数学物理公式速查

$\nabla\times(\nabla f)=0$

$\nabla\cdot(\nabla\times \mathbf{F})=0$

$\nabla\times(\nabla\times \mathbf{F})=\nabla(\nabla\cdot \mathbf{F})-\nabla^2\mathbf{F}$

$\nabla \cdot (\phi \mathbf{F}) = (\nabla \phi) \cdot \mathbf{F} + \phi (\nabla \cdot \mathbf{F})$

$\nabla \times (\mathbf{A}\times \mathbf{B}) = \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B}$

$\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$

$\nabla\cdot \mathbf{F} = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$

$\nabla \times \mathbf{F}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\;\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\;\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)$

$\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$

$\nabla^2 \mathbf{F} = \nabla (\nabla \cdot \mathbf{F}) - \nabla \times (\nabla \times \mathbf{F}) = \left(\frac{\partial^2 F_x}{\partial x^2} + \frac{\partial^2 F_x}{\partial y^2} + \frac{\partial^2 F_x}{\partial z^2}, \frac{\partial^2 F_y}{\partial x^2} + \frac{\partial^2 F_y}{\partial y^2} + \frac{\partial^2 F_y}{\partial z^2}, \frac{\partial^2 F_z}{\partial x^2} + \frac{\partial^2 F_z}{\partial y^2} + \frac{\partial^2 F_z}{\partial z^2}\right)$

梯度： $\nabla f = \hat{\mathbf{e}}_r \frac{\partial f}{\partial r} + \hat{\mathbf{e}}_\theta \frac{1}{r}\frac{\partial f}{\partial \theta} + \hat{\mathbf{e}}_z \frac{\partial f}{\partial z}$

散度： $\nabla\cdot \mathbf{F} = \frac{1}{r}\frac{\partial}{\partial r}(r F_r) + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}$

拉普拉斯（标量）： $\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}$

梯度： $\nabla f = \hat{\mathbf{e}}_r \frac{\partial f}{\partial r} + \hat{\mathbf{e}}_\theta \frac{1}{r}\frac{\partial f}{\partial \theta} + \hat{\mathbf{e}}_\varphi \frac{1}{r\sin\theta}\frac{\partial f}{\partial \varphi}$

散度： $\nabla\cdot \mathbf{F} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 F_r) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta F_\theta) + \frac{1}{r\sin\theta}\frac{\partial F_\varphi}{\partial \varphi}$

拉普拉斯（标量）： $\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}$

不可压缩流体： $\nabla \cdot \vec{u} = 0$

可压缩流体： $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$

$\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla) \vec{u} = -\frac{1}{\rho} \nabla p + \rho \vec{g}$

$\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla) \vec{u} = -\frac{1}{\rho} \nabla p + \nu \nabla^2 \vec{u} + \vec{g}$

$\rho \frac{D\vec{u}}{Dt} = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$

$\nabla p = \mu\nabla^2 \vec{u}$

$Re = \frac{\rho U L}{\mu} = \frac{U L}{\nu}$

直角坐标：$u_x = \frac{\partial \Psi}{\partial y}, u_y = -\frac{\partial \Psi}{\partial x}$

球坐标：$u_r = \frac{1}{r^2\sin\theta}\frac{\partial \Psi}{\partial \theta}, u_\theta = -\frac{1}{r\sin\theta}\frac{\partial \Psi}{\partial r}$

$\Psi(r,\theta) = f(r) \sin^2\theta = (Ar + \frac{B}{r} + Cr^2 + Dr^4)\sin^2\theta$