Final answer:
The Jacobian of the transformation with x and y given by the specified equations is found by taking the partial derivatives of x and y with respect to r, which results in a product of these derivatives. Since there's a potential typo in the given options, none of the options provided match the calculation, and thus no option is selected.
Step-by-step explanation:
The question asks for the Jacobian of the transformation given by the equations x = 6e^(-4r) sin(3), y = e^(4r) cos(3). To find the Jacobian, we need to consider the partial derivatives of x and y with respect to r and 3 (theta, in this case denoted as 3 due to a possible typo). However, since theta seems to be a constant (sin(3) and cos(3) are not variables), its derivatives will be zero.
The partial derivative of x with respect to r is -24e^(-4r) sin(3) and the partial derivative of y with respect to r is 4e^(4r) cos(3). Since the partial derivatives with respect to theta are zero, the Jacobian determinant will be the product of the derivatives of x and y with respect to r:
Jacobian = (-24e^(-4r) sin(3)) * (4e^(4r) cos(3)) = -96e^0 sin(3) cos(3) = -96 sin(3) cos(3)
However, none of the given options match this result, which suggests there may be a misunderstanding in the question or a typo in the provided options.
Since we are not confident in the correctness of any of the provided options, we decline to choose one. Instead, we explain how to compute the Jacobian properly given the transformation equations.