Vector Laplacian/Proofs

From Wikipedia, the free encyclopedia

Main article: Vector Laplacian

The following is the proof that

\nabla ^2 \left( {\mathbf{u}} \right) = \nabla \left( {\nabla \cdot{\mathbf{u}}} \right) - \nabla \times \left( {\nabla \times {\mathbf{u}}} \right) = \left\langle {\nabla ^2 u_x ,\nabla ^2 u_y ,\nabla ^2 u_z } \right\rangle.

This is a proof in Cartesian coordinates. This identity is only valid for Cartesian coordinates; completing the calculations in other coordinate systems will result in other solutions.

Let {\mathbf{u}} = u\,{\mathbf{\hat e}}_{\mathbf{x}} + v\,{\mathbf{\hat e}}_{\mathbf{y}} + w\,{\mathbf{\hat e}}_{\mathbf{z}} = \left\langle {u,v,w} \right\rangle
where u = u \left( {x,y,z} \right) , v = v \left( {x,y,z} \right) and w= w \left( {x,y,z} \right)


\begin{align} \nabla ^2 \left( {\mathbf{u}} \right) &= \nabla \left( {\nabla \cdot \left\langle {u,v,w} \right\rangle } \right) - \nabla \times \left( {\nabla \times \left\langle {u,v,w} \right\rangle } \right) \\ &= \nabla \left( {\frac{{\partial u}} {{\partial x}} + \frac{{\partial v}} {{\partial y}} + \frac{{\partial w}} {{\partial z}}} \right) - \nabla \times \left| {\begin{array}{*{20}c} {{\mathbf{\hat e}}_{\mathbf{x}} } & {{\mathbf{\hat e}}_{\mathbf{y}} } & {{\mathbf{\hat e}}_{\mathbf{z}} } \\ {\frac{\partial } {{\partial x}}} & {\frac{\partial } {{\partial y}}} & {\frac{\partial } {{\partial z}}} \\ u & v & w \\ \end{array} } \right| \\ &= \left\langle {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 v}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial x\partial z}},\frac{{\partial ^2 v}} {{\partial y^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial y\partial z}},\frac{{\partial ^2 w}} {{\partial z^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial z}} + \frac{{\partial ^2 v}} {{\partial y\partial z}}} \right\rangle - \nabla \times \left\langle {\frac{{\partial w}} {{\partial y}} - \frac{{\partial v}} {{\partial z}},\frac{{\partial u}} {{\partial z}} - \frac{{\partial w}} {{\partial x}},\frac{{\partial v}} {{\partial x}} - \frac{{\partial u}} {{\partial y}}} \right\rangle \\ &= \left( {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 v}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial x\partial z}}} \right){\mathbf{\hat e}}_{\mathbf{x}} + \left( {\frac{{\partial ^2 v}} {{\partial y^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial y\partial z}}} \right){\mathbf{\hat e}}_{\mathbf{y}} + \left( {\frac{{\partial ^2 w}} {{\partial z^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial z}} + \frac{{\partial ^2 v}} {{\partial y\partial z}}} \right){\mathbf{\hat e}}_{\mathbf{z}} - \left| {\begin{array}{*{20}c} {{\mathbf{\hat e}}_{\mathbf{x}} } & {{\mathbf{\hat e}}_{\mathbf{y}} } & {{\mathbf{\hat e}}_{\mathbf{z}} } \\ {\frac{\partial } {{\partial x}}} & {\frac{\partial } {{\partial y}}} & {\frac{\partial } {{\partial z}}} \\ {\frac{{\partial w}} {{\partial y}} - \frac{{\partial v}} {{\partial z}}} & {\frac{{\partial u}} {{\partial z}} - \frac{{\partial w}} {{\partial x}}} & {\frac{{\partial v}} {{\partial x}} - \frac{{\partial u}} {{\partial y}}} \\ \end{array} } \right| \\ &= \left( {\begin{array}{*{20}c} {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 v}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial x\partial z}}} \\ {\frac{{\partial ^2 v}} {{\partial y^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial y}} + \frac{{\partial ^2 w}} {{\partial y\partial z}}} \\ {\frac{{\partial ^2 w}} {{\partial z^2 }} + \frac{{\partial ^2 u}} {{\partial x\partial z}} + \frac{{\partial ^2 v}} {{\partial y\partial z}}} \\ \end{array} } \right)\left\{ {{\mathbf{\hat e}}_{\mathbf{x}} ,{\mathbf{\hat e}}_{\mathbf{y}} ,{\mathbf{\hat e}}_{\mathbf{z}} } \right\} + \left( {\begin{array}{*{20}c} { - \frac{{\partial ^2 v}} {{\partial x\partial y}} + \frac{{\partial ^2 u}} {{\partial y^2 }} + \frac{{\partial ^2 u}} {{\partial z^2 }} - \frac{{\partial ^2 w}} {{\partial x\partial z}}} \\ {\frac{{\partial ^2 v}} {{\partial x^2 }} - \frac{{\partial ^2 u}} {{\partial x\partial y}} - \frac{{\partial ^2 w}} {{\partial y\partial z}} + \frac{{\partial ^2 v}} {{\partial z^2 }}} \\ { - \frac{{\partial ^2 u}} {{\partial x\partial z}} + \frac{{\partial ^2 w}} {{\partial x^2 }} + \frac{{\partial ^2 w}} {{\partial y^2 }} - \frac{{\partial ^2 v}} {{\partial y\partial z}}} \\ \end{array} } \right)\,\left\{ {{\mathbf{\hat e}}_{\mathbf{x}} ,\,{\mathbf{\hat e}}_{\mathbf{y}} ,\,{\mathbf{\hat e}}_{\mathbf{z}} } \right\} \\ &= \left( {\begin{array}{*{20}c} {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 u}} {{\partial y^2 }} + \frac{{\partial ^2 u}} {{\partial z^2 }}} \\ {\frac{{\partial ^2 v}} {{\partial x^2 }} + \frac{{\partial ^2 v}} {{\partial z^2 }} + \frac{{\partial ^2 v}} {{\partial y^2 }}} \\ {\frac{{\partial ^2 w}} {{\partial x^2 }} + \frac{{\partial ^2 w}} {{\partial y^2 }} + \frac{{\partial ^2 w}} {{\partial z^2 }}} \\ \end{array} } \right)\left\{ {{\mathbf{\hat e}}_{\mathbf{x}} ,{\mathbf{\hat e}}_{\mathbf{y}} ,{\mathbf{\hat e}}_{\mathbf{z}} } \right\} \\ &= \nabla ^2 u\,{\mathbf{\hat e}}_{\mathbf{x}} + \nabla ^2 v\,{\mathbf{\hat e}}_{\mathbf{y}} + \nabla ^2 w\,{\mathbf{\hat e}}_{\mathbf{z}} \\ \end{align}