physref

Vector identities (WIP)

  1. Differential vector identities
  2. Integral vector identities

Differential vector identities

  1. ∇(φψ)=φ∇ψ+ψ∇φ\nabla(\varphi\psi)=\varphi\nabla\psi+\psi\nabla\varphi
  2. ∇(a⋅b)=(a⋅∇)b+(b⋅∇)a+a×(∇×b)+b×(∇×a)\nabla(\mathbf{a}\cdot\mathbf{b})=(\mathbf{a}\cdot\nabla)\mathbf{b}+(\mathbf{b}\cdot\nabla)\mathbf{a}+\mathbf{a}\times(\nabla\times\mathbf{b})+\mathbf{b}\times(\nabla\times\mathbf{a})
  3. ∇⋅(φa)=a⋅∇φ+φ∇⋅a\nabla\cdot(\varphi\mathbf{a})=\mathbf{a}\cdot\nabla\varphi+\varphi\nabla\cdot\mathbf{a}
  4. ∇⋅(a×b)=b⋅(∇×a)−a⋅(∇×b)\nabla\cdot(\mathbf{a}\times\mathbf{b})=\mathbf{b}\cdot(\nabla\times\mathbf{a})-\mathbf{a}\cdot(\nabla\times\mathbf{b})
  5. ∇×(φa)=∇φ×a+φ∇×a\nabla\times(\varphi\mathbf{a})=\nabla\varphi\times\mathbf{a}+\varphi\nabla\times\mathbf{a}
  6. ∇×(a×b)=a(∇⋅b)−b(∇⋅a)+(b⋅∇)a−(a⋅∇)b\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}
  7. ∇×∇φ=0\nabla\times\nabla\varphi=0
  8. ∇⋅(∇×a)=0\nabla\cdot(\nabla\times\mathbf{a})=0
  9. ∇×(∇×a)=∇(∇⋅a)−∇2a\nabla\times(\nabla\times\mathbf{a})=\nabla(\nabla\cdot\mathbf{a})-{\nabla^2}\mathbf{a}

Integral vector identities

  1. ∫ab(∇φ)⋅dl=φ(b)−φ(a)\int_{\mathbf{a}}^{\mathbf{b}} (\nabla\varphi)\cdot d\mathbf{l} = \varphi(\mathbf{b})-\varphi(\mathbf{a}) (Gradient theorem)
  2. ∫V∇⋅A dv=∫SA⋅da\int_V \nabla\cdot\mathbf{A}\,dv = \int_S\mathbf{A}\cdot d\mathbf{a}
  3. ∫V∇φ dv=∫Sφ da\int_V {\nabla \varphi } \,dv = \int_S {\varphi \,d{\mathbf{a}}}
  4. ∫V∇×A dv=∫Sda×A\int_V {\nabla \times {\mathbf{A}}} \,dv = \int_S {d{\mathbf{a}} \times {\mathbf{A}}} (Divergence/Gauss' theorem)
  5. ∫V(ψ∇2φ−φ∇2ψ) dv=∫S(ψ∇φ−φ∇ψ)⋅da\int_V (\psi\nabla^2\varphi-\varphi\nabla^2\psi)\,dv = \int_S(\psi\nabla\varphi-\varphi\nabla\psi)\cdot d\mathbf{a}
  6. ∫V(ψ∇2φ+∇ψ⋅∇φ) dv=∫Sψ∇φ⋅da\int_V (\psi\nabla^2\varphi+\nabla\psi\cdot\nabla\varphi)\,dv = \int_S\psi\nabla\varphi\cdot d\mathbf{a} (Green's theorem)
  7. ∫S(∇×A)⋅da=∮CA⋅dl\int_S {(\nabla \times {\mathbf{A}}) \cdot d{\mathbf{a}}} = \oint_C {{\mathbf{A}} \cdot d{\mathbf{l}}}
  8. ∫Sda×∇φ=∮Cφ dl\int_S {d{\mathbf{a}} \times \nabla \varphi } = \oint_C {\varphi \,d{\mathbf{l}}}
  9. ∫S(da×∇)×A=∮Cdl×A\int_S (d\mathbf{a}\times\nabla)\times\mathbf{A} = \oint_C d\mathbf{l}\times\mathbf{A} (Curl/Stokes' theorem)

See also:

Trigonometric identities