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:
- <math> \nabla \times \mathbf{v} = 0, <math>
- <math> \nabla \cdot \mathbf{v} = 0. <math>
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 :
- <math> \mathbf{v} = \nabla \phi \qquad \qquad (1) <math>.
Then, since the divergence of v is also zero, it follows from equation (1) that
- <math> \nabla \cdot \nabla \mathbf{v} = 0 <math>
which is equivalent to
- <math> \nabla^2 \phi = 0 <math>.
Therefore, the potential of a Laplacian field satisfies Laplace's equation.
See also: potential flow, harmonic function