L24 Math 233 Lecture Notes - Lecture 44: Piecewise, Simply Connected Space, Curve
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L24 math 233 lecture 44- proof and application of green"s theorem. Theorem: if c is a simple, piecewise, smooth, closed curve oriented positively, and f (x, ) y + q y (x, )j y = p (x, )i then. Note: if we start with an open, connected, and simply connected domain d with a piecewise smooth boundary, Proof: special case where d is open, connected, simply connected, and both type i and. Type ii with piecewise smooth boundary functions call such d is a simple domain. { x 1 1 x2 y 1 x2. { 1 y 1 1 y2 x 1 y2. Proof of claim: write d as a type i domain a. D (x, )dydx [p (x, b a b a g (x) Y g2 (x)} (x, g1 (x)]dx. For the line integral, write the boundary curve as c = c1 . Divide any not simple domain into simple domains.
