3次元デカルト座標での定義式。
\nabla =
\begin{pmatrix}
{\partial / \partial x} \\ 
{\partial / \partial y} \\ 
{\partial / \partial z}
\end{pmatrix}
\mathrm{grad}\phi =
\nabla \phi =
\begin{pmatrix}
{\partial \phi / \partial x} \\ 
{\partial \phi / \partial y} \\ 
{\partial \phi / \partial z}
\end{pmatrix}
\mathrm{div}\mathbf{F} =
\nabla\cdot\mathbf{F} =
\partial F_x/\partial x + \partial F_y/\partial y + \partial F_z/\partial z
\mathrm{curl}\mathbf{F} =
\mathrm{rot}\mathbf{F} =
\nabla\times\mathbf{F} =
\begin{pmatrix}
{\partial F_z / \partial y} - {\partial F_y / \partial z} \\ 
{\partial F_x / \partial z} - {\partial F_z / \partial x}\\ 
{\partial F_y / \partial x} - {\partial F_x / \partial y}
\end{pmatrix}
\Delta=\nabla^2 = 
{\partial^2 \over \partial x^2 } +
{\partial^2 \over \partial y^2 } +
{\partial^2 \over \partial z^2 } 
\Delta\phi=\nabla^2 \phi = \nabla \cdot ( \nabla \phi )
= \mathrm{div}(\mathrm{grad}\phi)
\operatorname{div}( a\mathbf{F} + b\mathbf{G} ) = a\;\operatorname{div}( \mathbf{F} ) + b\;\operatorname{div}( \mathbf{G} )
\operatorname{div}(\phi \mathbf{F}) = \operatorname{grad}(\phi) \cdot \mathbf{F} + \phi \;\operatorname{div}(\mathbf{F})
\nabla\cdot(\phi \mathbf{F}) = (\nabla\phi) \cdot \mathbf{F} + \phi \;(\nabla\cdot\mathbf{F})
\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl}(\mathbf{F})\cdot\mathbf{G} \;-\; \mathbf{F} \cdot \operatorname{curl}(\mathbf{G})
\operatorname{div}(\operatorname{curl}\mathbf{F}) =
\operatorname{div}(\nabla\times\mathbf{F}) =
\operatorname{curl}(\nabla)\cdot\mathbf{F} - \nabla\cdot\operatorname{curl}(\mathbf{F})
ここで
\left[\operatorname{curl}(\nabla)\right]_x =
\frac{\partial^2}{\partial z \partial y} - \frac{\partial^2}{\partial y \partial z} = 0
(演算対象の関数が連続でなめらかな場合)
であるので
\operatorname{div}(\operatorname{curl}\mathbf{F}) =
- \nabla\cdot\operatorname{curl}(\mathbf{F}) =
- \operatorname{div}(\operatorname{curl}\mathbf{F})
結局
\operatorname{div}(\operatorname{curl}\mathbf{F}) = 0
\operatorname{curl}(\operatorname{curl}(\mathbf{F})) =
-\Delta\mathbf{F} + \operatorname{grad}(\operatorname{div}\mathbf{F})
x成分をとって証明する。
\left[\operatorname{curl}(\operatorname{curl}(\mathbf{F}))\right]_x =
\left[\nabla\times(\nabla\times\mathbf{F})\right]_x =
\frac{\partial}{\partial y}\left[\nabla\times\mathbf{F}\right]_z -
\frac{\partial}{\partial z}\left[\nabla\times\mathbf{F}\right]_y
= \frac{\partial}{\partial y}(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}) -
\frac{\partial}{\partial z}(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x})
= - (\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})F_x
+ \frac{\partial}{\partial x}(\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z})
= - \Delta F_x + \frac{\partial}{\partial x}\operatorname{div}\mathbf{F}
= \left[-\Delta\mathbf{F} + \operatorname{grad}(\operatorname{div}\mathbf{F})\right]_x