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[[Category:MA271Fall2020Walther]]
 
[[Category:MA271Fall2020Walther]]
 +
==Vector Laplacian==
 +
 +
The Laplace operator is originally an operation where you input a scalar function and it returns a scalar function. However, there is an alternate version of the Laplace operator that can be performed on vector fields.
 +
 +
The vector Laplacian is defined as:
 +
 +
<math>\Delta F = \nabla^2 F = \nabla (\nabla \cdot F) - \nabla \times (\nabla \times F) \\</math>
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 +
where F is a vector field. In Cartesian coordinates, the vector Laplacian simplifies to the following:
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<math>
 +
F = \left[\begin{array}{1}
 +
M \\
 +
N \\
 +
P
 +
\end{array}\right] \\
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\Delta F =
 +
\nabla
 +
\left(\left[\begin{array}{1}
 +
\frac{\partial}{\partial x} \\
 +
\frac{\partial}{\partial y} \\
 +
\frac{\partial}{\partial z}
 +
\end{array}\right]
 +
\cdot \left[\begin{array}{1}
 +
M \\
 +
N \\
 +
P
 +
\end{array}\right] \right)
 +
-\nabla \times
 +
\left(\left[\begin{array}{1}
 +
\frac{\partial}{\partial x} \\
 +
\frac{\partial}{\partial y} \\
 +
\frac{\partial}{\partial z}
 +
\end{array}\right]
 +
\times \left[\begin{array}{1}
 +
M \\
 +
N \\
 +
P
 +
\end{array}\right] \right) \\
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 +
\Delta F =
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\nabla (M_x + N_y + P_z) -
 +
\left[\begin{array}{1}
 +
\frac{\partial}{\partial x} \\
 +
\frac{\partial}{\partial y} \\
 +
\frac{\partial}{\partial z}
 +
\end{array}\right]
 +
\times
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\left[\begin{array}{1}
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P_y - N_z \\
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M_z - P_x \\
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N_x - M_y
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\end{array}\right] \\
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 +
\Delta F = \left[\begin{array}{1}
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M_{xx} + N_{xy} + P_{xz} \\
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M_{xy} + N_{yy} + P_{yz} \\
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M_{xz} + N_{yz} + P_{zz}
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\end{array}\right]
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- \left[\begin{array}{1}
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N_{xy} + P_{xz} - M_{yy} - M_{zz} \\
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M_{xy} + P_{yz} - N_{xx} - N_{zz} \\
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M_{xz} + N_{yz} - P_{xx} - P_{yy}
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\end{array}\right] \\
 +
 +
\Delta F = \left[\begin{array}{1}
 +
M_{xx} + M_{yy} + M_{zz} \\
 +
N_{xx} + N_{yy} + N_{zz} \\
 +
P_{xx} + P_{yy} + P_{zz}
 +
\end{array}\right] \\
 +
 +
\Delta F = \left[\begin{array}{1}
 +
\Delta M \\
 +
\Delta N \\
 +
\Delta P
 +
\end{array}\right] \\
 +
</math>
 +
 +
The formulas, let alone the derivations, for the vector Laplacian in other coordinate systems are a bit too complex for the level of this article. However, if you wanted to see the formulas, they can be found [https://mathworld.wolfram.com/VectorLaplacian.html here].
  
 
[[Walther_MA271_Fall2020_topic9|Back to main page]]
 
[[Walther_MA271_Fall2020_topic9|Back to main page]]

Latest revision as of 00:34, 7 December 2020

Vector Laplacian

The Laplace operator is originally an operation where you input a scalar function and it returns a scalar function. However, there is an alternate version of the Laplace operator that can be performed on vector fields.

The vector Laplacian is defined as:

$ \Delta F = \nabla^2 F = \nabla (\nabla \cdot F) - \nabla \times (\nabla \times F) \\ $

where F is a vector field. In Cartesian coordinates, the vector Laplacian simplifies to the following:

$ F = \left[\begin{array}{1} M \\ N \\ P \end{array}\right] \\ \Delta F = \nabla \left(\left[\begin{array}{1} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array}\right] \cdot \left[\begin{array}{1} M \\ N \\ P \end{array}\right] \right) -\nabla \times \left(\left[\begin{array}{1} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array}\right] \times \left[\begin{array}{1} M \\ N \\ P \end{array}\right] \right) \\ \Delta F = \nabla (M_x + N_y + P_z) - \left[\begin{array}{1} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array}\right] \times \left[\begin{array}{1} P_y - N_z \\ M_z - P_x \\ N_x - M_y \end{array}\right] \\ \Delta F = \left[\begin{array}{1} M_{xx} + N_{xy} + P_{xz} \\ M_{xy} + N_{yy} + P_{yz} \\ M_{xz} + N_{yz} + P_{zz} \end{array}\right] - \left[\begin{array}{1} N_{xy} + P_{xz} - M_{yy} - M_{zz} \\ M_{xy} + P_{yz} - N_{xx} - N_{zz} \\ M_{xz} + N_{yz} - P_{xx} - P_{yy} \end{array}\right] \\ \Delta F = \left[\begin{array}{1} M_{xx} + M_{yy} + M_{zz} \\ N_{xx} + N_{yy} + N_{zz} \\ P_{xx} + P_{yy} + P_{zz} \end{array}\right] \\ \Delta F = \left[\begin{array}{1} \Delta M \\ \Delta N \\ \Delta P \end{array}\right] \\ $

The formulas, let alone the derivations, for the vector Laplacian in other coordinate systems are a bit too complex for the level of this article. However, if you wanted to see the formulas, they can be found here.

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