Line 11: Line 11:
  
 
<math>
 
<math>
 +
F = \left[\begin{array}{1}
 +
M \\
 +
N \\
 +
P
 +
\end{array}\right] \\
 
\Delta F =  
 
\Delta F =  
\left[\begin{array} {1}
+
\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] \\
 +
</math>
  
 
[[Walther_MA271_Fall2020_topic9|Back to main page]]
 
[[Walther_MA271_Fall2020_topic9|Back to main page]]

Revision as of 00:29, 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] \\ $

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

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