Final answer:
The Jacobian of the transformation with x = 2u⁶v and y = 4u⁸v is -16u¹³v, which indicates a scaling of areas in the u-v plane when transformed to the x-y plane. The provided options do not include the correct answer, suggesting a possible error or typo in the provided choices.
Step-by-step explanation:
The Jacobian of a transformation is a determinant that gives us a measure of the change of variables in a multiple integral. When dealing with a Jacobian for a transformation from variables (u, v) to (x, y), where x and y are defined as x = 2u⁶v and y = 4u⁸v, we need to construct a matrix with the partial derivatives of x and y with respect to u and v and then calculate the determinant of this matrix.
The matrix for the Jacobian is:
- ∂x/∂u = 12u⁵v
- ∂x/∂v = 2u⁶
- ∂y/∂u = 32u⁷v
- ∂y/∂v = 4u⁸
Calculating the determinant of this matrix gives us:
J = (∂x/∂u)(∂y/∂v) - (∂x/∂v)(∂y/∂u) = (12u⁵v × 4u⁸) - (2u⁶ × 32u⁷v) = 48u¹³v - 64u¹³v = -16u¹³v.
Therefore, the Jacobian is -16u¹³v, which is not present in options a), b), c), or d). Therefore, it appears there may be a typo within the provided choices since the correct Jacobian is not listed.