Final answer:
To calculate the Jacobian determinant for the transformation given by x = 2u - 1, y = 3v - 4, z = (u w) / 2, we find the partial derivatives, construct the Jacobian matrix, and calculate its determinant, which results in a Jacobian determinant of 3u.
Step-by-step explanation:
The question requires us to calculate the Jacobian determinant of the transformation given by x = 2u - 1, y = 3v - 4, z = (u w) / 2. The Jacobian determinant reflects how a function transforms the area (or volume in three dimensions) near a point in space. To calculate the Jacobian determinant, we construct the Jacobian matrix by taking the partial derivatives of each function with respect to each variable (u, v, w) and then calculate the determinant of that matrix.
First, let's find the partial derivatives and construct the Jacobian matrix:
- ∂x/∂u = 2, ∂x/∂v = 0, ∂x/∂w = 0
- ∂y/∂u = 0, ∂y/∂v = 3, ∂y/∂w = 0
- ∂z/∂u = w/2, ∂z/∂v = 0, ∂z/∂w = u/2
The resulting Jacobian matrix is:
J =
| 2 0 0 |
| 0 3 0 |
| w/2 0 u/2 |
Now we calculate the determinant:
det(J) = 2 * (3 * (u/2)) - 0 - 0
det(J) = 3u
Thus, the Jacobian determinant of the given transformation is 3u.