Surfaces x^2+y^2+z^2+w^2=1 and x+2y+3z+4w=2 define implicitly functions x=x(y,z) and w=w(y,z). Show that (deltax)/(deltay)=(2x−4y)/(4x−w) when 4x!=w

Stephany Wilkins

Stephany Wilkins

Answered question

2022-10-30

Surfaces x 2 + y 2 + z 2 + w 2 = 1 and x + 2 y + 3 z + 4 w = 2 define implicitly functions x = x ( y , z ) and w = w ( y , z ). Show that δ x δ y = 2 x 4 y 4 x w when 4 x w

Attempt to solve
Now i am not quite sure what i am suppose to do here ? I suppose compute δ x δ y via implicit-differentiation since both of these surfaces are given in implicit form. Another problem is that i am suppose to form system of equations from two of these surfaces since x = x ( y , z ) and w = w ( y , z ) are referring to same function ? Something like this ?
F ( x ( y , z ) , y , z , w ( y , z ) ) = { x 2 + y 2 + z 2 + w 2 = 1 x + 2 y + 3 z + 4 w = 2
I don't think i have very good understanding on what i am actually suppose to do here so if someone can hint me in right direction / maybe try to explain on what i am suppose to do. I don't expect someone to handover complete solutions.

Answer & Explanation

yorbakid2477w6

yorbakid2477w6

Beginner2022-10-31Added 12 answers

Hint:

(1) Write x , w as functions and the other variables as... variables:
x ( y , z ) 2 + y 2 + z 2 + w ( y , z ) 2 = 1 ,
x ( y , z ) + 2 y + 3 z + 4 w ( y , z ) = 2.
(2) Apply y ,... to each equation:
2 x ( y , z ) x y ( y , x ) + 2 y + 2 w ( y , z ) w y ( y , x ) = 0 ,

(3) Solve the (linear) system with unknowns x y , w y , x z , w z .

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