The following identities are important in vector calculus :
Operator notations
Gradient
Gradient of a tensor ,
T
{\displaystyle \mathbf {\mathfrak {T}} }
, of order n , is generally written as
grad
(
T
)
=
∇
T
{\displaystyle \operatorname {grad} (\mathbf {\mathfrak {T}} )=\nabla \mathbf {\mathfrak {T}} }
and is a tensor of order n+1 . In particular, if the tensor is order 0 (i.e. a scalar),
ψ
{\displaystyle \psi }
, the resulting gradient,
grad
(
ψ
)
=
∇
ψ
{\displaystyle \operatorname {grad} (\psi )=\nabla \psi }
is a vector field.
Divergence
Divergence of a tensor ,
T
{\displaystyle {\stackrel {\mathbf {\mathfrak {T}} }{}}}
, of non-zero order n , is generally written as
div
(
T
)
=
∇
⋅
T
{\displaystyle \operatorname {div} (\mathbf {\mathfrak {T}} )=\nabla \cdot \mathbf {\mathfrak {T}} }
and is a contraction to a tensor of order n-1 . Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor may be found by decomposing the tensor into a sum of outer products, thereby allowing the use of the identity,
∇
⋅
(
a
⊗
T
^
)
=
T
^
(
∇
⋅
a
)
+
(
a
⋅
∇
)
T
^
{\displaystyle \nabla \cdot (\mathbf {a} \otimes {\hat {\mathbf {\mathfrak {T}} }})={\hat {\mathbf {\mathfrak {T}} }}(\nabla \cdot \mathbf {a} )+(\mathbf {a} \cdot \nabla ){\hat {\mathbf {\mathfrak {T}} }}}
where
a
⋅
∇
{\displaystyle \mathbf {a} \cdot \nabla }
is the directional derivative in the direction of
a
{\displaystyle \mathbf {a} }
multiplied by its magnitude. Specifically, for the outer product of two vectors,
∇
⋅
(
a
b
T
)
=
b
(
∇
⋅
a
)
+
(
a
⋅
∇
)
b
{\displaystyle \nabla \cdot (\mathbf {a} \mathbf {b^{T}} )=\mathbf {b} (\nabla \cdot \mathbf {a} )+(\mathbf {a} \cdot \nabla )\mathbf {b} }
Curl
For a 3-dimensional vector field
v
{\displaystyle \mathbf {v} }
, curl is generally written as:
curl
(
v
)
=
∇
×
v
{\displaystyle \operatorname {curl} (\mathbf {v} )=\nabla \times \mathbf {v} }
and is also a 3-dimensional vector field.
Laplacian
For a tensor ,
T
{\displaystyle \mathbf {\mathfrak {T}} }
, the laplacian is generally written as:
Δ
T
=
∇
2
T
=
(
∇
⋅
∇
)
T
{\displaystyle \Delta \mathbf {\mathfrak {T}} =\nabla ^{2}\mathbf {\mathfrak {T}} =(\nabla \cdot \nabla )\mathbf {\mathfrak {T}} }
and is a tensor of the same order.
Integrals
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Special notations
In Feynman subscript notation ,
∇
B
(
A
⋅
B
)
=
A
×
(
∇
×
B
)
+
(
A
⋅
∇
)
B
{\displaystyle \nabla _{B}\left(\mathbf {A\cdot B} \right)=\mathbf {A\ \times } \left(\mathbf {\nabla \times B} \right)+\left(\mathbf {A\cdot \nabla } \right)\mathbf {B} }
where the notation ∇ B means the subscripted gradient operates on only the factor B .[ 1] [ 2]
A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed.[ 3] The above identity is then expressed as:
∇
˙
(
A
⋅
B
˙
)
=
A
×
(
∇
×
B
)
+
(
A
⋅
∇
)
B
{\displaystyle {\dot {\nabla }}\left(\mathbf {A} \cdot {\dot {\mathbf {B} }}\right)=\mathbf {A\ \times } \left(\mathbf {\nabla \times 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 }
and
ϕ
{\displaystyle \phi }
follows the same form as the 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
)
=
A
⋅
∇
ψ
+
ψ
∇
⋅
A
{\displaystyle \nabla \cdot (\psi \mathbf {A} )=\mathbf {A} \cdot \nabla \psi +\psi \nabla \cdot \mathbf {A} }
∇
×
(
ψ
A
)
=
ψ
∇
×
A
+
∇
ψ
×
A
{\displaystyle \nabla \times (\psi \mathbf {A} )=\psi \nabla \times \mathbf {A} +\nabla \psi \times \mathbf {A} }
Vector dot product
∇
(
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} )\ .}
Alternatively, using Feynman subscript notation,
∇
(
A
⋅
B
)
=
∇
A
(
A
⋅
B
)
+
∇
B
(
A
⋅
B
)
.
