Today we went over line integrals and a new method of evaluating them. Usually, we find the integral and break it down into a piecewise curve. However, if we think of this curve as a region, we can then use a double integral to find it out. Green's theorem can be defined as follows:
Let R be a simply-connected region with a piecewise smooth boundary C, with positive orientation. If L and M have continuous partial derivatives, then
Let R be a simply-connected region with a piecewise smooth boundary C, with positive orientation. If L and M have continuous partial derivatives, then
- simply-connected meaning region has no holes in it and doesn't cross itself
- piecewise smooth meaning the region is continuous with a finite number of cusps
- positive orientation meaning counter clockwise
A question that was posed was, "Why does positive orientation mean counterclockwise direction?" A possible answer may be because that's the order of the four quadrants (quadrant 1 (top right) to quadrant 4 (bottom right)). This is reasonable, but apparently, there is no answer to that question. It's just conventional.
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