Compute the Jacobian $\mathrm{J}(\mathrm{u}, \mathrm{v})$ for the following transformation. \[ T: x=4 u \cos (\pi v), y=4 u \sin (\pi v) \] Choose the correct Jacobian determinant of $\mathrm{T}$ below. A. $J(u, v)=\frac{\partial}{\partial u}\left(4 u \sin (\pi v) \frac{\partial}{\partial v}\left(4 u \sin (\pi v)-\frac{\partial}{\partial v}(4 u \cos (\pi v)) \frac{\partial}{\partial u}(4 u \cos (\pi v))\right.\right.$ B. $J(u, v)=\frac{\partial}{\partial u}(4 u \cos (\pi v)) \frac{\partial}{\partial v}(4 u \sin (\pi v))-\frac{\partial}{\partial v}(4 u \cos (\pi v)) \frac{\partial}{\partial u}(4 u \sin (\pi v))$ C. $J(u, v)=\frac{\partial}{\partial u}(4 u \cos (\pi v)) \frac{\partial}{\partial v}(4 u \cos (\pi v))-\frac{\partial}{\partial v}\left(4 u \sin (\pi v) \frac{\partial}{\partial u}(4 u \sin (\pi v))\right.$ D. $J(u, v)=\frac{\partial}{\partial u}\left(4 u \sin (\pi v) \frac{\partial}{\partial v}(4 u \cos (\pi v))-\frac{\partial}{\partial v}\left(4 u \sin (\pi v) \frac{\partial}{\partial u}(4 u \cos (\pi v))\right.\right.$ Compute the Jacobian. \[ J(u, v)= \]

Step 1 ：Given the transformation T: \(x=4 u \cos (\pi v)\), \(y=4 u \sin (\pi v)\)

Step 2 ：The Jacobian matrix of a vector-valued function is a matrix of all its first-order partial derivatives. In this case, we have a transformation T from (u, v) to (x, y). The Jacobian matrix J(u, v) is then a 2x2 matrix where the first row is the partial derivatives of x with respect to u and v, and the second row is the partial derivatives of y with respect to u and v.

Step 3 ：The determinant of this matrix is the Jacobian determinant, which is what we're asked to compute.

Step 4 ：The Jacobian determinant is given by the formula: \(J(u, v) = \frac{\partial x}{\partial u} * \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} * \frac{\partial y}{\partial u}\)

Step 5 ：Compute these partial derivatives directly from the given transformation equations.

Step 6 ：For \(x = 4u\cos(\pi v)\), we have \(\frac{\partial x}{\partial u} = 4\cos(\pi v)\) and \(\frac{\partial x}{\partial v} = -4\pi u\sin(\pi v)\)

Step 7 ：For \(y = 4u\sin(\pi v)\), we have \(\frac{\partial y}{\partial u} = 4\sin(\pi v)\) and \(\frac{\partial y}{\partial v} = 4\pi u\cos(\pi v)\)

Step 8 ：Substitute these into the formula for the Jacobian determinant to get \(J = 16\pi u\sin(\pi v)^2 + 16\pi u\cos(\pi v)^2\)

Step 9 ：\(J(u, v) = 16\pi u\sin^2(\pi v) + 16\pi u\cos^2(\pi v)\) is the final answer.

Step 10 ：\(\boxed{J(u, v) = 16\pi u\sin^2(\pi v) + 16\pi u\cos^2(\pi v)}\) does not match any of the given options A, B, C, or D. Therefore, the correct answer is not listed.