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
:
\nabla \times \mathbf{v} = 0,
\nabla \cdot \mathbf{v} = 0.
Since the
curl
of
v
is zero, it follows that
v
can be expressed as the gradient of a
scalar potential
(see
irrotational field
)
φ
:
\mathbf{v} = \nabla \phi \qquad \qquad (1)
.
Then, since the
divergence
of
v
is also zero, it follows from equation (1) that
\nabla \cdot \nabla \mathbf{v} = 0
which is equivalent to
\nabla^2 \phi = 0
.
Therefore, the potential of a Laplacian field satisfies
Laplace's equation
.
See also:
potential flow
,
harmonic function
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