Green’s Theorem  (continued)

 

 

Green’s theorem can also be expressed in its “divergence form”.

 

 

 

 

Let   n  be the unit outward normal to the curve, C. 

Here  n  =  dx i  -  dy j

 

  and                         F  =  P i  +  Q  j

 

So   F  .  n  =   P dx  -  Q dy  

 

 ∫ F  .  n  ds  =   ∫  P dx – Q dy  =   ∫   ∫  [∂Q/∂x  +  ∂P/∂y]

           C                     C                             R

 

Finally    ∫ F  .  n  ds   =   ∫  ∫  div F  dA

               C                        R

 

 

 

 

 

 

 

    

Summary:

 

 

 

Green's Theorem in Standard Form:  

 

 

 

    ∫ P dx + Q dy  =  ∫    ∫  [ ∂Q/∂x  ˗  ∂P/∂y ] dA

   C                            R

 

 

 

 

 

Green's Theorem in Curl Form:

       (vector form)

 

  

    ∫ F  .  dr   =    ∫ F  . T ds   =    ∫    ∫  curlz F   dA     

   C                     C                        R

 

 

    ∫ F  .  dr     =    ∫    ∫  (curl F ) ·  k  dA     

   C                         R

 

 

 

Green's Theorem in Divergence Form:

             (vector form)

 

 

 

     ∫ F  .  n  ds   =   ∫   ∫  div F  dA

    C                         R

 

 

 

Click here for four examples involving Green’s Theorem.

 



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