Engineering Calculus III

Question 1. Consider the transformation T defined by u(x, y) = 2x +
2x
x2+y2 , v(x, y) = 2y −
2y
x2+y2 .
(i) Show that T maps the exterior of the disk of radius 2, D := {(x, y) | x
2 + y
2 ≥ 4}, onto the exterior
of the ellipse {(u, v) | u
2/25 + v
2/9 ≥ 1}. Compute the Jacobian of the this transformation.
(ii) Use this transformation to compute the area enclosed between the curves u
2/25 + v
2/9 = 1,
u
2/(17/2)2 + v
2/(15/2)2 = 1.