Gradient
For a function
f
(
x
,
y
,
z
)
{\displaystyle f(x,y,z)}
Cartesian coordinate variables, the gradient is the vector field:
grad
(
f
)
=
∇
f
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
f
=
∂
f
∂
x
i
+
∂
f
∂
y
j
+
∂
f
∂
z
k
{\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}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 for the x , y , z -axes. More generally, for a function of n variables
ψ
(
x
1
,
…
,
x
n
)
{\displaystyle \psi (x_{1},\ldots ,x_{n})}
scalar field, the gradient is the vector field :
∇
ψ
=
(
∂
∂
x
1
,
…
,
∂
∂
x
n
)
ψ
=
∂
ψ
∂
x
1
e
1
+
⋯
+
∂
ψ
∂
x
n
e
n
{\displaystyle \nabla \psi ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\end{pmatrix}}\psi ={\frac {\partial \psi }{\partial x_{1}}}\mathbf {e} _{1}+\dots +{\frac {\partial \psi }{\partial x_{n}}}\mathbf {e} _{n}}
e
i
(
i
=
1
,
2
,
.
.
.
,
n
)
{\displaystyle \mathbf {e} _{i}\,(i=1,2,...,n)}
As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.
For a vector field
A
=
(
A
1
,
…
,
A
n
)
{\displaystyle \mathbf {A} =\left(A_{1},\ldots ,A_{n}\right)}
total derivative is the n × n Jacobian matrix :
J
A
=
d
A
=
(
∇
A
)
T
=
(
∂
A
i
∂
x
j
)
i
j
.
{\displaystyle \mathbf {J} _{\mathbf {A} }=d\mathbf {A} =(\nabla \!\mathbf {A} )^{\textsf {T}}=\left({\frac {\partial A_{i}}{\partial x_{j}}}\right)_{\!ij}.}
For a tensor field
T
{\displaystyle \mathbf {T} }
k , the gradient
grad
(
T
)
=
d
T
=
(
∇
T
)
T
{\displaystyle \operatorname {grad} (\mathbf {T} )=d\mathbf {T} =(\nabla \mathbf {T} )^{\textsf {T}}}
k + 1.
For a tensor field
T
{\displaystyle \mathbf {T} }
k > 0, the tensor field
∇
T
{\displaystyle \nabla \mathbf {T} }
k + 1 is defined by the recursive relation
(
∇
T
)
⋅
C
=
∇
(
T
⋅
C
)
{\displaystyle (\nabla \mathbf {T} )\cdot \mathbf {C} =\nabla (\mathbf {T} \cdot \mathbf {C} )}
C
{\displaystyle \mathbf {C} }
Divergence
In 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} }
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} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\cdot {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.}
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The divergence of a tensor field
T
{\displaystyle \mathbf {T} }
k is written as
div
(
T
)
=
∇
⋅
T
{\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} }
contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,
∇
⋅
(
A
⊗
T
)
=
T
(
∇
⋅
A
)
+
(
A
⋅
∇
)
T
{\displaystyle \nabla \cdot \left(\mathbf {A} \otimes \mathbf {T} \right)=\mathbf {T} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {T} }
A
⋅
∇
{\displaystyle \mathbf {A} \cdot \nabla }
directional derivative in the direction of
A
{\displaystyle \mathbf {A} }
∇
⋅
(
A
B
T
)
=
B
(
∇
⋅
A
)
+
(
A
⋅
∇
)
B
.
{\displaystyle \nabla \cdot \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=\mathbf {B} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {B} .}
For a tensor field
T
{\displaystyle \mathbf {T} }
k > 1, the tensor field
∇
⋅
T
{\displaystyle \nabla \cdot \mathbf {T} }
k − 1 is defined by the recursive relation
(
∇
⋅
T
)
⋅
C
=
∇
⋅
(
T
⋅
C
)
{\displaystyle (\nabla \cdot \mathbf {T} )\cdot \mathbf {C} =\nabla \cdot (\mathbf {T} \cdot \mathbf {C} )}
C
{\displaystyle \mathbf {C} }
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
=
∇
×
F
=
(
∂
∂
x
,
∂
∂
y
,
∂
∂
z
)
×
(
F
x
,
F
y
,
F
z
)
=
|
i
j
k
∂
∂
x
∂
∂
y
∂
∂
z
F
x
F
y
F
z
|
=
(
∂
F
z
∂
y
−
∂
F
y
∂
z
)
i
+
(
∂
F
x
∂
z
−
∂
F
z
∂
x
)
j
+
(
∂
F
y
∂
x
−
∂
F
x
∂
y
)
k
{\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\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}}\\[1em]&=\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} \end{aligned}}}
i , j , and k are the unit vectors for the x -, y -, and z -axes, respectively.
