differential forms

Tutorial comes from https://www.youtube.com/watch?v=jFe6dMnpQho
0-forms : f:R3→Rf : \mathbb R^3 \rightarrow \mathbb R
1-forms : α=adx+bdy+cdz=w<a,b,c> \alpha =adx + bdy + cdz = w_{}
2-forms : β=ady∧dz+bdz∧dx+cdx∧dy=Φ<a,b,c> \beta = ady\wedge dz + bdz \wedge dx + c dx \wedge dy = \Phi_{}
3-forms : γ=gdx∧dy∧dz\gamma = gdx\wedge dy \wedge dz

1 . suppose α1\alpha_1, α2\alpha_2 both are 1-forms
α1∧α2=−α2∧α1\alpha_1 \wedge \alpha_2 = - \alpha_2 \wedge \alpha_1

2 . wA⃗ ∧wB⃗ =ΦA⃗ ×B⃗ w_\vec{A} \wedge w_\vec{B} = \Phi_{\vec{A}\times \vec{B}}

3 . A⃗ ×(B⃗ ×C⃗ )≠(A⃗ ×B⃗ )×C⃗ \vec A \times (\vec B \times \vec C) \neq (\vec A \times \vec B) \times \vec C
but
wA⃗ ∧(wB⃗ ∧wC⃗ )=(wA⃗ ∧wB⃗ )∧wC⃗ w_\vec A \wedge (w_ \vec B \wedge w _\vec C) = (w_\vec A \wedge w_ \vec B) \wedge w _\vec C

4 . ΦA⃗ ∧wB⃗ =(A⃗ ⋅B⃗ )dx∧dy∧dz\Phi _{\vec A} \wedge w_\vec B = (\vec A \cdot \vec B)dx \wedge dy \wedge dz

5 . wA⃗ ∧wB⃗ ∧wC⃗ =ΦA⃗ ×B⃗ ∧wC⃗ =(A⃗ ×B⃗ )⋅C⃗ dx∧dy∧dzw _\vec A \wedge w _\vec B \wedge w _\vec C = \Phi _{\vec A \times \vec B} \wedge w_\vec C = (\vec A \times \vec B) \cdot \vec C \ dx \wedge dy \wedge dz

6 . Hodge duality

∗wF⃗ =ΦF⃗ * w_ \vec F = \Phi_ \vec F
∗ΦF⃗ =wF⃗ * \Phi _ \vec F = w_ \vec F
∗(g dx∧dy∧dz)=g*(g\ dx \wedge dy \wedge dz) = g
∗(f)=f dx∧dy∧dz*(f) = f \ dx \wedge dy \wedge dz

p↦∗(3−p)formp \overset{*}{\mapsto} (3-p) form

7 . α=∑iαidxi\alpha = \displaystyle \sum_i \alpha_i dx_i
8 . dα=∑idαi∧dxid \alpha = \displaystyle \sum_i d\alpha_i \wedge dx_i

d:p-forms↦(p+1)-formsd :\text{p-forms} \mapsto \text{(p+1)-forms}

9 . ex
0 : f=x2+yzf = x^2 + yz
1 : df=2xdx+zdy+ydz=w<2x,z,y>=w∇fdf = 2xdx + zdy + ydz = w_{} = w_{\nabla f}

所以 df=w∇fdf = w_{\nabla f}

10 . ex

α=yzdx−zdy+z2dz\alpha = yz dx - zdy + z^2 dz

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>

\hspace{-430px} \begin{align} d \alpha &= d(yz) \wedge dx - dz \wedge dy + d(z^2) \wedge dz \\ & = (zdy + ydz) \wedge dx - dz \wedge dy + 2z dz \wedge dz \\ & = dy \wedge dz + y dz \wedge dx - z dx \wedge dy \\ & = \Phi_{} \end{align}

∇×<yz,−z,z2>=det∣∣∣∣∣x^∂xyzy^∂y−zz^∂zz2∣∣∣∣∣=<1,y,−z>\nabla \times = det \begin{vmatrix}\hat x & \hat y & \hat z \\ \partial _x & \partial _y & \partial _z \\ yz& -z &z^2\end{vmatrix} =

所以

dwF⃗ =Φ∇×F⃗ dw_\vec F = \Phi _{\nabla \times \vec F}

11 .

β=x2dy∧dz+(y+z)dz∧dx+zdx∧dy\beta = x^2 dy \wedge dz + (y+z)dz \wedge dx + z dx \wedge dy

dβ=2xdx∧dy∧dz+(dy+dz)∧dz∧dx+dz∧dx∧dy=(2x+1+1) dx∧dy∧dz

\hspace{-430px} \begin{align} d\beta & = 2xdx\wedge dy \wedge dz + (dy + dz) \wedge dz \wedge dx + dz \wedge dx \wedge dy\\ & = (2x + 1 + 1)\ dx \wedge dy \wedge dz \end{align}

dΦ<x2,y+z,z>=(2x+1+1) dx∧dy∧dz=∇⋅<x2,y+z,z> dx∧dy∧dzd \Phi_{} = (2x + 1+ 1)\ dx \wedge dy \wedge dz = \nabla \cdot \ dx \wedge dy \wedge dz

