1. Find the image of the region S under the given transformation T. (a) S = {(u, v) |0

1. Find the image of the region S under the given transformation T. (a) S = {01, v) O < u 1, O 2} and T (u, v) = (u + v, (b) S is the parallelogram with the vertices (1, 3), (1, 3), (3, 1), (1, 5) and (c) S = { (u, v) O < u 1, 1} and T (u, v) = 2. Compute the Jacobian matrix and the Jacobian of each transformation = (uev, veu), (v sin(u2), cos(u2)). 3. Let R be the parallelogram in the .ry-plane with the vertices (O, O), (4, 3), (2, 4), (2, 1) Find a transformation T : (u, v) (:r, y) so that R is the image under a linear trasformation T of a rectangular region in the uv-plaue (with sides parallel to the u, v-axes). 4. Recall that a transformation is linear if it is of the form T (u, v) = (au + bv, cu -F dv). Justify the claim that "If ad bc O, then the image of any parallelogram under T is a parallelogram" , or give a counterexample to the claim. 5. (a) Determine where functions w = w(y, z) and = z) are defined implicitly by the system wa:yz = 1 Your answer should be in the form of some condition(s) on the variables w, x, y, z that must be satisfied in order to have w(y, z), .r(y, z) implicitly defined. (b) Compute , where they exist.