Laplacian vector field

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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 \phi = 0

which is equivalent to

Failed to parse (Cannot write to or create math output directory): \nabla^2 \phi = 0

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

[edit] See also

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