Consider the equations

3xyz = w^3 , xy = z^2 + w^2 + 1 (name those as (1))

and let

S := {(x, y, z, w) R^4 : 3xyz = w^3 , xy = z^2 + w^2 + 1}

C := {(x, y, 0, 0) R^4 : xy = 1} S.

(a) Fix p S \ C. Explain, using the Implicit Function Theorem applied to f(x, y, z, w) := (3xyz w^3 , z^2 xy + w^2 ), that it is possible to solve (1) for z and w in terms of x and y near p. Call the resulting functions z(x, y) and w(x, y) and compute z/y and w/y in terms of x, y, z(x, y) and w(x, y).

(b) Show that it is not possible to solve (1) for x and y in terms of differentiable functions of z and w near any point p S.

(c) Fix p C. Show that it is possible to solve (1) for y and z in terms of x and w near p.

Remark. Parts (a) and (c) show that the rank of f(x, y, z, w) is equal to 2 at all points of S and that therefore S is a submanifold of R^4 of dimension 2 (=4-2), i.e., S is a regular surface in R^4 . This remark is meant to reinforce Proposition 6.5.5, but you do not need to understand this proposition to answer this question.