• Source: Laplacian vector field

In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:








∇
×

v




=

0

,




∇
⋅

v




=
0.






{\displaystyle {\begin{aligned}\nabla \times \mathbf {v} &=\mathbf {0} ,\\\nabla \cdot \mathbf {v} &=0.\end{aligned}}}


From the vector calculus identity




∇

2



v

≡
∇
(
∇
⋅

v

)
−
∇
×
(
∇
×

v

)


{\displaystyle \nabla ^{2}\mathbf {v} \equiv \nabla (\nabla \cdot \mathbf {v} )-\nabla \times (\nabla \times \mathbf {v} )}

it follows that





∇

2



v

=

0



{\displaystyle \nabla ^{2}\mathbf {v} =\mathbf {0} }


that is, that the field v satisfies Laplace's equation.
However, the converse is not true; not every vector field that satisfies Laplace's equation is a Laplacian vector field, which can be a point of confusion. For example, the vector field





v


=
(
x
y
,
y
z
,
z
x
)


{\displaystyle {\bf {v}}=(xy,yz,zx)}

satisfies Laplace's equation, but it has both nonzero divergence and nonzero curl and is not a Laplacian vector field.
A Laplacian vector field in the plane satisfies the Cauchy–Riemann equations: it is holomorphic.
Since the curl of v is zero, it follows that (when the domain of definition is simply connected) v can be expressed as the gradient of a scalar potential (see irrotational field) φ :





v

=
∇
ϕ
.


(
1
)


{\displaystyle \mathbf {v} =\nabla \phi .\qquad \qquad (1)}


Then, since the divergence of v is also zero, it follows from equation (1) that




∇
⋅
∇
ϕ
=
0


{\displaystyle \nabla \cdot \nabla \phi =0}


which is equivalent to





∇

2


ϕ
=
0.


{\displaystyle \nabla ^{2}\phi =0.}


Therefore, the potential of a Laplacian field satisfies Laplace's equation.




See also


Potential flow
Harmonic function




References

Kata Kunci Pencarian:

• Kalkulus vektor
• Laplacian vector field
• Laplace operator
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• Vector operator
• List of multivariable calculus topics
• Laplacian matrix
• Vector calculus
• Vector calculus identities
• Laplace–Beltrami operator
• Vector (mathematics and physics)