Given $x,y,z$, I calculated the $3\times3$ matrix determinant. My assignment wants it in the form
$$dx\,dy\,dz = \text{__________ }du\,dv\,dw$$
Would I just put the Jocobian into this?
$$\begin{align} x&=u\,sin (6v)\\ y&=-6w+1\\ z&=u\,cos(6v) \end{align}$$
Jacobian I calculated: $$-6(6u\,sin^2(6v)+6u\,cos^2(6v))$$
$ J= det\left[ {\begin{array}{ccc} {\partial x / \partial u} & {\partial x / \partial v} & {\partial x / \partial w} \\ {\partial y / \partial u} & {\partial y / \partial v} & {\partial y / \partial w} \\ {\partial z / \partial u} & {\partial z / \partial v} & {\partial z / \partial w} \\ \end{array} } \right] = det\left[ {\begin{array}{ccc} {sin(6v)} & {6ucos(6v)} & {0} \\ {0} & {0} & {-6} \\ {cos(6v)} & {-6usin(6v)} & {0} \\ \end{array} } \right]$
$ J = (-6)(-1)^{2+3}det \left[ {\begin{array}{cc} {sin(6v)} & {6ucos(6v)} \\ {cos(6v)} & {-6usin(6v)} \\ \end{array} } \right] = 6 (-6usin^2(6v) - 6ucos^2(6v)) = -36u$