As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.
In Einstein notation , the vector field
F
=
(
F
1
,
F
2
,
F
3
)
{\displaystyle \mathbf {F} ={\begin{pmatrix}F_{1},\ F_{2},\ F_{3}\end{pmatrix}}}
∇
×
F
=
ε
i
j
k
e
i
∂
F
k
∂
x
j
{\displaystyle \nabla \times \mathbf {F} =\varepsilon ^{ijk}\mathbf {e} _{i}{\frac {\partial F_{k}}{\partial x_{j}}}}
ε
{\displaystyle \varepsilon }
Levi-Civita parity symbol .
For a tensor field
T
{\displaystyle \mathbf {T} }
k > 1, the tensor field
∇
×
T
{\displaystyle \nabla \times \mathbf {T} }
k is defined by the recursive relation
(
∇
×
T
)
⋅
C
=
∇
×
(
T
⋅
C
)
{\displaystyle (\nabla \times \mathbf {T} )\cdot \mathbf {C} =\nabla \times (\mathbf {T} \cdot \mathbf {C} )}
C
{\displaystyle \mathbf {C} }
A tensor field of order greater than one may be decomposed into a sum of outer products , and then the following identity may be used:
∇
×
(
A
⊗
T
)
=
(
∇
×
A
)
⊗
T
−
A
×
(
∇
T
)
.
{\displaystyle \nabla \times \left(\mathbf {A} \otimes \mathbf {T} \right)=(\nabla \times \mathbf {A} )\otimes \mathbf {T} -\mathbf {A} \times (\nabla \mathbf {T} ).}
∇
×
(
A
B
T
)
=
(
∇
×
A
)
B
T
−
A
×
(
∇
B
)
.
{\displaystyle \nabla \times \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=(\nabla \times \mathbf {A} )\mathbf {B} ^{\textsf {T}}-\mathbf {A} \times (\nabla \mathbf {B} ).}
Laplacian
In Cartesian coordinates , the Laplacian of a function
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}}}.}
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
When the Laplacian is equal to 0, the function is called a harmonic function . That is,
Δ
f
=
0.
{\displaystyle \Delta f=0.}
For a tensor field ,
T
{\displaystyle \mathbf {T} }
Δ
T
=
∇
2
T
=
(
∇
⋅
∇
)
T
{\displaystyle \Delta \mathbf {T} =\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} }
For a tensor field
T
{\displaystyle \mathbf {T} }
k > 0, the tensor field
∇
2
T
{\displaystyle \nabla ^{2}\mathbf {T} }
k is defined by the recursive relation
(
∇
2
T
)
⋅
C
=
∇
2
(
T
⋅
C
)
{\displaystyle \left(\nabla ^{2}\mathbf {T} \right)\cdot \mathbf {C} =\nabla ^{2}(\mathbf {T} \cdot \mathbf {C} )}
C
{\displaystyle \mathbf {C} }
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} }
B B .[1] [2]
Less general but similar is the Hestenes overdot notation in geometric algebra .[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} }
B , is differentiated, while the (undotted) A is held constant.
The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C ⋅(A ×B ) = (C ×A )⋅B :
∇
⋅
(
A
×
B
)
=
∇
A
⋅
(
A
×
B
)
+
∇
B
⋅
(
A
×
B
)
=
(
∇
A
×
A
)
⋅
B
+
(
∇
B
×
A
)
⋅
B
=
(
∇
A
×
A
)
⋅
B
−
(
A
×
∇
B
)
⋅
B
=
(
∇
A
×
A
)
⋅
B
−
A
⋅
(
∇
B
×
B
)
=
(
∇
×
A
)
⋅
B
−
A
⋅
(
∇
×
B
)
{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\nabla _{\mathbf {A} }\cdot (\mathbf {A} \times \mathbf {B} )+\nabla _{\mathbf {B} }\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} +(\nabla _{\mathbf {B} }\times \mathbf {A} )\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \nabla _{\mathbf {B} })\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla _{\mathbf {B} }\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}}
An alternative method is to use the Cartesian components of the del operator as follows:
∇
⋅
(
A
