Coordinate system

• Cartesian system $(\vec i,\vec j,\vec k)$
  • orthogonal: $\vec i\cdot\vec j=\vec j\cdot\vec k=\vec k\cdot\vec i=0$
  • right-handed screw relation: $\vec i\times\vec j=\vec k$, $\vec j\times\vec k=\vec i$, $\vec k\times \vec i=\vec j$
  • $A\cdot B = A_1B_1+A_2B_2+A_3B_3$
  • $$A\times B=\det(\left[ \begin{matrix} \vec i & \vec j & \vec k\newline A_1 & A_2 & A_3\newline B_1 & B_2 & B_3 \end{matrix} \right])$$
• polar coordinate systems $(\vec e_\rho, \vec e_\phi)$
  • moving frames
  • position vector $\vec\rho=\rho\vec e_\rho$
  • $\vec e_\rho=cos\varphi\vec i+sin\varphi\vec j$
  • $\frac{d\vec e_\rho}{dt}=\dot{\varphi}\vec e_\varphi$
  • $\frac{d\vec e_\varphi}{dt}=-\dot{\varphi}\vec e_\rho$
• Spherical systems $(\vec e_\rho, \vec e_\phi, \vec k)$
• Cylindrical systems
• intrinsic system

temp

• In sperical coordinate system: $\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})+\frac{1}{r^2}[\frac{1}{sin\theta}\frac{\partial}{\partial \theta}(sin\theta\frac{\partial}{\partial \theta})+\frac{1}{sin^2\theta}\frac{\partial^2}{\partial \varphi^2}]$

场论

• $\nabla\times(\nabla\times A) = \nabla(\nabla\cdot A) - \nabla^2 A$
• 柱坐标系： $\nabla = \frac{\partial}{\partial\rho}+\frac{1}{\rho}\cdot\frac{\partial}{\partial\phi}+\frac{\partial}{\partial z}$