# Changing the direction of an integral

1. Feb 16, 2009

### rsq_a

Very simple question: Suppose I have a region, D, defined by an outer boundary $$\partial D$$, but with a hole (like an annulus, for example), defined by $$\partial D_2$$.

If I were to apply Green's Theorem to this region, coupled with a vector field, it would give me something like,

$$\text{Area Integral} = \int_{\partial D} + \int_{\partial D_2}$$

where by the conditions of the theorem, the outer contour should go counterclockwise and the inner contour should go clockwise.

My question is, can we do this?

$$\text{Area Integral} = \int_{\partial D} - \int_{\partial D_3}$$

where now $$\partial D_3$$ is the same contour as $$\partial D_2$$ but going clockwise.