Other ways of writing Green's theorem

In the introductory reading on Green's theorem, we wrote Greens theorem as

$\displaystyle \int_{{C}}^{}$F . ds = $\displaystyle \iint_{D}^{}$(curl F) . k dA    

or

$\displaystyle \int_{{C}}^{}$F . ds = $\displaystyle \iint_{D}^{}$$\displaystyle \left(\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right.$$\displaystyle {\frac{{\partial F_2}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial F_1}}{{\partial y}}}$$\displaystyle \left.\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right)$dA,    

where F(x, y) = (F1(x, y), F2(x, y)) is a two-dimensional vector field, D is a region in plane, and C is its positively oriented boundary. I used this notation to tie Green's theorem in with the concept of the curl of a vector field.

We often denote C by $ \partial$D to make it explicit that the curve C is the (positively oriented) boundary of D. This notation is also more natural when the region D has more than one boundary component. Then, Green's theorem can look like, for example,

$\displaystyle \int_{{\partial D}}^{}$F . ds = $\displaystyle \iint_{D}^{}$$\displaystyle \left(\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right.$$\displaystyle {\frac{{\partial F_2}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial F_1}}{{\partial y}}}$$\displaystyle \left.\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right)$dA,    

Note that the notation $ \partial$D simply means the boundary of D. It has nothing to do with a partial derivative.

However, people frequently write Green's theorem differently. First, they like to change the formula by writing the line integral at the left in terms of components:

$\displaystyle \int_{{\partial D}}^{}$F1 dx + F2 dy = $\displaystyle \iint_{D}^{}$$\displaystyle \left(\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right.$$\displaystyle {\frac{{\partial F_2}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial F_1}}{{\partial y}}}$$\displaystyle \left.\vphantom{ \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}}\right)$dA.    

Then, they like to let the vector field be F(x, y) = (P(x, y), Q(x, y)), so that Green's theorem becomes

$\displaystyle \int_{{\partial D}}^{}$P dx + Q dy = $\displaystyle \iint_{D}^{}$$\displaystyle \left(\vphantom{ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right.$$\displaystyle {\frac{{\partial Q}}{{\partial x}}}$ - $\displaystyle {\frac{{\partial P}}{{\partial y}}}$$\displaystyle \left.\vphantom{ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right)$dA.    

Sometimes, F = (P, Q) won't be referred to as a vector field. Instead, one can discuss the above version of Green's theorem as applied to the two scalar valued functions Q : D$ \to$R and P : D$ \to$R.

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Duane Nykamp
nykamp@math.umn.edu
2007-03-19
06036 hits since
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