Header: 数学物理方法

高频公式表

2021-04-05

名书镇贴

东西慢慢加进来好了,顺便看看公式渲染的情况。

 
柱坐标系下的梯度散度旋度公式 #
(u)_r=ur, (u)_θ=1ruθ, (u)_z=uz(\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}A=1r(rAr)r+1rAθθ+Azz\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}×A=1r[Azθ(rAθ)z]er+[ArzAzr]eθ+1r[(rAθ)rArθ]ez\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 #
(u)_r=ur, (u)_θ=1ruθ, (u)_ϕ=1rsinθuϕ(\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}A=1r2(r2Ar)r+1rsinθ(sinθAθ)θ+1rsinθAϕϕ\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}×A=1rsinθ[(sinθAϕ)θAθϕ]er+1r[1sinθArϕ(rAϕ)r]eθ+1r[(rAθ)rArθ]eϕ\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}}
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