Difference between revisions of "Green's Theorem"
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Latest revision as of 11:12, 30 May 2019
Green's Theorem is a result in real analysis. It is a special case of Stokes' Theorem.
Statement
Let 
be a bounded subset of 
with positively
oriented boundary 
, and let 
and 
be functions with
continuous partial derivatives mapping an open set containing
into 
. Then
![\[\int_C \bigl[ L(x,y)dx + M(x,y)dy \bigr] = \iint_D \left( \frac{\partial M} {\partial x} - \frac{\partial L}{\partial y} \right) dA.\]](http://latex.artofproblemsolving.com/7/d/6/7d61382f7bcb33ab07d9e764ec3c788184718e0f.png)
Proof
It suffices to show that the theorem holds when is a square,
since can always be approximated arbitrarily well with
a finite collection of squares.
Then let be a square with vertices 
, 
, 
,
, with 
and 
. Then
![\begin{align*} \int_C \bigl[ L(x,y)dx + M(x,y)dy \bigr] ={}& \int_{a}^b L(x,c) dx + \int_{c}^d M(b,y)dy \\ &+ \int_b^a L(x,d) dx + \int_{d}^c M(a,y) dy \\ ={}& \int_c^d \bigl[ M(b,y) - M(a,y) \bigr] dy + \int_a^b \bigl[ L(x,d) - L(x,c) \bigr] dx . \end{align*}](http://latex.artofproblemsolving.com/0/4/0/040b5d9e15a5a66e0e005286bb1cdd8089dc6d3e.png)
Now, by the Fundamental Theorem of Calculus,
![\[L(x,d) - L(x,c) = \int_c^d \frac{\partial L}{\partial y}(x) dy\]](http://latex.artofproblemsolving.com/0/c/2/0c220a4abfd97459d709a3aedf19d796f80b72ba.png)
and
![\[M(b,y) - M(a,y) = \int_a^b \frac{\partial M}{\partial x}(y) dx .\]](http://latex.artofproblemsolving.com/e/a/a/eaa107544a06a0147325a0db85a5c59ac0020df9.png)
Hence
![\begin{align*} \int_c^d \bigl[ M(b,y) - M(a,y) \bigr] dx - \int_a^b \bigl[ L(x,d) - L(x,c) \bigr] dy &= \int_c^d \int_a^b \frac{\partial M}{\partial x} dx dy - \int_a^b \int_c^d \frac{\partial L}{\partial y} dy dx \\ &= \iint_D \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA , \end{align*}](http://latex.artofproblemsolving.com/1/e/1/1e1ba3c617c9d26da07565f44d14c3a4afb923e9.png)
as desired. 
