Editing Laplace operator (section)

==Vector Laplacian==
The '''vector Laplace operator''', also denoted by <math>\nabla^2</math>, is a [[differential operator]] defined over a [[vector field]].<ref>{{cite web | url = http://mathworld.wolfram.com/VectorLaplacian.html | title = Vector Laplacian 
| author = MathWorld}}</ref> The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a [[scalar field]] and returns a scalar quantity, the vector Laplacian applies to a [[vector field]], returning a vector quantity. When computed in [[orthonormal]] [[Cartesian coordinates]], the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

The '''vector Laplacian''' of a [[vector field]] <math> \mathbf{A} </math> is defined as
<math display="block"> \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}). </math>
This definition can be seen as the [[Helmholtz decomposition]] of the vector Laplacian.

In [[Cartesian coordinate]]s, this reduces to the much simpler expression
<math display="block"> \nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z), </math>
where <math>A_x</math>, <math>A_y</math>, and <math>A_z</math> are the components of the vector field <math>\mathbf{A}</math>, and <math> \nabla^2 </math> just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see [[Vector triple product]].

For expressions of the vector Laplacian in other coordinate systems see [[Del in cylindrical and spherical coordinates]].

===Generalization===
The Laplacian of any [[tensor field]] <math>\mathbf{T}</math> ("tensor" includes scalar and vector) is defined as the [[divergence]] of the [[gradient]] of the tensor:
<math display="block">\nabla ^2\mathbf{T} = (\nabla \cdot \nabla) \mathbf{T}.</math>

For the special case where <math>\mathbf{T}</math> is a [[scalar (mathematics)|scalar]] (a tensor of degree zero), the [[Laplacian]] takes on the familiar form.

If <math>\mathbf{T}</math> is a vector (a tensor of first degree), the gradient is a [[covariant derivative]] which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the [[Jacobian matrix]] shown below for the gradient of a vector:
<math display="block">\nabla \mathbf{T}= (\nabla T_x, \nabla T_y, \nabla T_z) = \begin{bmatrix}
T_{xx} & T_{xy} & T_{xz} \\
T_{yx} & T_{yy} & T_{yz} \\
T_{zx} & T_{zy} & T_{zz}
\end{bmatrix} ,
\text{ where } T_{uv} \equiv \frac{\partial T_u}{\partial v}.</math>

And, in the same manner, a [[dot product]], which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices:
<math display="block"> \mathbf{A} \cdot \nabla \mathbf{B}
= \begin{bmatrix} A_x & A_y & A_z \end{bmatrix} \nabla \mathbf{B}
= \begin{bmatrix} \mathbf{A} \cdot \nabla B_x & \mathbf{A} \cdot \nabla B_y & \mathbf{A} \cdot \nabla B_z \end{bmatrix}.</math>
This identity is a coordinate dependent result, and is not general.

===Use in physics===
An example of the usage of the vector Laplacian is the [[Navier-Stokes equations]] for a [[Newtonian fluid|Newtonian]] [[incompressible flow]]:
<math display="block">\rho \left(\frac{\partial \mathbf{v}}{\partial t}+ ( \mathbf{v} \cdot \nabla ) \mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\nabla ^2 \mathbf{v}\right),</math>
where the term with the vector Laplacian of the [[velocity]] field <math>\mu\left(\nabla ^2 \mathbf{v}\right)</math> represents the [[viscosity|viscous]] [[Stress (physics)|stress]]es in the fluid.

Another example is the wave equation for the electric field that can be derived from [[Maxwell's equations]] in the absence of charges and currents:
<math display="block">\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0.</math>

This equation can also be written as:
<math display="block">\Box\, \mathbf{E} = 0,</math>
where <math display="block">\Box\equiv\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2,</math> is the [[D'Alembertian]], used in the [[Klein–Gordon equation]].