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...R rr oliver knill harvard summer school c f dr y dydx x example in class find the area of region en closed by sin t chapter integral theorems h i for to do so use greens theorem with vector eld xi section why is true look rst at a small square g if hp q two dimensions and hx line qi along boundary p dt curve rp rq this measures circulation place because z b about same as curl dxdy all identities hold limit prove statement general chop it into squares size summing up integrals around boundaries gives work done path interior cancel vorticities on rectangles riemann sum approximation double dimensional dened scalar thecurl vorticity one can write cross product examples we have then zero clairot s yx xy green which parametrized that left allows express coordinates centroid center mass xda da george was physicist nineteenth century he self taught mathematician miller his greatly contributed modern physics using lets special case when gradient both sides are fundamental gcurl an important ap...