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Field-Theory

2018-11-15·University Physics

Coordinate system

  • Cartesian system (i⃗,j⃗,k⃗)(\vec i,\vec j,\vec k)
    • orthogonal: i⃗⋅j⃗=j⃗⋅k⃗=k⃗⋅i⃗=0\vec i\cdot\vec j=\vec j\cdot\vec k=\vec k\cdot\vec i=0
    • right-handed screw relation: i⃗×j⃗=k⃗\vec i\times\vec j=\vec k, j⃗×k⃗=i⃗\vec j\times\vec k=\vec i, k⃗×i⃗=j⃗\vec k\times \vec i=\vec j
    • A⋅B=A1B1+A2B2+A3B3A\cdot B = A_1B_1+A_2B_2+A_3B_3
    • A×B=det⁡([i⃗j⃗k⃗A1A2A3B1B2B3])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 (e⃗ρ,e⃗ϕ)(\vec e_\rho, \vec e_\phi)
    • moving frames
    • position vector ρ⃗=ρe⃗ρ\vec\rho=\rho\vec e_\rho
    • e⃗ρ=cosφi⃗+sinφj⃗\vec e_\rho=cos\varphi\vec i+sin\varphi\vec j
    • de⃗ρdt=φ˙e⃗φ\frac{d\vec e_\rho}{dt}=\dot{\varphi}\vec e_\varphi
    • de⃗φdt=−φ˙e⃗ρ\frac{d\vec e_\varphi}{dt}=-\dot{\varphi}\vec e_\rho
  • Spherical systems (e⃗ρ,e⃗ϕ,k⃗)(\vec e_\rho, \vec e_\phi, \vec k)
  • Cylindrical systems
  • intrinsic system

temp

  • In sperical coordinate system: ∇2=1r2∂∂r(r2∂∂r)+1r2[1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ∂2∂φ2]\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

  • ∇×(∇×A)=∇(∇⋅A)−∇2A\nabla\times(\nabla\times A) = \nabla(\nabla\cdot A) - \nabla^2 A
  • 柱坐标系： ∇=∂∂ρ+1ρ⋅∂∂ϕ+∂∂z\nabla = \frac{\partial}{\partial\rho}+\frac{1}{\rho}\cdot\frac{\partial}{\partial\phi}+\frac{\partial}{\partial z}