Amap of the arctic may be defined by a stereographic projection from the south pole as follows. each point on the surface s(0, r) has coordinates (x, y, z) such that x2 + y2 + z2-r" earth's surface may be approximated by a sphere, s(0,r), with radius r > 0 or r = 1 centered at the origin elsewhere, but doing so only complicates the formulae. each point on the map r2 has coordinates (u, v) or (u, v,0) (z, 3, 2) on the map r? is the point (u, v,0) -(u(, v(x, y, z),0) that is at the intersection of the equatorial for each point (u, v) on the map r2, the inverse stereographic projection of (u, v) is the point (x, y, z) 0. the map may be scaled and moved the map is the equatorial plane, r2, which is the plane in space where z except for the south pole (0,0, -r), for each other point (x, y, z) on s (0, r), the stereographic projection of plane r2 and the straight line through the south pole (0, 0, -r) and the point (x, y,z) plu, v) = (x(u, u), y(u, u), z(u, u)) that is the intersection of the sphere s(0, r) and the straight line through the south pole (0, 0,-r) and the point (u, u, 0). problem 1 find formulae for p(u, v) (x(u, v), y (u, v), z(u, v)) find a formula for ds-1 ×噐11 dudu. if d is the map of a country c, then the area of d on the map r2 is given by the integral formula 1 dudv. however, the area of d on the map r2 is not necessarily the area of the country c on the sphere s(0, r) find an integral formula for the area of c of the type jjd f(u, v) dudv. find f(u, v). in other words, if you are looking at the map, how can you compute the area of the country c from its map d?