Given F_1(x, y, z, u, v, w) | = | 0 F_2(x, y, z, u, v, w) | = | 0 F_3(x, y, z, u, v, w) | = | 0, if the determinantof the Jacobian left bracketing bar J F(u, v, w) right bracketing bar = left bracketing bar (d(F_1, F_2, F_3))/(d(u, v, w)) right bracketing bar !=0, then u, v, and w can be solved for in terms of x, y, and z and partial derivatives of u, v, w with respect to x, y, and z can be found by differentiating implicitly.

change of variables theorem | Jacobian