Ethan's Notes

    Gradient, Divergence, And Curl In General Coordinates

    #Physics
    Based on this pdf

    Coordinate$\displaystyle h_{1}$$\displaystyle h_{2}$$\displaystyle h_{3}$$\displaystyle u_{1}$$\displaystyle u_{2}$$\displaystyle u_{3}$$\displaystyle n_{1}$$\displaystyle n_{2}$$\displaystyle n_{3}$
    Cartesian$\displaystyle 1$$\displaystyle 1$$\displaystyle 1$$\displaystyle x$$\displaystyle y$$\displaystyle z$$\displaystyle \hat{x}$$\displaystyle \hat{y}$$\displaystyle \hat{z}$
    Cylindrical$\displaystyle 1$$\displaystyle s$$\displaystyle 1$$\displaystyle s$$\displaystyle \phi$$\displaystyle z$$\displaystyle \hat{s}$$\displaystyle \hat{\phi}$$\displaystyle \hat{z}$
    Spherical$\displaystyle 1$$\displaystyle \rho$$\displaystyle \rho\sin \theta$$\displaystyle \rho$$\displaystyle \theta$$\displaystyle \phi$$\displaystyle \hat{\rho}$$\displaystyle \hat{\theta}$$\displaystyle \hat{\phi}$

    $\displaystyle \nabla f=\frac{1}{h_{i}}\partial_{u_{i}}fn_{i}$

    $\displaystyle \nabla \cdot \vec{f}=\frac{1}{h_{1}h_{2}h_{3}}\partial_{u_{i}}(h_{j}h_{k}f_{i})$

    $\displaystyle \nabla \times \vec{f}=\frac{1}{h_{j}h_{k}}\varepsilon_{ijk}\partial_{u_{j}}(h_{k}f_{k})n_{i}$

    Cartesian

    $\displaystyle \nabla f={\left\langle{\partial_{x}}f,\partial_{y}f,\partial_{z}f\right\rangle}$

    Cylindrical

    Spherical

    $\displaystyle \nabla f=\partial_{\rho}f\hat{\rho}+\frac{1}{\rho}\partial_{\theta}f\hat{\theta}+\frac{1}{\rho \sin \theta}\partial_{\phi}f\hat{\phi}$

    $\displaystyle \nabla \times \vec{f}=\frac{1}{\rho\sin \theta}[\partial_{\theta}(f_{\phi}\sin \theta)-\partial_{\phi}f_{\theta}]\hat{\rho}+\frac{1}{\rho}\left[ \frac{1}{\sin \theta}\partial_{\phi}f_{\rho}-\partial_{\rho}(\rho f_{\phi}) \right]\hat{\theta}+\frac{1}{\rho}[\partial_{\rho}(\rho f_{\theta})-\partial_{\theta}f_{\rho}]\hat{\phi}$