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Obtain the Jacobian via the Context Panel

$\left[r \cos \left(\mathrm{\θ}\right)\,r \sin \left(\mathrm{\θ}\right)\,z\right]$$\stackrel{\text{Jacobian}}{\to }$${\cos \left(\mathrm{\θ}\right)}^{2}r\+r{\sin \left(\mathrm{\θ}\right)}^2$$\stackrel{\text{simplify}}{\=}$$r$$$

$X\=r \cos \left(\mathrm{\θ}\right)$$\stackrel{\text{assign}}{\to }$

$Y\=r \sin \left(\mathrm{\θ}\right)$$\stackrel{\text{assign}}{\to }$

$\left[\begin{array}{ccc}\frac{\partial }{\partial r} X& \frac{\partial }{\partial \mathrm{\θ}} X& \frac{\partial }{\partial z} X\\ \frac{\partial }{\partial r} Y& \frac{\partial }{\partial \mathrm{\θ}} Y& \frac{\partial }{\partial z} Y\\ \frac{\partial }{\partial r} z& \frac{\partial }{\partial \mathrm{\θ}} z& \frac{\partial }{\partial z} z\end{array}\right]$ = $\stackrel{\text{determinant}}{\to }$${\cos \left(\mathrm{\θ}\right)}^{2}r\+r{\sin \left(\mathrm{\θ}\right)}^2$$\stackrel{\text{simplify}}{\=}$$r$$$

Table 7.4.15(a)   From first principles, the Jacobian for cylindrical coordinates

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right)\:$

$\mathrm{simplify}\left(\mathrm{Jacobian}\left(\left[r \cos \left(\mathrm{\theta }\right)\,r \sin \left(\mathrm{\theta }\right)\,z\right]\,\left[r\,\mathrm{\theta }\,z\right]\,\mathrm{output}\=\mathrm{determinant}\right)\right)$ = $r$$$