Question

A26.4 (i) (4 marks) When u = xy and v= y/x, compute the Jacobian determinants ə(u, v) Ə(x, y) (x, y > 0). Ə(x, y)' ə(u, v) (ii) (6 marks) Find the area of the region R in the positive quadrant that is bounded by the curves xy = a, xy = b; y = (1/2)x, y = 2x, where 0 < a < b are constants.

Answer 1

To compute the Jacobian determinants a(u, v) and a(x, y), we need to find the partial derivatives of u and v with respect to x and y. Let's start with the first part:

Given:
u = xy
v = y/x

To find a(u, v) / a(x, y), we need to compute the following partial derivatives:

∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y

∂u/∂x = y
∂u/∂y = x
∂v/∂x = -y/x^2
∂v/∂y = 1/x

Now, let's compute the Jacobian determinant a(u, v) / a(x, y):

a(u, v) / a(x, y) = (∂u/∂x * ∂v/∂y) - (∂u/∂y * ∂v/∂x)
= (y * 1/x) - (x * (-y/x^2))
= y/x + y/x
= 2y/x

For the second part, we need to find the area of the region R bounded by the curves xy = a, xy = b, y = (1/2)x, and y = 2x, where a