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MIT18_024S11_Pset7

# MIT18_024S11_Pset7 - PSET 7 DUE MARCH 31 1 9.8:7(6 points...

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PSET 7 - DUE MARCH 31 1. 9.8:7 (6 points) Hint: It might help to define a scalar field F ( x, y, z ) = f ( u ( x, y, z ) , v ( x, y, z )) where u, v are as needed. 2. Let f : R m + n R m be continuously differentiable and let x R n , y R m . De- note by D f = D f x + D f y the decomposition of the Jacobian such that for h R n , k R m , D f ( x , y )( h , k ) = D f x ( x , y ) h + D f y ( x , y ) k . (That is D f x ( x , y ) : R n R m and D f y ( x , y ) : R m R m as mentioned in class.) Suppose for a R n , b R m , f ( a , b ) = 0 and det ( D f y ( a , b )) = 0. We consider a few steps of the implicit function theorem in this setting. Let F ( x , y ) : R m + n R m + n such that F ( x , y ) = ( x , f ( x , y )). Write down the matrix DF as we described it in class. You may write it in block decomposition, but also explain how you produce each block! (2 points) Prove DF is invertible at ( a , b ). (3 points) Using the inverse function theorem, we know there exists ( a , b ) V open and F ( a , b ) W open such that F : V W is invertible with continuously differ- entiable inverse G . Let U = { x R n | ( x , 0 ) W } . Prove that U is open. (3 points) BONUS

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