Final answer:
The Jacobian determinant is found by computing partial derivatives of x, y, and z with respect to u, v, and w, organizing them into a matrix, and calculating the determinant of that matrix.
Step-by-step explanation:
The student is asking to find the Jacobian matrix of the transformation given by the functions for x, y, and z with respect to the variables u, v, w. The Jacobian determinant is a measure of how a multivariable function, such as the one given, changes as the input variables change. We compute this by finding the partial derivatives of each of the functions x, y, and z with respect to each of the variables u, v, and w, and then arranging these derivatives in a matrix format.
The provided functions are:
x=-(6 u+4 u v), y=-(3 u v+8 u v w), and z=5 u v w. Following that, the partial derivatives for each function with respect to each variable are found:
- ∂x/∂u = -6 - 4v
- ∂x/∂v = -4u
- ∂x/∂w = 0
- ∂y/∂u = -3v - 8vw
- ∂y/∂v = -3u - 8uw
- ∂y/∂w = -8uv
- ∂z/∂u = 5vw
- ∂z/∂v = 5uw
- ∂z/∂w = 5uv
The determinant of the matrix composed of these partial derivatives gives us ∂(x, y, z)/∂(u, v, w), which is the required solution. To compute the Jacobian determinant, we would arrange these into a matrix and then calculate the determinant of that matrix.