proof of general Stokes theorem
We divide the proof in several steps.
Step One.
Suppose M=(0,1]×(0,1)n-1 and
ω(x1,…,xn)=f(x1,…,xn)dx1∧⋯∧^dxj∧⋯∧dxn |
(i.e. the term dxj is missing). Hence we have
dω(x1,…,xn) | = | (∂f∂x1dx1+⋯+∂f∂xndxn)∧dx1∧⋯∧^dxj∧⋯∧dxn | ||
= | (-1)j-1∂f∂xjdx1∧⋯∧dxn |
and from the definition of integral on a manifold we get
∫M𝑑ω=∫10⋯∫10(-1)j-1∂f∂xj𝑑x1⋯𝑑xn. |
From the fundamental theorem of Calculus we get
∫M𝑑ω=(-1)j-1∫10⋯ˆ∫10⋯∫10f(x1,…,1,…,xn)-f(x1,…,0,…,xn)dx1⋯^dxj⋯dxn. |
Since ω and hence f have compact support in M we obtain
∫M𝑑ω={∫10⋯∫10f(1,x2,…,xn)𝑑x2⋯𝑑xnifj=10ifj>1. |
On the other hand we notice that
∫∂Mω
is to be understood as ∫∂Mi*ω where
i:∂M→M is the inclusion map.
Hence it is trivial to verify that when j≠1 then i*ω=0
while if j=1 it holds
i*ω(x)=f(1,x2,…,xn)dx2∧…∧dxn |
and hence, as wanted
∫∂Mi*ω=∫10⋯∫10f(1,x2,…,xn)𝑑x2⋯𝑑xn. |
Step Two.
Suppose now that M=(0,1]×(0,1)n-1 and let ω be any differential form.
We can always write
ω(x)=∑jfj(x)dx1∧⋯∧^dxj∧⋯∧dxn |
and by the additivity of the integral we can reduce ourself to the previous case.
Step Three.
When M=(0,1)n we could follow the proof as in the first case and end up with ∫M𝑑ω=0 while, in fact, ∂M=∅.
Step Four.
Consider now the general case.
First of all we consider an oriented atlas (Ui,ϕi) such that either Ui is the cube (0,1]×(0,1)n-1 or Ui=(0,1)n. This is always possible. In fact given any open set U in [0,+∞)×ℝn-1 and a point x∈U up to translations and rescaling it is possible to find a “cubic” neighbourhood of x contained in U.
Then consider a partition of unity αi for this atlas.
From the properties of the integral on manifolds we have
∫M𝑑ω | = | ∑i∫Uiαiϕ*𝑑ω=∑i∫Uiαid(ϕ*ω) | ||
= | ∑i∫Uid(αi⋅ϕ*ω)-∑i∫Ui(dαi)∧(ϕ*ω). |
The second integral in the last equality is zero since ∑idαi=d∑iαi=0, while applying the previous steps to the first integral we have
∫M𝑑ω=∑i∫∂Uiαi⋅ϕ*ω. |
On the other hand, being (∂Ui,ϕ|∂Ui) an oriented atlas for ∂M and being αi|∂Ui a partition of unity, we have
∫∂Mω=∑i∫∂Uiαiϕ*ω |
and the theorem is proved.
Title | proof of general Stokes theorem |
---|---|
Canonical name | ProofOfGeneralStokesTheorem |
Date of creation | 2013-03-22 13:41:43 |
Last modified on | 2013-03-22 13:41:43 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 9 |
Author | paolini (1187) |
Entry type | Proof |
Classification | msc 58C35 |