332 FOR F&Auml;&deg;NAL1

# 332 FOR F&Auml;&deg;NAL1 - T Liggett Mathematics...

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T. Liggett Mathematics 131C – Midterm Solutions May 5, 2010 (25) 1. Consider solving the equations u 3 + xv - y = 0 v 3 + yu - x = 0 (1) for u,v in terms of x,y . (a) Show that there are diﬀerentiable functions u ( x,y ) and v ( x,y ) deﬁned in a neighborhood N of ( x,y ) = (0 , 1) so that u (0 , 1) = 1 ,v (0 , 1) = - 1, and u = u ( x,y ) ,v = v ( x,y ) solve (1). Let h ( x,y,u,v ) = ( u 3 + xv - y,v 3 + yu - x ) . Then h 0 = ± v - 1 3 u 2 x - 1 u y 3 v 2 ² . The assumption in the implicit function theorem is that the matrix ± 3 u 2 x y 3 v 2 ² be invertible at ( x,y ) = (0 , 1). At this point, the determinant of this matrix is 9, so the assumption is satisﬁed. (b) Let f ( x,y ) = ( u ( x,y ) ,v ( x,y )) for ( x,y ) N . Compute f 0 ( x,y ) in N . Since h (( x,y ) ,f ( x,y )) = 0 in N , the chain rule gives ± v - 1 - 1 u ² + ± 3 u 2 x y 3 v 2 ² f 0 ( x,y ) = 0 . Solving gives f 0 ( x,y ) = 1 9 u 2 v 2 - xy ± - 3 v 3 - x 3 v 2 + ux 3 u 2 + vy - 3 u 3 - y ² . (30) 2. In each case, decide whether the statement is true or false. If true,

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332 FOR F&Auml;&deg;NAL1 - T Liggett Mathematics...

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