= 7. This problem can be conceptually framed in terms of either physics or economics. (Physics version) In four-dimensional spacetime, a surface is specified as the intersection of the hypersphere x2 + y2 + 2 = 2 2 and the hyperplane 3.x + 2y + 2 - 2t = 2. (Economics version) A resource is consumed at rate t to produce goods at rates x, y, and z; production is constrained by the equation r + y2 + 22 = t2 2. Furthermore, the expense of extracting the resource is met by selling the goods, so that 2t 3.r + 2y + 2 - 2. = - In either case, we have a manifold that is the locus of F 0-67232) = (x2 + y2 + 22 +2 +2 3.0 + 2y +z - 2t - 2 = = 0. (a) Show that this surface is a smooth 2-dimensional manifold. (b) One point on the manifold is Y 2 t 2 3 4 Near this point the manifold is the graph of a function g that expresses x and y as functions of z and t. Using the implicit function theorem, determine (Dg) at the point z = 3,t = 4