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.
