Monday, May 23, 2011

How do we turn line integrals into region integrals?

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 

  • 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
This provides us with a much easier method to solve some problems. Go try it out on some old homework! An efficient method of turning 40 minutes of long winded problems into 5 minutes of integration!

1 comment:

  1. 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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