×
B
)
=
e
i
∂
i
⋅
(
A
×
B
)
=
e
i
⋅
(
∂
i
A
×
B
+
A
×
∂
i
B
)
=
e
i
⋅
(
∂
i
A
×
B
)
+
e
i
⋅
(
A
×
∂
i
B
)
=
(
e
i
×
∂
i
A
)
⋅
B
+
(
e
i
×
A
)
⋅
∂
i
B
=
(
e
i
×
∂
i
A
)
⋅
B
−
(
A
×
e
i
)
⋅
∂
i
B
=
(
e
i
×
∂
i
A
)
⋅
B
−
A
⋅
(
e
i
×
∂
i
B
)
=
(
e
i
∂
i
×
A
)
⋅
B
−
A
⋅
(
e
i
∂
i
×
B
)
=
(
∇
×
A
)
⋅
B
−
A
⋅
(
∇
×
B
)
{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\mathbf {e} _{i}\partial _{i}\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=\mathbf {e} _{i}\cdot (\partial _{i}\mathbf {A} \times \mathbf {B} +\mathbf {A} \times \partial _{i}\mathbf {B} )\\[2pt]&=\mathbf {e} _{i}\cdot (\partial _{i}\mathbf {A} \times \mathbf {B} )+\mathbf {e} _{i}\cdot (\mathbf {A} \times \partial _{i}\mathbf {B} )\\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} +(\mathbf {e} _{i}\times \mathbf {A} )\cdot \partial _{i}\mathbf {B} \\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \mathbf {e} _{i})\cdot \partial _{i}\mathbf {B} \\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\mathbf {e} _{i}\times \partial _{i}\mathbf {B} )\\[2pt]&=(\mathbf {e} _{i}\partial _{i}\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\mathbf {e} _{i}\partial _{i}\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}}
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term (i.e., the operators must be nested). The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule.
For example, from the identity A ⋅(B ×C ) = (A ×B )⋅C
we may derive A ⋅(∇×C ) = (A ×∇)⋅C but not ∇⋅(B ×C ) = (∇×B )⋅C ,
nor from A ⋅(B ×A ) = 0 may we derive A ⋅(∇×A ) = 0.
On the other hand, a subscripted del operates on all occurrences of the subscript in the term, so that A ⋅(∇A A ) = ∇A A ×A ) = ∇⋅(A ×A ) = 0.
Also, from A ×(A ×C ) = A (A ⋅C ) − (A ⋅A )C we may derive ∇×(∇×C ) = ∇(∇⋅C ) − ∇2 C ,
but from (A ψ )⋅(A φ ) = (A ⋅A )(ψφ ) we may not derive (∇ψ )⋅(∇φ ) = ∇2 (ψφ ).
For the remainder of this article, Feynman subscript notation will be used where appropriate.
For scalar fields
ψ
{\displaystyle \psi }
ϕ
{\displaystyle \phi }
A
{\displaystyle \mathbf {A} }
B
{\displaystyle \mathbf {B} }
Distributive properties
∇
(
ψ
+
ϕ
)
=
∇
ψ
+
∇
ϕ
∇
(
A
+
B
)
=
∇
A
+
∇
B
∇
⋅
(
A
+
B
)
=
∇
⋅
A
+
∇
⋅
B
∇
×
(
A
+
B
)
=
∇
×
A
+
∇
×
B
{\displaystyle {\begin{aligned}\nabla (\psi +\phi )&=\nabla \psi +\nabla \phi \\\nabla (\mathbf {A} +\mathbf {B} )&=\nabla \mathbf {A} +\nabla \mathbf {B} \\\nabla \cdot (\mathbf {A} +\mathbf {B} )&=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} \\\nabla \times (\mathbf {A} +\mathbf {B} )&=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} \end{aligned}}}
First derivative associative properties
(
A
⋅
∇
)
ψ
=
A
⋅
(
∇
ψ
)
(
A
⋅
∇
)
B
=
A
⋅
(
∇
B
)
(
A
×
∇
)
ψ
=
A
×
(
∇
ψ
)
(
A
×
∇
)
B
=
A
×
(
∇
B
)
{\displaystyle {\begin{aligned}(\mathbf {A} \cdot \nabla )\psi &=\mathbf {A} \cdot (\nabla \psi )\\(\mathbf {A} \cdot \nabla )\mathbf {B} &=\mathbf {A} \cdot (\nabla \mathbf {B} )\\(\mathbf {A} \times \nabla )\psi &=\mathbf {A} \times (\nabla \psi )\\(\mathbf {A} \times \nabla )\mathbf {B} &=\mathbf {A} \times (\nabla \mathbf {B} )\end{aligned}}}
Product rule for multiplication by a scalar
We have the following generalizations of the product rule in single-variable calculus .
