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