Math454 - Feb7

Math454 - Feb7 - Math 454 7 February 2008 Isoperimetric...

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Unformatted text preview: Math 454 7 February, 2008 Isoperimetric Inequality : 4 πA ≤ L 2 . This is for a simple closed curve (positively oriented). Jordan Curve Theorem : Such a curve (simple closed) divides the plane into two regions: abounded interior and unbounded exterior . Using Green's Theorem, we see that A = ´ xdy =- ´ ydx = 1 2 ´ ( xdy- ydx ) . Let ( x,y ) = ( x ( t ) , y ( t )) be periodic functions of t with some period; Then Intgerate over the period of t. Proof of the Isoperimetric Inequality : (E. Schmidt) Suppose C is a simple closed curve that is at least C 1 and is positively oriented. Then, one can compare C with an appropriate circle. Establish and upper bound for A + πR 2 in terms of L using Green's Theorem. Step 1 : parametrize C wth arc length S with x ( s ) , y ( s ) with unit length parametrization (i.e., x 2 + y 2 = 1 ). Then, express the y-coordinate of the circle in terms of the x-coordinate so that: Step 2 : A = ´ xdy ⇒ πR 2 =- ´ yd x , if (¯ x ( s ) , ¯ y ( s )) parametrizes the circle.parametrizes the circle....
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Math454 - Feb7 - Math 454 7 February 2008 Isoperimetric...

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