∇
(
ψ
ϕ
)
=
ϕ
∇
ψ
+
ψ
∇
ϕ
∇
(
ψ
A
)
=
(
∇
ψ
)
A
T
+
ψ
∇
A
=
∇
ψ
⊗
A
+
ψ
∇
A
∇
⋅
(
ψ
A
)
=
ψ
∇
⋅
A
+
(
∇
ψ
)
⋅
A
∇
×
(
ψ
A
)
=
ψ
∇
×
A
+
(
∇
ψ
)
×
A
∇
2
(
ψ
ϕ
)
=
ψ
∇
2
ϕ
+
2
∇
ψ
⋅
∇
ϕ
+
ϕ
∇
2
ψ
{\displaystyle {\begin{aligned}\nabla (\psi \phi )&=\phi \,\nabla \psi +\psi \,\nabla \phi \\\nabla (\psi \mathbf {A} )&=(\nabla \psi )\mathbf {A} ^{\textsf {T}}+\psi \nabla \mathbf {A} \ =\ \nabla \psi \otimes \mathbf {A} +\psi \,\nabla \mathbf {A} \\\nabla \cdot (\psi \mathbf {A} )&=\psi \,\nabla {\cdot }\mathbf {A} +(\nabla \psi )\,{\cdot }\mathbf {A} \\\nabla {\times }(\psi \mathbf {A} )&=\psi \,\nabla {\times }\mathbf {A} +(\nabla \psi ){\times }\mathbf {A} \\\nabla ^{2}(\psi \phi )&=\psi \,\nabla ^{2\!}\phi +2\,\nabla \!\psi \cdot \!\nabla \phi +\phi \,\nabla ^{2\!}\psi \end{aligned}}}
Quotient rule for division by a scalar
∇
(
ψ
ϕ
)
=
ϕ
∇
ψ
−
ψ
∇
ϕ
ϕ
2
∇
(
A
ϕ
)
=
ϕ
∇
A
−
∇
ϕ
⊗
A
ϕ
2
∇
⋅
(
A
ϕ
)
=
ϕ
∇
⋅
A
−
∇
ϕ
⋅
A
ϕ
2
∇
×
(
A
ϕ
)
=
ϕ
∇
×
A
−
∇
ϕ
×
A
ϕ
2
∇
2
(
ψ
ϕ
)
=
ϕ
∇
2
ψ
−
2
ϕ
∇
(
ψ
ϕ
)
⋅
∇
ϕ
−
ψ
∇
2
ϕ
ϕ
2
{\displaystyle {\begin{aligned}\nabla \left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla \psi -\psi \,\nabla \phi }{\phi ^{2}}}\\[1em]\nabla \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla \mathbf {A} -\nabla \phi \otimes \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \cdot \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\cdot }\mathbf {A} -\nabla \!\phi \cdot \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \times \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\times }\mathbf {A} -\nabla \!\phi \,{\times }\,\mathbf {A} }{\phi ^{2}}}\\[1em]\nabla ^{2}\left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla ^{2\!}\psi -2\,\phi \,\nabla \!\left({\frac {\psi }{\phi }}\right)\cdot \!\nabla \phi -\psi \,\nabla ^{2\!}\phi }{\phi ^{2}}}\end{aligned}}}
Dot product rule
∇
(
A
⋅
B
)
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
=
A
⋅
J
B
+
B
⋅
J
A
=
(
∇
B
)
⋅
A
+
(
∇
A
)
⋅
B
{\displaystyle {\begin{aligned}\nabla (\mathbf {A} \cdot \mathbf {B} )&\ =\ (\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} )\\&\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {B} }+\mathbf {B} \cdot \mathbf {J} _{\mathbf {A} }\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,+\,(\nabla \mathbf {A} )\cdot \mathbf {B} \end{aligned}}}
where
J
A
=
(
∇
A
)
T
=
(
∂
A
i
/
∂
x
j
)
i
j
{\displaystyle \mathbf {J} _{\mathbf {A} }=(\nabla \!\mathbf {A} )^{\textsf {T}}=(\partial A_{i}/\partial x_{j})_{ij}}
Jacobian matrix of the vector field
A
=
(
A
1
,
…
,
A
n
)
{\displaystyle \mathbf {A} =(A_{1},\ldots ,A_{n})}
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} )\ .}
See these notes.[4]
As a special case, when A = B
1
2
∇
(
A
⋅
A
)
=
A
⋅
J
A
=
(
∇
A
)
⋅
A
=
(
A
⋅
∇
)
A
+
A
×
(
∇
×
A
)
=
A
∇
A
.
{\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {A} }\ =\ (\nabla \mathbf {A} )\cdot \mathbf {A} \ =\ (\mathbf {A} {\cdot }\nabla )\mathbf {A} \,+\,\mathbf {A} {\times }(\nabla {\times }\mathbf {A} )\ =\ A\nabla A.}
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection , which differentiates a vector field to give a vector-valued 1-form .
