Find the Jacobian for the given transformation
-x = 6u cos 5v, y = 6u sin 5v
A) 180u
B) 150v
C) 180v
D) 150u

Question

Quizplus · Accepted Answer

The Jacobian is given by the determinant of the matrix of partial derivatives:$$\begin{align*}J &= \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} \\&= \begin{vmatrix} 6\cos 5v & -30u\sin 5v \\ 6\sin 5v & 30u\cos 5v \end{vmatrix} \\&= (6\cos 5v)(30u\cos 5v) - (-30u\sin 5v)(6\sin 5v) \\&= 180u\cos^2 5v + 180u\sin^2 5v \\&= 180u.\end{align*}$$

Find the Jacobian for the Given Transformation