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 ≠ wAttempt to solveNow 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.

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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  ≠  wAttempt to solveNow 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&#039;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&#039;t expect someone to handover complete solutions.

yorbakid2477w6 · Accepted Answer

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  .

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

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