Cross product rule
∇
(
A
×
B
)
=
(
∇
A
)
×
B
−
(
∇
B
)
×
A
∇
⋅
(
A
×
B
)
=
(
∇
×
A
)
⋅
B
−
A
⋅
(
∇
×
B
)
∇
×
(
A
×
B
)
=
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
=
A
(
∇
⋅
B
)
+
(
B
⋅
∇
)
A
−
(
B
(
∇
⋅
A
)
+
(
A
⋅
∇
)
B
)
=
∇
⋅
(
B
A
T
)
−
∇
⋅
(
A
B
T
)
=
∇
⋅
(
B
A
T
−
A
B
T
)
A
×
(
∇
×
B
)
=
∇
B
(
A
⋅
B
)
−
(
A
⋅
∇
)
B
=
A
⋅
J
B
−
(
A
⋅
∇
)
B
=
(
∇
B
)
⋅
A
−
A
⋅
(
∇
B
)
=
A
⋅
(
J
B
−
J
B
T
)
(
A
×
∇
)
×
B
=
(
∇
B
)
⋅
A
−
A
(
∇
⋅
B
)
=
A
×
(
∇
×
B
)
+
(
A
⋅
∇
)
B
−
A
(
∇
⋅
B
)
(
A
×
∇
)
⋅
B
=
A
⋅
(
∇
×
B
)
{\displaystyle {\begin{aligned}\nabla (\mathbf {A} \times \mathbf {B} )&\ =\ (\nabla \mathbf {A} )\times \mathbf {B} \,-\,(\nabla \mathbf {B} )\times \mathbf {A} \\[5pt]\nabla \cdot (\mathbf {A} \times \mathbf {B} )&\ =\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\\[5pt]\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} \\[2pt]&\ =\ \mathbf {A} (\nabla {\cdot }\mathbf {B} )\,+\,(\mathbf {B} {\cdot }\nabla )\mathbf {A} \,-\,(\mathbf {B} (\nabla {\cdot }\mathbf {A} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} )\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\right)\,-\,\nabla {\cdot }\left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\,-\,\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[5pt]\mathbf {A} \times (\nabla \times \mathbf {B} )&\ =\ \nabla _{\mathbf {B} }(\mathbf {A} {\cdot }\mathbf {B} )\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {B} }\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} \cdot (\nabla \mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \cdot (\mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}})\\[5pt](\mathbf {A} \times \nabla )\times \mathbf {B} &\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \times (\nabla \times \mathbf {B} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[5pt](\mathbf {A} \times \nabla )\cdot \mathbf {B} &\ =\ \mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\end{aligned}}}
J
B
−
J
B
T
{\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}}}
Divergence of curl is zero
The divergence of the curl of any continuously twice-differentiable vector field A is always zero:
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex .
Divergence of gradient is Laplacian
The Laplacian of a scalar field is the divergence of its gradient:
Δ
ψ
=
∇
2
ψ
=
∇
⋅
(
∇
ψ
)
{\displaystyle \Delta \psi =\nabla ^{2}\psi =\nabla \cdot (\nabla \psi )}
Divergence of divergence is not defined
The divergence of a vector field A is a scalar, and the divergence of a scalar quantity is undefined. Therefore,
∇
⋅
(
∇
⋅
A
)
is undefined.
{\displaystyle \nabla \cdot (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}}
Curl of 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 .
Curl of divergence is not defined
The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore,
∇
×
(
∇
⋅
A
)
is undefined.
{\displaystyle \nabla \times (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}}
Second derivative associative properties
(
∇
⋅
∇
)
ψ
=
∇
⋅
(
∇
ψ
)
=
∇
2
ψ
(
∇
⋅
∇
)
A
=
∇
⋅
(
∇
A
)
=
∇
2
A
(
∇
×
∇
)
ψ
=
∇
×
(
∇
ψ
)
=
0
(
∇
×
∇
)
A
=
∇
×
(
∇
A
)
=
0
{\displaystyle {\begin{aligned}(\nabla \cdot \nabla )\psi &=\nabla \cdot (\nabla \psi )=\nabla ^{2}\psi \\(\nabla \cdot \nabla )\mathbf {A} &=\nabla \cdot (\nabla \mathbf {A} )=\nabla ^{2}\mathbf {A} \\(\nabla \times \nabla )\psi &=\nabla \times (\nabla \psi )=\mathbf {0} \\(\nabla \times \nabla )\mathbf {A} &=\nabla \times (\nabla \mathbf {A} )=\mathbf {0} \end{aligned}}}
DCG chart:
Some rules for second derivatives.
