■勾配(grad)
直交座標 \(\nabla f=\displaystyle\frac{\partial f}{\partial x}\mathbf{e_x}+\frac{\partial f}{\partial y}\mathbf{e_y}+\frac{\partial f}{\partial z}\mathbf{e_z}\)
球座標 \(\nabla f=\displaystyle\frac{\partial f}{\partial r}\mathbf{e_r}+\frac{1}{r}\frac{\partial f}{\partial \theta}\mathbf{e_{\theta}}+\frac{1}{rsin\theta}\frac{\partial f}{\partial \phi}\mathbf{e_{\phi}}\)
円筒座標 \(\nabla f=\displaystyle\frac{\partial f}{\partial \rho}\mathbf{e_{\rho}}+\frac{1}{r}\frac{\partial f}{\partial \phi}\mathbf{e_{\phi}}+\frac{\partial f}{\partial z}\mathbf{e_z}\)
■発散(div)
直交座標 \(\nabla･\mathbf{A}=\displaystyle\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\)
球座標 \(\nabla･\mathbf{A}=\displaystyle\frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r}+\frac{1}{rsin\theta}\frac{\partial (A_{\theta}sin\theta)}{\partial \theta}+\frac{1}{rsin\theta}\frac{\partial A_{\phi}}{\partial \phi}\)
円筒座標 \(\nabla･\mathbf{A}=\displaystyle\frac{1}{\rho}\frac{\partial (\rho A_{\rho})}{\partial \rho}+\frac{1}{\rho}\frac{\partial A_{\phi}}{\partial \phi}+\frac{\partial A_z}{\partial z}\)
■回転(rot)
直交座標 \(\nabla×\mathbf{A}=\begin{vmatrix} \mathbf{e_x} & \mathbf{e_y} & \mathbf{e_z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}\)
球座標 \(\nabla×\mathbf{A}=\begin{vmatrix} \frac{\mathbf{e_r}}{r^2sin\theta} & \frac{\mathbf{e_{\theta}}}{rsin\theta} & \frac{\mathbf{e_{\phi}}}{r} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ A_r & A_{\theta} & rA_{\phi}sin\theta \end{vmatrix}\)
円筒座標 \(\nabla×\mathbf{A}=\begin{vmatrix} \frac{\mathbf{e_{\rho}}}{\rho} & \mathbf{e_{\phi}} & \frac{\mathbf{e_z}}{\rho} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \phi} & \frac{\partial}{\partial z} \\ A_{\rho} & \rho A_{\phi} & A_z \end{vmatrix}\)
■ラプラシアン
直交座標 \(\nabla^2f=\displaystyle\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}\)
球座標 \(\nabla^2f=\displaystyle\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial f}{\partial r} \right)+\frac{1}{r^2sin\theta}\frac{\partial}{\partial \theta}\left( sin\theta \frac{\partial f}{\partial \theta} \right)+\frac{1}{r^2sin^2\theta}\frac{\partial^2f}{\partial \phi^2}\)
円筒座標 \(\nabla^2f=\displaystyle\frac{1}{\rho}\frac{\partial}{\partial \rho}\left( \rho\frac{\partial f}{\partial \rho} \right)+\frac{1}{\rho^2}\frac{\partial^2 f}{\partial \phi^2}+\frac{\partial^2 f}{\partial z^2}\)