)dV}
(Divergence theorem )
∂
V
{\displaystyle \scriptstyle \partial V}
ψ
d
S
=
∭
V
∇
ψ
d
V
{\displaystyle \psi d\mathbf {S} =\iiint _{V}\nabla \psi \,dV}
∂
V
{\displaystyle \scriptstyle \partial V}
(
n
^
×
A
)
d
S
=
∭
V
(
∇
×
A
)
d
V
{\displaystyle \left({\hat {\mathbf {n} }}\times \mathbf {A} \right)dS=\iiint _{V}\left(\nabla \times \mathbf {A} \right)dV}
∂
V
{\displaystyle \scriptstyle \partial V}
ψ
(
∇
φ
⋅
n
^
)
d
S
=
∭
V
(
ψ
∇
2
φ
+
∇
φ
⋅
∇
ψ
)
d
V
{\displaystyle \psi \left(\nabla \varphi \cdot {\hat {\mathbf {n} }}\right)dS=\iiint _{V}\left(\psi \nabla ^{2}\varphi +\nabla \varphi \cdot \nabla \psi \right)dV}
(Green's first identity )
∂
V
{\displaystyle \scriptstyle \partial V}
[
(
ψ
∇
φ
−
φ
∇
ψ
)
⋅
n
^
]
d
S
=
{\displaystyle \left[\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot {\hat {\mathbf {n} }}\right]dS=\,\!}
∂
V
{\displaystyle \scriptstyle \partial V}
[
ψ
∂
φ
∂
n
−
φ
∂
ψ
∂
n
]
d
S
{\displaystyle \left[\psi {\frac {\partial \varphi }{\partial n}}-\varphi {\frac {\partial \psi }{\partial n}}\right]dS}
=
∭
V
(
ψ
∇
2
φ
−
φ
∇
2
ψ
)
d
V
{\displaystyle \displaystyle =\iiint _{V}\left(\psi \nabla ^{2}\varphi -\varphi \nabla ^{2}\psi \right)dV\,\!}
(Green's second identity )
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve ):
∮
∂
S
A
⋅
d
ℓ
=
∬
S
(
∇
×
A
)
⋅
d
s
{\displaystyle \oint _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}=\iint _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {s} }
(Stokes' theorem )
∮
∂
S
ψ
d
ℓ
=
∬
S
(
n
^
×
∇
ψ
)
d
S
{\displaystyle \oint _{\partial S}\psi d{\boldsymbol {\ell }}=\iint _{S}\left({\hat {\mathbf {n} }}\times \nabla \psi \right)dS}
Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral ):
∂
S
{\displaystyle {\scriptstyle \partial S}}
A
⋅
d
ℓ
=
−
{\displaystyle \mathbf {A} \cdot {\rm {d}}{\boldsymbol {\ell }}=-}
∂
S
{\displaystyle {\scriptstyle \partial S}}
A
⋅
d
ℓ
.
{\displaystyle \mathbf {A} \cdot {\rm {d}}{\boldsymbol {\ell }}.}
See also
References
^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics . Addison-Wesley. Vol II, p. 27–4. ISBN 0-8053-9049-9 ^ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". arXiv :physics/0504223 ^ Doran, C. ; Lasenby, A. (2003). Geometric algebra for physicists . Cambridge University Press. p. 169. ISBN 978-0-521-71595-9
Further reading
Balanis, Constantine A. Advanced Engineering Electromagnetics . ISBN 0-471-62194-3 Schey, H. M. (1997). Div Grad Curl and all that: An informal text on vector calculus . W. W. Norton & Company. ISBN 0-393-96997-5 Griffiths, David J. (1999). Introduction to Electrodynamics . Prentice Hall. ISBN 0-13-805326-X
![{\displaystyle \left[\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot {\hat {\mathbf {n} }}\right]dS=\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1470e43dc80bf8600b28ce6f82bc271b01036c)
![{\displaystyle \left[\psi {\frac {\partial \varphi }{\partial n}}-\varphi {\frac {\partial \psi }{\partial n}}\right]dS}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d09e9241f712c4ab4f74dba58c6cddabd74710f)