A mnemonic
The figure to the right is a mnemonic for some of these identities. The abbreviations used are:
D: divergence,
C: curl,
G: gradient,
L: Laplacian,
CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.
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 }
∇
(
ψ
A
)
=
∇
ψ
⊗
A
+
ψ
∇
A
{\displaystyle \nabla (\psi \mathbf {A} )=\nabla \psi \otimes \mathbf {A} +\psi \nabla \mathbf {A} }
∇
(
A
⋅
B
)
=
(
A
⋅
∇
)
B
+
(
B
⋅
∇
)
A
+
A
×
(
∇
×
B
)
+
B
×
(
∇
×
A
)
{\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=(\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} )}
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
)
=
(
∇
×
A
)
⋅
B
−
(
∇
×
B
)
⋅
A
{\displaystyle \nabla \cdot \left(\mathbf {A} \times \mathbf {B} \right)=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -(\nabla \times \mathbf {B} )\cdot \mathbf {A} }
Curl
∇
×
(
A
+
B
)
=
∇
×
A
+
∇
×
B
{\displaystyle \nabla \times (\mathbf {A} +\mathbf {B} )=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} }
∇
×
(
ψ
A
)
=
ψ
(
∇
×
A
)
−
(
A
×
∇
)
ψ
=
ψ
(
∇
×
A
)
+
(
∇
ψ
)
×
A
{\displaystyle \nabla \times \left(\psi \mathbf {A} \right)=\psi \,(\nabla \times \mathbf {A} )-(\mathbf {A} \times \nabla )\psi =\psi \,(\nabla \times \mathbf {A} )+(\nabla \psi )\times \mathbf {A} }
∇
×
(
ψ
∇
ϕ
)
=
∇
ψ
×
∇
ϕ
{\displaystyle \nabla \times \left(\psi \nabla \phi \right)=\nabla \psi \times \nabla \phi }
∇
×
(
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} }
[5] Vector-dot-Del Operator
(
A
⋅
∇
)
B
=
1
2
[
∇
(
A
⋅
B
)
−
∇
×
(
A
×
B
)
−
B
×
(
∇
×
A
)
−
A
×
(
∇
×
B
)
−
B
(
∇
⋅
A
)
+
A
(
∇
⋅
B
)
]
{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} ={\frac {1}{2}}{\bigg [}\nabla (\mathbf {A} \cdot \mathbf {B} )-\nabla \times (\mathbf {A} \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+\mathbf {A} (\nabla \cdot \mathbf {B} ){\bigg ]}}
[6]
(
A
⋅
∇
)
A
=
1
2
∇
|
A
|
2
−
A
×
(
∇
×
A
)
=
1
2
∇
|
A
|
2
+
(
∇
×
A
)
×
A
{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} ={\frac {1}{2}}\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2}}\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }
A
⋅
∇
(
B
⋅
B
)
=
2
B
⋅
(
A
⋅
∇
)
B
{\displaystyle \mathbf {A} \cdot \nabla (\mathbf {B} \cdot \mathbf {B} )=2\mathbf {B} \cdot (\mathbf {A} \cdot \nabla )\mathbf {B} }
Second derivatives
∇
⋅
(
∇
×
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
)
+
(
∇
2
ϕ
)
ψ
{\displaystyle \nabla ^{2}(\phi \psi )=\phi \nabla ^{2}\psi +2(\nabla \phi )\cdot (\nabla \psi )+\left(\nabla ^{2}\phi \right)\psi }
∇
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 \left(\nabla ^{2}\psi \right)}
∇
2
(
∇
⋅
A
)
=
∇
⋅
(
∇
(
∇
⋅
A
)
)
=
∇
⋅
(
∇
2
A
)
{\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla (\nabla \cdot \mathbf {A} ))=\nabla \cdot \left(\nabla ^{2}\mathbf {A} \right)}
∇
2
(
∇
×
A
)
=
−
∇
×
(
∇
×
(
∇
×
A
)
)
=
∇
×
(
∇
2
A
)
{\displaystyle \nabla ^{2}(\nabla \times \mathbf {A} )=-\nabla \times (\nabla \times (\nabla \times \mathbf {A} ))=\nabla \times \left(\nabla ^{2}\mathbf {A} \right)}
Integration
Below, the curly symbol ∂ means "boundary of " a surface or solid.
