高频公式表

[AllanChain]

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名书镇贴

东西慢慢加进来好了，顺便看看公式渲染的情况。

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柱坐标系下的梯度散度旋度公式

$$ (\nabla u)_r = \frac{\partial u}{\partial r},\ (\nabla u)_{\theta} = \frac 1 r \frac{\partial u}{\partial \theta},\ (\nabla u)_z = \frac{\partial u}{\partial z} $$

$$ \nabla \cdot \mathbf{A} = \frac 1 r \frac{\partial (r A_r)}{\partial r} + \frac 1 r \frac{\partial A_{\theta}}{\partial \theta} + \frac{\partial A_z}{\partial z} $$

$$ \nabla \times \mathbf{A} = \frac 1 r \left[\frac{\partial A_z}{\partial \theta} - \frac{\partial (rA_{\theta})}{\partial z}\right]\mathbf{e_r} + \left[\frac{\partial A_r}{\partial z} - \frac{\partial A_z}{\partial r}\right]\mathbf{e_{\theta}} + \frac 1 r \left[\frac{\partial (rA_{\theta})}{\partial r} - \frac{\partial A_r}{\partial \theta}\right]\mathbf{e_z} $$

球坐标系下的梯度散度旋度公式

$$ (\nabla u)_r = \frac{\partial u}{\partial r},\ (\nabla u)_{\theta} = \frac 1 r \frac{\partial u}{\partial \theta},\ (\nabla u)_{\phi} = \frac{1}{r \sin \theta} \frac{\partial u}{\partial \phi} $$

$$ \nabla \cdot \mathbf{A} = \frac{1}{r^2} \frac{\partial (r^2 A_r)}{\partial r} + \frac{1}{r \sin \theta} \frac{\partial (\sin \theta A_{\theta})}{\partial \theta} + \frac{1}{r \sin \theta} \frac{\partial A_{\phi}}{\partial \phi} $$

$$ \nabla \times \mathbf{A} = \frac{1}{r \sin \theta} \left[\frac{\partial (\sin \theta A_{\phi})}{\partial \theta} - \frac{\partial A_{\theta}}{\partial \phi}\right]\mathbf{e_r} + \frac 1 r \left[\frac{1}{\sin \theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial (r A_{\phi})}{\partial r}\right]\mathbf{e_{\theta}} + \frac 1 r \left[\frac{\partial (rA_{\theta})}{\partial r} - \frac{\partial A_r}{\partial \theta}\right]\mathbf{e_{\phi}} $$