所以
dΦF⃗ =(∇⋅F⃗ ) dx∧dy∧dzd \Phi _\vec F = (\nabla \cdot \vec F)\ dx \wedge dy \wedge dz

12 . Identities

假设 α\alpha 为 p-form(包括0-form)

d(α+β)=dα+dβd(\alpha + \beta) = d \alpha + d \beta
d(α∧β)=dα∧β+(−1)pα∧dβd(\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^p \alpha \wedge d \beta
d(dα)=0d(d \alpha) = 0

13 . d2=0d^2 = 0

对于α\alpha为0-form
d(df)=0d(df) = 0
d(w∇f)=0d(w_{\nabla f}) = 0
Φ∇×∇f=0⟹∇×∇f=0 \Phi _{\nabla \times \nabla f} = 0 \Longrightarrow \nabla \times \nabla f = 0

对于α\alpha为1-form
d(dwF⃗ )=0d(dw_\vec F) = 0
dΦ∇×F⃗ =0d \Phi _ {\nabla \times \vec F} = 0
∇⋅(∇×F⃗ ) dxdydz=0⟹∇⋅(∇×F⃗ )=0 \nabla \cdot (\nabla \times \vec F) \ dxdydz= 0 \Longrightarrow \nabla \cdot (\nabla \times \vec F) = 0

14 .

d(wF⃗ ∧wG⃗ )=dwF⃗ ∧wG⃗ −wF⃗ ∧dwG⃗ d(w_\vec F \wedge w _ \vec G) = d w _ \vec F \wedge w _ \vec G - w_ \vec F \wedge dw_\vec G
dΦF⃗ ×G⃗ =Φ∇×F∧wG⃗ −wF⃗ ∧Φ∇×G⃗ d \Phi _ {\vec F \times \vec G} = \Phi _{\nabla \times F} \wedge w _\vec G - w_ \vec F \wedge \Phi _{\nabla \times \vec G}
∇⋅(F⃗ ×G⃗ )dx∧dy∧dz=((∇×F⃗ )⋅G⃗ −F⃗ ⋅(∇×G⃗ ))dx∧dy∧dz\nabla \cdot (\vec F \times \vec G) dx \wedge dy \wedge dz = ((\nabla \times \vec F) \cdot \vec G - \vec F \cdot (\nabla \times \vec G))dx \wedge dy \wedge dz
⟹∇⋅(F⃗ ×G⃗ )=((∇×F⃗ )⋅G⃗ −F⃗ ⋅(∇×G⃗ ))\Longrightarrow \nabla \cdot (\vec F \times \vec G) = ((\nabla \times \vec F) \cdot \vec G - \vec F \cdot (\nabla \times \vec G))

15 . integral

0-form : ∫{a,b}f=f(b)−f(a)\displaystyle \int_{\{a,b\}}f = f(b) - f(a)
1-form : ∫CwF⃗ =∫CF⃗ ⋅dr⃗ \displaystyle \int_C w_\vec F = \int _C \vec F \cdot d\vec r
2-form : ∫SΦF⃗ =∬SF⃗ ⋅dS⃗ \displaystyle \int_S \Phi _\vec F = \iint_S \vec F \cdot d\vec S
3-form : ∫Eg dx∧dy∧dz=∭Eg dxdydz\displaystyle \int _ \mathcal E g \ dx \wedge dy \wedge dz = \iiint _ \mathcal E g\ dxdydz

16 . Generalized stokes’ theorems

对于任何 p-form α\alpha
∫Mdα=∫∂Mα\displaystyle \int _ M d\alpha = \int _ {\partial M} \alpha

对于 M=CM = C
∫Cdf=∫{a,b}f=f(b)−f(a)(=∫Cw∇f=∫C∇f⋅dr⃗ )\displaystyle \int_C df = \int_{\{a,b\}}f = f(b) - f(a) ( = \int _C w_{\nabla f} = \int _C \nabla f \cdot d \vec r)

对于 M=SM = S

∫∂SwF⃗ =∫SdwF⃗ =∫SΦ∇×F⃗ \displaystyle \int _{\partial S} w_\vec F = \int_S d w_\vec F = \int_S \Phi_{\nabla \times \vec F}

∫∂SF⃗ ⋅dr⃗ =∬S(∇×F⃗ )⋅dS⃗ \displaystyle \int _{\partial S} \vec F \cdot d\vec r = \iint _ S (\nabla \times \vec F) \cdot d\vec S

对于 M=EM = \mathcal E
∫∂EΦF⃗ =∫EdΦF⃗ \displaystyle \int _{\partial \mathcal E} \Phi _ \vec F = \int _ \mathcal E d \Phi_\vec F
∫∂EF⃗ ⋅dS⃗ =∭E(∇⋅F⃗ ) dxdydz\displaystyle \int _{\partial \mathcal E} \vec F \cdot d \vec S = \iiint _ \mathcal E(\nabla \cdot \vec F)\ dxdydz

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