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface ):
∂
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}
A
⋅
d
S
=
∭
V
∇
⋅
A
d
V
{\displaystyle \mathbf {A} \cdot d\mathbf {S} \ =\ \iiint _{V}\nabla \cdot \mathbf {A} \,dV}
(divergence theorem )
∂
V
{\displaystyle \scriptstyle \partial V}
A
×
d
S
=
−
∭
V
∇
×
A
d
V
{\displaystyle \mathbf {A} \times d\mathbf {S} \ =\ -\iiint _{V}\nabla \times \mathbf {A} \,dV}
∂
V
{\displaystyle \scriptstyle \partial V}
ψ
∇
φ
⋅
d
S
=
∭
V
(
ψ
∇
2
φ
+
∇
φ
⋅
∇
ψ
)
d
V
{\displaystyle \psi \nabla \!\varphi \cdot d\mathbf {S} \ =\ \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
)
⋅
d
S
=
{\displaystyle \left(\psi \nabla \!\varphi -\varphi \nabla \!\psi \right)\cdot d\mathbf {S} \ =\ }
∂
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 )
∭
V
A
⋅
∇
ψ
d
V
=
{\displaystyle \iiint _{V}\mathbf {A} \cdot \nabla \psi \,dV\ =\ }
∂
V
{\displaystyle \scriptstyle \partial V}
ψ
A
⋅
d
S
−
∭
V
ψ
∇
⋅
A
d
V
{\displaystyle \psi \mathbf {A} \cdot d\mathbf {S} -\iiint _{V}\psi \nabla \cdot \mathbf {A} \,dV}
integration by parts )
∭
V
ψ
∇
⋅
A
d
V
=
{\displaystyle \iiint _{V}\psi \nabla \cdot \mathbf {A} \,dV\ =\ }
∂
V
{\displaystyle \scriptstyle \partial V}
ψ
A
⋅
d
S
−
∭
V
A
⋅
∇
ψ
d
V
{\displaystyle \psi \mathbf {A} \cdot d\mathbf {S} -\iiint _{V}\mathbf {A} \cdot \nabla \psi \,dV}
integration by parts )
∭
V
A
⋅
(
∇
×
B
)
d
V
=
−
{\displaystyle \iiint _{V}\mathbf {A} \cdot \left(\nabla \times \mathbf {B} \right)\,dV\ =\ -}
∂
V
{\displaystyle \scriptstyle \partial V}
(
A
×
B
)
⋅
d
S
+
∭
V
(
∇
×
A
)
⋅
B
d
V
{\displaystyle \left(\mathbf {A} \times \mathbf {B} \right)\cdot d\mathbf {S} +\iiint _{V}\left(\nabla \times \mathbf {A} \right)\cdot \mathbf {B} \,dV}
integration by parts )
∂
V
{\displaystyle \scriptstyle \partial V}
A
×
(
d
S
⋅
(
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{\displaystyle \mathbf {A} \times \left(d\mathbf {S} \cdot \left(\mathbf {B} \mathbf {C} ^{\textsf {T}}\right)\right)\ =\ \iiint _{V}\mathbf {A} \times \left(\nabla \cdot \left(\mathbf {B} \mathbf {C} ^{\textsf {T}}\right)\right)\,dV+\iiint _{V}\mathbf {B} \cdot (\nabla \mathbf {A} )\times \mathbf {C} \,dV}
[7]
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{\displaystyle \iiint _{V}\left(\nabla \cdot \mathbf {B} +\mathbf {B} \cdot \nabla \right)\mathbf {A} \,dV\ =\ }
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{\displaystyle \left(\mathbf {B} \cdot d\mathbf {S} \right)\mathbf {A} }
[8] Tensor integrals
A tensor form of a vector integral theorem may be obtained by replacing the vector (or one of them) by a tensor, provided that the vector is first made to appear only as the right-most vector of each integrand. For example, Stokes' theorem becomes
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{\displaystyle \oint _{\partial S}d{\boldsymbol {\ell }}\cdot \mathbf {T} \ =\ \iint _{S}d\mathbf {S} \cdot \left(\nabla \times \mathbf {T} \right)}
A scalar field may also be treated as a vector and replaced by a vector or tensor. For example, Green's first identity becomes
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{\displaystyle \psi \,d\mathbf {S} \cdot \nabla \!\mathbf {A} \ =\ \iiint _{V}\left(\psi \nabla ^{2}\!\mathbf {A} +\nabla \!\psi \cdot \nabla \!\mathbf {A} \right)\,dV}
.Similar rules apply to algebraic and differentiation formulas. For algebraic formulas one may alternatively use the left-most vector position.
