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| In [[mathematics]] and [[physics]], the '''vector Laplace operator''', denoted by <math>\scriptstyle \nabla^2</math>, named after [[Pierre-Simon Laplace]], is a differential operator defined over a [[vector field]]. The vector Laplacian is similar to the [[Laplacian|scalar Laplacian]]. Whereas the scalar Laplacian applies to [[scalar field]] and returns a scalar quantity, the vector Laplacian applies to the vector fields and returns a vector quantity. When computed in rectangular cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied on the individual elements.
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| ==Definition==
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| The '''vector Laplacian''' of a vector field <math> \mathbf{A} </math> is defined as
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| :<math> \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}) </math>
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| In [[Cartesian coordinate]]s, this reduces to the much simpler form (This can be seen as a special case of Lagrange's formula, see [[Vector triple product]].)
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| :<math> \nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z) </math> | |
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| where <math>A_x</math>, <math>A_y</math>, and <math>A_z</math> are the components of <math>\mathbf{A}</math>.
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| For expressions of the vector Laplacian in other coordinate systems see [[Nabla in cylindrical and spherical coordinates]].
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| ===Generalization===
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| 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:
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| :<math>\nabla^2 \mathbf{T} = \nabla \cdot (\nabla \mathbf{T})</math>
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| For the special case where <math>\mathbf{T}</math> is a [[scalar (mathematics)|scalar]] (a tensor of rank zero), the [[Laplacian]] takes on the familiar form.
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| If <math>\mathbf{T}</math> is a vector (a tensor of first rank), the gradient is a [[covariant derivative]] which results in a tensor of second rank, 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 expression shown below for the gradient of a vector:
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| :<math>\nabla \mathbf{T}= (\nabla T_x, \nabla T_y, \nabla T_z)=\begin{bmatrix} T_{xx} & T_{xy} & T_{xz} \\
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| T_{yx} & T_{yy} & T_{yz} \\
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| T_{zx} & T_{zy} & T_{zz} \end{bmatrix} , \text{ where } T_{uv} \equiv \frac{\partial T_v}{\partial u}.</math>
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| 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 rank) can be seen as a product of matrices:
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| :<math> \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>
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| This identity is a coordinate dependent result, and is not general.
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| ==Use in physics==
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| {{unreferenced section|date=January 2010}}
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| An example of the usage of the vector Laplacian is the [[Navier-Stokes equations]] for a [[Newtonian fluid|Newtonian]] [[incompressible flow]]:
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| :<math>\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>
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| 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.
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| Another example is the wave equation for the electric field that can be derived from
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| the [[Maxwell equations]] in the absence of charges and currents:
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| :<math>\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0.</math>
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| The previous equation can also be written as:
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| :<math>\Box\, \mathbf{E} = 0,</math>
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| where
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| :<math>\Box\equiv\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2,</math>
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| is the [[D'Alembertian]], used in the [[Klein–Gordon equation]].
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| ==References==
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| *{{cite web | url = http://mathworld.wolfram.com/VectorLaplacian.html | title = Vector Laplacian
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| | author = MathWorld}}
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| *[http://farside.ph.utexas.edu/teaching/em/lectures/node23.html http://farside.ph.utexas.edu/teaching/em/lectures/node23.html]
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| [[Category:Multivariable calculus|Laplacian]]
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| [[he:לפלסיאן#הלפלסיאן הווקטורי]]
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The author is recognized by the title of Figures Wunder. California is exactly where her at home std testing; listen to this podcast, is but she needs to move simply because of her family. The preferred hobby for my children and me is to play baseball but I haven't made a dime with it. Since she was eighteen she's been operating as a meter reader but she's always wanted her personal business.