Partial Integration Partial Integration
Partial integration in  Rn  Partial integration in  Rn 
CAUCHY's theorem CAUCHY's theorem
CAUCHY's theorem Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany

GREEN's formula

english This is an english version of the script
You may switch to the original german version: german


GREEN's formula    [sect:6-9]
Let  Ω⊂Rn  be a bounded GAUSS domain (see definition definition:6-7). Then the following holds:
  • [sect:6-9-(1)] If  u,v∈C1(clos(Ω)) ,  u∈C2(Ω)  with  Δu∈L1(Ω) , then
     
    ∫
    Ω
    		 ∇v•∇u dLn =
     
    ∫
    ∂Ω
    		 v∇u•νΩ dHn-1 -
     
    ∫
    Ω
    		 vΔu dLn .
  • [sect:6-9-(2)] For  u,v∈C1(clos(Ω))∩C2(Ω)  with  Δu, Δv∈L1(Ω) 
     
    ∫
    Ω
    		 (u∇v - v∇u)•νΩ dHn-1 =
     
    ∫
    Ω
    		 (uΔv -vΔu) dLn .

Proof sect:6-9-(1). Let  f:=v∇u . Then  f  is continuously differentiable with
div (f) =
n
∑
i=1
 ∂i(v∂iu) =
n
∑
i=1
 (∂iv∂iu+v∂i2u) = ∇v•∇u+vΔu.
Then the assertion follows from the divergence theorem satz:6-7.
Proof sect:6-9-(2). Apply sect:6-9-(1) to  u,v  and to  v,u  and subtract.

Version 1.7
H.W. Alt - 02.01.2007

Partial Integration Partial Integration
Partial integration in  Rn  Partial integration in  Rn 
CAUCHY's theorem CAUCHY's theorem
CAUCHY's theorem Index
© 2001-2007 Prof. Dr. Hans Wilhelm Alt, University Bonn, Germany