![{\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\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}}\\[1em]&=\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} \end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/8579a43f51c457ddcad8e559206bf6e1696b1419)
![{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\nabla _{\mathbf {A} }\cdot (\mathbf {A} \times \mathbf {B} )+\nabla _{\mathbf {B} }\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} +(\nabla _{\mathbf {B} }\times \mathbf {A} )\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \nabla _{\mathbf {B} })\cdot \mathbf {B} \\[2pt]&=(\nabla _{\mathbf {A} }\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla _{\mathbf {B} }\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/6b2be73fb147d2d991b691c0322ec8602518462f)
![{\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&=\mathbf {e} _{i}\partial _{i}\cdot (\mathbf {A} \times \mathbf {B} )\\[2pt]&=\mathbf {e} _{i}\cdot (\partial _{i}\mathbf {A} \times \mathbf {B} +\mathbf {A} \times \partial _{i}\mathbf {B} )\\[2pt]&=\mathbf {e} _{i}\cdot (\partial _{i}\mathbf {A} \times \mathbf {B} )+\mathbf {e} _{i}\cdot (\mathbf {A} \times \partial _{i}\mathbf {B} )\\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} +(\mathbf {e} _{i}\times \mathbf {A} )\cdot \partial _{i}\mathbf {B} \\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} -(\mathbf {A} \times \mathbf {e} _{i})\cdot \partial _{i}\mathbf {B} \\[2pt]&=(\mathbf {e} _{i}\times \partial _{i}\mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\mathbf {e} _{i}\times \partial _{i}\mathbf {B} )\\[2pt]&=(\mathbf {e} _{i}\partial _{i}\times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\mathbf {e} _{i}\partial _{i}\times \mathbf {B} )\\[2pt]&=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -\mathbf {A} \cdot (\nabla \times \mathbf {B} )\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f74c1ad36123a7e8211b72753ee1143e28264e49)
![{\displaystyle {\begin{aligned}\nabla \left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla \psi -\psi \,\nabla \phi }{\phi ^{2}}}\\[1em]\nabla \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla \mathbf {A} -\nabla \phi \otimes \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \cdot \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\cdot }\mathbf {A} -\nabla \!\phi \cdot \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \times \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\times }\mathbf {A} -\nabla \!\phi \,{\times }\,\mathbf {A} }{\phi ^{2}}}\\[1em]\nabla ^{2}\left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla ^{2\!}\psi -2\,\phi \,\nabla \!\left({\frac {\psi }{\phi }}\right)\cdot \!\nabla \phi -\psi \,\nabla ^{2\!}\phi }{\phi ^{2}}}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/93a497c998c46163f4eb8615f7cf7c6793c2f475)
![{\displaystyle {\begin{aligned}\nabla (\mathbf {A} \times \mathbf {B} )&\ =\ (\nabla \mathbf {A} )\times \mathbf {B} \,-\,(\nabla \mathbf {B} )\times \mathbf {A} \\[5pt]\nabla \cdot (\mathbf {A} \times \mathbf {B} )&\ =\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\\[5pt]\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} \\[2pt]&\ =\ \mathbf {A} (\nabla {\cdot }\mathbf {B} )\,+\,(\mathbf {B} {\cdot }\nabla )\mathbf {A} \,-\,(\mathbf {B} (\nabla {\cdot }\mathbf {A} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} )\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\right)\,-\,\nabla {\cdot }\left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\,-\,\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[5pt]\mathbf {A} \times (\nabla \times \mathbf {B} )&\ =\ \nabla _{\mathbf {B} }(\mathbf {A} {\cdot }\mathbf {B} )\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {B} }\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} \cdot (\nabla \mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \cdot (\mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}})\\[5pt](\mathbf {A} \times \nabla )\times \mathbf {B} &\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \times (\nabla \times \mathbf {B} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[5pt](\mathbf {A} \times \nabla )\cdot \mathbf {B} &\ =\ \mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/48f38185cdae9d925d8c98642858cba0a27a1f2f)
![{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} ={\frac {1}{2}}{\bigg [}\nabla (\mathbf {A} \cdot \mathbf {B} )-\nabla \times (\mathbf {A} \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+\mathbf {A} (\nabla \cdot \mathbf {B} ){\bigg ]}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/8cd10d81d948e6fa928365812a1c43d2cb14330a)
![{\displaystyle \oint _{\partial S}(\mathbf {B} \cdot d{\boldsymbol {\ell }})\mathbf {A} =\iint _{S}(d\mathbf {S} \cdot \left[\nabla \times \mathbf {B} -\mathbf {B} \times \nabla \right])\mathbf {A} }](http://wikimedia.org/api/rest_v1/media/math/render/svg/20c48d9c590a9b81aca2c1184cea2b05aad96283)
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