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Unformatted text preview: Homework 6 – Math 118C, Spring 2010 Due on Tuesday, May 18th, 2010 1. Define f in R 2 by f ( x,y ) = 2 x 3 − 3 x 2 + 2 y 3 + 3 y 2 . (a) Find the four points in R 2 at which the gradient of f is zero. Show that f has exactly one local maximum and one local minimum in R 2 . (b) Let S be the set of all ( x,y ) ∈ R 2 at which f ( x,y ) = 0. Find those points of S that have no neighborhoods in which the equation f ( x,y ) = 0 can be solved for y in terms of x (or for x in terms of y ). Describe S as precisely as you can. 2. Define f ( x,y ) = braceleftBigg xy ( x 2 − y 2 ) x 2 + y 2 ( x,y ) negationslash = (0 , 0) , ( x,y ) = (0 , 0) . Prove that (a) f , D 1 f , and D 2 f are continuous in R 2 . (b) D 12 f and D 21 f exists at every point of R 2 , and are continuous except at (0 , 0). (c) D 12 f (0 , 0) = 1, and D 21 f (0 , 0) = − 1. 3. In this problem you are asked to prove the following existence theorem for Ordinary Differential Equations, known as Peano’s theorem: Theorem 0.1 (Peano’s Existence Theorem)Theorem 0....
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 Fall '09
 Garcia
 Math, Trigraph, Continuous function, dω, dy ∧ dz, dx ∧ dy

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