Homework 12

Homework 12 - APPM 2350 HOMEWORK 12 Due Friday, November...

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APPM 2350 HOMEWORK 12 FALL 2010 Due Friday, November 19, 2010 at 4 pm under your TA’s door. 1. Let ° F ( x, y )= y x 2 + y 2 ˆ i + x x 2 + y 2 ˆ j. (a) Is ° F ( x, y ) continuous for all x and y ? If not, then where is it discontinuous? (b) Using the convention ° F ( x, y )= M ( x, y ) ˆ i + N ( x, y ) ˆ j , show that M y = N x . (c) If C is any closed contour not around the origin, ±nd ° C ° F · d°r , and justify your answer. (d) Let C ° be x 2 + y 2 = ± 2 , ±> 0, where you move in a counter-clockwise fashion around the circle. Now ±nd ° C ° ° F · d°r . Note, your answer should be independent of ± . How can you reconcile your result with the fact that M y = N x ? 2. Now keep ° F the same as the previous problem, but let C be some arbitrary curve around the origin in the xy-plane. Imagine something like the ±gure shown immediately below. ! " # We are now going to ±gure out how to ±nd ° C ° F · d°r . To do this, we ±rst look at a similar but slightly di²erent curve as shown in the picture below.
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Homework 12 - APPM 2350 HOMEWORK 12 Due Friday, November...

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