{\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=\nabla _{A}(\mathbf {A} \cdot \mathbf {B} )+\nabla _{B}(\mathbf {A} \cdot \mathbf {B} )\ .}
As a special case, when A = B ,
1
2
∇
(
A
⋅
A
)
=
A
×
(
∇
×
A
)
+
(
A
⋅
∇
)
A
.
{\displaystyle {\frac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \times (\nabla \times \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {A} \ .}
Vector cross product
∇
⋅
(
A
×
B
)
=
B
⋅
(
∇
×
A
)
−
A
⋅
(
∇
×
B
)
.
{\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\mathbf {B} \cdot (\nabla \times \mathbf {A} )-\mathbf {A} \cdot (\nabla \times \mathbf {B} )\ .}
∇
×
(
A
×
B
)
=
A
(
∇
⋅
B
)
−
B
(
∇
⋅
A
)
+
(
B
⋅
∇
)
A
−
(
A
⋅
∇
)
B
.
{\displaystyle \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} \ .}
Second derivatives
Curl of the gradient
The curl of the gradient of any scalar field
ϕ
{\displaystyle \ \phi }
is always the zero vector :
∇
×
(
∇
ϕ
)
=
0
→
{\displaystyle \nabla \times (\nabla \phi )={\vec {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 )
Differentiation
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.
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 \mathbf {\nabla } \right)\mathbf {B} +\left(\mathbf {B} \cdot \mathbf {\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
−
A
×
∇
ψ
{\displaystyle \nabla \times \left(\psi \mathbf {A} \right)=\psi \nabla \times \mathbf {A} -\mathbf {A} \times \nabla \psi }
∇
×
(
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
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
∇
×
(
∇
ψ
)
=
0
{\displaystyle \nabla \times (\nabla \psi )=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 \nabla \times \mathbf {A} =\nabla ^{2}\mathbf {A} }
(vector Laplacian )
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)}
Integration
∬
S
⊂
⊃
A
⋅
d
s
=
∭
V
(
∇
⋅
A
)
d
v
{\displaystyle \iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \mathbf {A} \cdot d\mathbf {s} =\iiint \limits _{V}\left(\nabla \cdot \mathbf {A} \right)dv}
(Divergence theorem )
∬
S
⊂
⊃
ψ
d
s
=
∭
V
∇
ψ
d
v
{\displaystyle \iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset {\psi }d\mathbf {s} =\iiint \limits _{V}\nabla \psi \,dv}
∬
S
⊂
⊃
(
n
^
×
A
)
d
s
=
∭
V
(
∇
×
A
)
d
v
{\displaystyle \iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left({\hat {\mathbf {n} }}\times \mathbf {A} \right)ds=\iiint \limits _{V}\left(\nabla \times \mathbf {A} \right)dv}
∭
V
(
ψ
∇
2
φ
+
∇
φ
⋅
∇
ψ
)
d
v
=
∬
S
⊂
⊃
ψ
(
∇
φ
⋅
n
^
)
d
s
{\displaystyle \iiint \limits _{V}\left(\psi \nabla ^{2}\varphi +\nabla \varphi \cdot \nabla \psi \right)dv=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \psi \left(\nabla \varphi \cdot {\hat {\mathbf {n} }}\right)ds}
(Green's first identity )
∭
V
(
ψ
∇
2
φ
−
φ
∇
2
ψ
)
d
v
=
∬
S
⊂
⊃
[
(
ψ
∇
φ
−
φ
∇
ψ
)
⋅
n
^
]
d
s
=
∬
S
⊂
⊃
[
ψ
∂
φ
∂
n
−
φ
∂
ψ
∂
n
]
d
s
{\displaystyle \iiint \limits _{V}\left(\psi \nabla ^{2}\varphi -\varphi \nabla ^{2}\psi \right)dv=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left[\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot {\hat {\mathbf {n} }}\right]ds=\iint \limits _{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \left[\psi {\frac {\partial \varphi }{\partial n}}-\varphi {\frac {\partial \psi }{\partial n}}\right]ds}
(Green's second identity )
∮
C
A
⋅
d
l
=
∬
S
(
∇
×
A
)
⋅
d
s
{\displaystyle \oint \limits _{C}\mathbf {A} \cdot d\mathbf {l} =\iint \limits _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {s} }
(Stokes' theorem )
∮
C
ψ
d
l
=
∬
S
(
n
^
×
∇
ψ
)
d
s
{\displaystyle \oint \limits _{C}\psi d\mathbf {l} =\iint \limits _{S}\left({\hat {\mathbf {n} }}\times \nabla \psi \right)ds}
See also
Notes and references
^ Feynman, R. P.; Leighton, R. B.; Sands, M. (1964). The Feynman Lecture on Physics . Addison-Wesley. Vol II, p. 27–4. ISBN 0805390499 .
^ Kholmetskii, A. L.; Missevitch, O. V. (2005). "The Faraday induction law in relativity theory". p. 4. 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 0471621943 .
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 .