1. Chapter 18 Section 18.5 Green’s Theorem

2. Closed Curves and Simple Closed Curves Closed curves are figures that can be drawn so that you begin and end at the same place. The “curve” can be either a straight line or a curve. A simple closed curve is a closed curve that does not cross over itself. This is also known as a Jordan curve. Closed curve, not simple Simple curve, not closed Curve, not simple and not closed Simple closed curve Positively Oriented Curves A simple closed curve separates the plane into 3 parts, the interior, the exterior and the curve itself. A curve is positively oriented if you “walk” along the curve in one direction the interior will be on your left. Sometimes this is called a “counterclockwise” orientation but for some curves this can be misleading like spirals. If an enclosed region has a boundary that is made up of more than one simple closed curve the boundary is positively oriented if the region is on the left as you “walk” along the curve. A negatively oriented curve goes in the opposite direction. Positively Oriented Curves

3. This can be applied to make the computation of paths integrals easier and quicker. This is especially true when the boundary of the region is made up of multiple paths. This condenses it down to a single double integral instead of parameterizing multiple paths and then integrating.

4. x y

5. x y Notice that the answer is the same as before, but we only needed to compute one double integral instead of three single integrals and we did not need to parameterize any of the curves.

8. Notice again the value of the single integral is equal to the value of its equivalent double integral.

9. Convert to Polar !!!

10. x y a b A parameterization is as follows: