Vector laplacian

Differential operator noted \nabla^2 that applies to vector field. The vector laplacian transform a vector field into an other scalar field.

Source:Université en ligne Unisciel[1]

Formula

Let \overrightarrow V(M) be a vector field, its laplacian is given by the equation : \nabla^2\overrightarrow V(M)=\overrightarrow{grad}~[div~\overrightarrow V(M)]-\overrightarrow{rot}~[\overrightarrow{rot}~\overrightarrow V(M)]

Expression of the vector laplacian in a cartesian frame of reference: \nabla^2\overrightarrow V=\left| \begin{array}{lc} \nabla^2V_x=\frac{\partial^2V_x}{\partial x^2}+\frac{\partial^2V_x}{\partial y^2}+\frac{\partial^2V_x}{\partial z^2} \\\\ \nabla^2V_y=\frac{\partial^2V_y}{\partial x^2}+\frac{\partial^2V_y}{\partial y^2}+\frac{\partial^2V_y}{\partial z^2} \\\\ \nabla^2V_z=\frac{\partial^2V_z}{\partial x^2}+\frac{\partial^2V_z}{\partial y^2}+\frac{\partial^2V_z}{\partial z^2}\end{array} \right.

  1. Université en ligne Unisciel

    Traduction par Unisciel