Prelim2 nov. 2003 - A&EP 321 Mathematical Physics...

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Mathematical Physics Prelim #2 November 20, 2003 7:30-9:30 p.m. CLOSED BOOK NO CALCULATORS
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1. For this problem use Schwartz-Christoffel mapping techniques. Consider the 30% mapping function z = z ( w ) with dz dw = 1 ( w - 1) 2 / 3 ( w + 1) 2 / 3 and the mapping of the real w -axis onto the z -plane. a. Show that the expression z = Z ω - 1 dx ( x - 1) 2 / 3 ( x + 1) 2 / 3 will satisfy the above differential equation. b. Using the expression in part (a) above, locate the mapping of the point w = w 1 = - 1 onto the z -plane. Call this point z 1 . c. Let w = w 2 = +1 with w 2 mapping to z 2 . z 2 will lie along the mapping of the line segment from w 1 to w 2 along the real w -axis. How does this line segment map onto the z -plane? Note that so far the length of this mapped line segment and consequently the exact location of z 2 in the z -plane is unknown. d. Show that the distance form z 1 to z 2 is given by: Z 1 0 2 dx (1 - x 2 ) 2 / 3 . This integral is approximately equal to 4.
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This note was uploaded on 11/18/2008 for the course A&EP 321 taught by Professor Kusse during the Fall '07 term at Cornell University (Engineering School).

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Prelim2 nov. 2003 - A&EP 321 Mathematical Physics...

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