test1 rev sol

Test1 rev sol - Math 136 Answers to Review Problems for Test 1 1 We calculate D f using the chain rule and the fact that F x,y,f x,y = 0 We define

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 136, Answers to Review Problems for Test 1 1. We calculate D f using the chain rule and the fact that F ( x,y,f ( x,y )) = 0. We define G ( x,y ) = x y f ( x,y ) and note that D g = 1 1 ∂f ∂x ∂f ∂y . We know G is differentiable since its coordinate functions are C 1 (the first two coordinate functions are C ∞ , and the third, f , is C 1 since F and ( x ,y ,z ) satisfy the Implicit Function Theorem). Using the chain rule on the equation F ( x,y,f ( x,y )) = 0, we get = D 0 = D ( F ( x,y,f ( x,y ) ) = D ( F ◦ G ) = D F · D G where · is matrix multiplication. Now, using the fact that D F ( x,y,z ) = ( ∂F ∂x , ∂F ∂y , ∂F ∂z ) we see (1) 0 = ∂F ∂x + ∂F ∂z ∂f ∂x , ∂F ∂y + ∂f ∂y ∂F ∂z . We can solve (1) for ∂f ∂x and ∂f ∂y and this give ∂f ∂x =- ∂F ∂x ∂F ∂z and ∂f ∂y =- ∂F ∂y ∂F ∂z evaluated at ( x ,y ,f ( x ,y )). Here we used that ∂F ∂z ( x ,y ,z ) 6 = 0 (and z = f ( x ,y )). 2. The function F in this problem satisfies the hypotheses of the Implicit Function Theo- rem, so we can assert there is an open neighborhood U of x and an open neighborhood V of y and a C 1 function f : U → V such that F ( x,f ( x )) = F ( x ,y ) for all x ∈ U . (a) So, as there are an infinite number of points in U , and for each x ∈ U , F ( x,f ( x )) = F ( x ,y ), there are an infinite number of points that satisfy F ( x,y ) = F ( x ,y )....
View Full Document

This note was uploaded on 03/27/2008 for the course MATH 136 taught by Professor Quinto during the Spring '08 term at Tufts.

Page1 / 3

Test1 rev sol - Math 136 Answers to Review Problems for Test 1 1 We calculate D f using the chain rule and the fact that F x,y,f x,y = 0 We define

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online