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hw4 - Modern Analysis Homework 4 1 Suppose that f R2 R2 is...

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Modern Analysis, Homework 4 1. Suppose that f : R 2 R 2 is defined by f ( x, y ) = ( x 2 + xy, y ). Prove directly from the definition that f is differentiable at each point of R 2 . 2. Rudin, Ch 9 # 14: Define f (0 , 0) = 0 and f ( x, y ) = x 3 x 2 + y 2 if ( x, y ) 6 = (0 , 0) . (a) Prove that D 1 f and D 2 f are bounded functions in R 2 . (Hence f is continuous.) (b) Let ~u be any unit vector in R 2 . Show that the directional derivative ( D ~u f )(0 , 0) exists, and that is absolute value is at most 1. (c) Let γ be a differentiable mapping of R into R 2 (in other words, γ is a differentiable curve in R 2 ), with γ (0) = (0 , 0) and | γ 0 (0) | > 0. Put g ( t ) = f ( γ ( t )) and prove that g is differentiable for every t R . If γ C 1 ( R , R 2 ), prove that g C 1 ( R , R ). ( Note: Rudin does not assume the following, but you may: for all t R at which γ ( t ) = (0 , 0), we have | γ 0 ( t ) | > 0. I suspect, that by looking at a function such as the monster in 4a, that this assumption really is necessary.) (d) In spite of this, prove that f is not differentiable at (0 , 0). ( Hint: Writing ~u in components: ~u = u i e i , the formula ( D ~u f )( x ) = X ( D i f )( x ) u i does not hold.) 3. Rudin, Ch 9 #14 (mostly): Define f (0 , 0) = 0, and put f ( x, y ) = x 2 + y 2 - 2 x 2 y - 4 x 6 y 2 ( x 4 + y 2 ) 2 if ( x, y ) 6 = (0 , 0). (a) Prove that for all ( x, y ) R 2 we have 4 x 4 y 2 ( x 4 + y 2 ) 2 . Conclude that f is continuous. (b) For 0 θ 2 π , -∞ < t < , define g θ ( t ) = f ( t cos θ, t sin θ ) .
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