Vector Products

a×b=x^y^z^axayazbxbybz\bm{a}\times\bm{b} = \begin{vmatrix} \bm{\hat{x}} & \bm{\hat{y}} & \bm{\hat{z}} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

where

x^,y^,z^\bm{\hat{x}}, \bm{\hat{y}}, \bm{\hat{z}}

are unit vectors.

a×(b×c)=b(ac)c(ab)\bm{a}\times(\bm{b}\times\bm{c}) = \bm{b}(\bm{a}\cdot \bm{c}) - \bm{c}(\bm{a} \cdot \bm{b})
a×(b×c)+b×(c×a)+c×(a×b)=0\bm{a}\times(\bm{b}\times\bm{c}) + \bm{b}\times(\bm{c}\times\bm{a}) + \bm{c}\times(\bm{a}\times\bm{b}) = 0
(a×b)(c×d)=(ac)(bd)(ad)(bc)(\bm{a}\times\bm{b})\cdot(\bm{c}\times\bm{d}) = (\bm{a}\cdot\bm{c})(\bm{b}\cdot\bm{d}) - (\bm{a}\cdot\bm{d})(\bm{b}\cdot\bm{c})
(a×b)×(c×d)=c((a×b)d)d((a×b)c)(\bm{a}\times\bm{b})\times(\bm{c}\times\bm{d}) = \bm{c}((\bm{a}\times\bm{b})\cdot\bm{d}) - \bm{d}((\bm{a}\times\bm{b})\cdot\bm{c})

Gradient, Divergence, Curl and The Laplace Operator

gradf=f=(fx,fy,fz)=(fr,1rfθ,1rsinθfφ)\textup{grad}f = \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) = \left(\frac{\partial f}{\partial r}, \frac{1}{r}\frac{\partial f}{\partial \theta}, \frac{1}{r\sin\theta}\frac{\partial f}{\partial \varphi}\right)
{x=rsinθcosφy=rsinθsinφz=rcosθ\begin{cases} x = r\sin\theta\cos\varphi \\ y = r\sin\theta\sin\varphi \\ z = r\cos\theta \end{cases}
diva=a=axx+ayy+azz=1r2r(r2ar)+1rsinθθ(sinθaθ)+1rsinθaφφ\begin{split} \textup{div}\bm{a} &= \nabla \cdot \bm{a} = \frac{\partial a_x}{\partial x} + \frac{\partial a_y}{\partial y} + \frac{\partial a_z}{\partial z} \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 a_r\right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta a_\theta\right) + \frac{1}{r\sin\theta}\frac{\partial a_\varphi}{\partial \varphi} \end{split}
rota=×a=(azyayz,axzazx,ayxaxy)=(1rsinθ(θ(sinθaφ)aθφ),1rsinθarφ1rr(raφ),1rr(raθ)1rarθ)\begin{split} \textup{rot} \bm{a} &= \nabla\times\bm{a} = \left(\frac{\partial a_z}{\partial y} - \frac{\partial a_y}{\partial z}, \frac{\partial a_x}{\partial z} - \frac{\partial a_z}{\partial x}, \frac{\partial a_y}{\partial x} - \frac{\partial a_x}{\partial y}\right) \\ &= \left(\frac{1}{r\sin\theta}\left(\frac{\partial}{\partial\theta}(\sin\theta a_\varphi) - \frac{\partial a_\theta}{\partial \varphi}\right), \frac{1}{r\sin\theta}\frac{\partial a_r}{\partial \varphi} - \frac{1}{r}\frac{\partial}{\partial r}(r a_\varphi), \frac{1}{r}\frac{\partial}{\partial r}(r a_\theta) - \frac{1}{r}\frac{\partial a_r}{\partial \theta}\right) \end{split}
Δ=2x2+2y2+2z2=1r2r(r2r)+1r2sinθθ(sinθθ)+1r2sin2θ2φ2\begin{split} \Delta &= \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} + \frac{\partial ^2}{\partial z^2} \\ &= \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \end{split}
Δf(r)=1rd2dr2(rf),r0\Delta f(r) = \frac{1}{r}\frac{d^2}{dr^2}(rf), \quad r \neq 0
×(U)=0(U)=U(×A)=0×(×A)=(A)ΔA(UV)=UΔV+2UV+VΔU(UV)=UΔV+2(U)V+VΔU×(UV)=U×V+(U)×V(A×B)=B(×A)A(×B)(AB)=A×(×B)+B×(×A)+(B)A+(A)B×(A×B)=(B)A(A)B+A(B)B(A)\begin{gathered} \nabla\times(\nabla\mathcal{U}) = 0 \\ \nabla\cdot(\nabla\mathcal{U}) = \nabla\mathcal{U} \\ \nabla\cdot(\nabla\times\bm{A}) = 0 \\ \nabla\times(\nabla\times\bm{A}) = \nabla(\nabla\cdot\bm{A}) - \Delta\bm{A} \\ \nabla\cdot(\mathcal{U}V) = \mathcal{U}\Delta V + 2\nabla\mathcal{U}\cdot\nabla V + V\Delta\mathcal{U} \\ \nabla\cdot(\mathcal{U}\bm{V}) = \mathcal{U}\Delta \bm{V} + 2(\nabla\mathcal{U}\cdot\nabla) \bm{V} + \bm{V}\Delta\mathcal{U} \\ \nabla\times(\mathcal{U}\bm{V }) = \mathcal{U}\nabla\times\bm{V} + (\nabla\mathcal{U}) \times \bm{V} \\ \nabla\cdot(\bm A\times \bm B) = \bm B \cdot(\nabla \times \bm A) - \bm A \cdot(\nabla \times \bm B) \\ \nabla(\bm A \cdot \bm B) = \bm A \times ( \nabla \times \bm B) + \bm B \times ( \nabla \times \bm A) + (\bm B \cdot \nabla)\bm A + (\bm A \cdot \nabla)\bm B \\ \nabla\times(\bm A \times \bm B) = (\bm B \cdot \nabla) \bm A - (\bm A \cdot \nabla)\bm B + \bm A(\nabla\cdot \bm B) - \bm B (\nabla \cdot \bm A) \end{gathered}

Gauss' theorem

S(V)adS=V(a)dV\oint_{S(V)} \bm a \cdot d \bm S = \int_V(\nabla \cdot \bm a) d V

Where

dVdV

in polar coordinates are

r2sinθ  dr  dθ  dφr^2\sin\theta \;dr\; d\theta \;d\varphi

Stoke's theorem

C(S)adl=S(×a)dS\oint_{C(S)} \bm a\cdot d \bm l = \int_S(\nabla \times \bm a) \cdot d\bm S

Where

SS

is an arbitrary surface with border

C(S)C(S)

Green's theorem

S(V)(ΨφφΨ)dS=V(ΨΔφφΔΨ)dV\oint_{S(V)} (\Psi\nabla \varphi - \varphi\nabla\Psi) \cdot d\bm S = \int_V (\Psi\Delta\varphi - \varphi \Delta\Psi)dV