Tutorial comes from https://www.youtube.com/watch?v=jFe6dMnpQho 0-forms : f:R3→R 1-forms : α=adx+bdy+cdz=w<a,b,c> 2-forms : β=ady∧dz+bdz∧dx+cdx∧dy=Φ<a,b,c> 3-forms : γ=gdx∧dy∧dz
1 . suppose α1, α2 both are 1-forms α1∧α2=−α2∧α1
2 . wA⃗ ∧wB⃗ =ΦA⃗ ×B⃗
3 . A⃗ ×(B⃗ ×C⃗ )≠(A⃗ ×B⃗ )×C⃗ but wA⃗ ∧(wB⃗ ∧wC⃗ )=(wA⃗ ∧wB⃗ )∧wC⃗
4 . ΦA⃗ ∧wB⃗ =(A⃗ ⋅B⃗ )dx∧dy∧dz
5 . wA⃗ ∧wB⃗ ∧wC⃗ =ΦA⃗ ×B⃗ ∧wC⃗ =(A⃗ ×B⃗ )⋅C⃗ dx∧dy∧dz
6 . Hodge duality
∗wF⃗ =ΦF⃗ ∗ΦF⃗ =wF⃗ ∗(g dx∧dy∧dz)=g ∗(f)=f dx∧dy∧dz
p↦∗(3−p)form
7 . α=∑iαidxi 8 . dα=∑idαi∧dxi
d:p-forms↦(p+1)-forms
9 . ex 0 : f=x2+yz 1 : df=2xdx+zdy+ydz=w<2x,z,y>=w∇f
所以 df=w∇f
10 . ex
α=yzdx−zdy+z2dz dα=d(yz)∧dx−dz∧dy+d(z2)∧dz=(zdy+ydz)∧dx−dz∧dy+2zdz∧dz=dy∧dz+ydz∧dx−zdx∧dy=Φ<1,y,−z>
∇×<yz,−z,z2>=det∣∣∣∣∣x^∂xyzy^∂y−zz^∂zz2∣∣∣∣∣=<1,y,−z>
所以
dwF⃗ =Φ∇×F⃗
11 .
β=x2dy∧dz+(y+z)dz∧dx+zdx∧dy dβ=2xdx∧dy∧dz+(dy+dz)∧dz∧dx+dz∧dx∧dy=(2x+1+1) dx∧dy∧dz
dΦ<x2,y+z,z>=(2x+1+1) dx∧dy∧dz=∇⋅<x2,y+z,z> dx∧dy∧dz
所以 dΦF⃗ =(∇⋅F⃗ ) dx∧dy∧dz
12 . Identities
假设 α 为 p-form(包括0-form)
d(α+β)=dα+dβ d(α∧β)=dα∧β+(−1)pα∧dβ d(dα)=0
13 . d2=0
对于α为0-form d(df)=0 d(w∇f)=0 Φ∇×∇f=0⟹∇×∇f=0
对于α为1-form d(dwF⃗ )=0 dΦ∇×F⃗ =0 ∇⋅(∇×F⃗ ) dxdydz=0⟹∇⋅(∇×F⃗ )=0
14 .
d(wF⃗ ∧wG⃗ )=dwF⃗ ∧wG⃗ −wF⃗ ∧dwG⃗ dΦF⃗ ×G⃗ =Φ∇×F∧wG⃗ −wF⃗ ∧Φ∇×G⃗ ∇⋅(F⃗ ×G⃗ )dx∧dy∧dz=((∇×F⃗ )⋅G⃗ −F⃗ ⋅(∇×G⃗ ))dx∧dy∧dz ⟹∇⋅(F⃗ ×G⃗ )=((∇×F⃗ )⋅G⃗ −F⃗ ⋅(∇×G⃗ ))
15 . integral
0-form : ∫{a,b}f=f(b)−f(a) 1-form : ∫CwF⃗ =∫CF⃗ ⋅dr⃗ 2-form : ∫SΦF⃗ =∬SF⃗ ⋅dS⃗ 3-form : ∫Eg dx∧dy∧dz=∭Eg dxdydz
16 . Generalized stokes’ theorems
对于任何 p-form α ∫Mdα=∫∂Mα
对于 M=C ∫Cdf=∫{a,b}f=f(b)−f(a)(=∫Cw∇f=∫C∇f⋅dr⃗ )
对于 M=S
∫∂SwF⃗ =∫SdwF⃗ =∫SΦ∇×F⃗
∫∂SF⃗ ⋅dr⃗ =∬S(∇×F⃗ )⋅dS⃗
对于 M=E ∫∂EΦF⃗ =∫EdΦF⃗ ∫∂EF⃗ ⋅dS⃗ =∭E(∇⋅F⃗ ) dxdydz
