$x = r sin \theta cos \phi \; y = r sin \theta sin \phi \; and z= r cos \theta \\ \text{ According to the definition,} \\ {\delta (x,y,z) \over \delta (r, \theta, \phi)} = \begin{vmatrix} {\delta x \over \delta r} & {\delta x \over \delta \theta} & {\delta x \over \delta \phi} \\ {\delta y \over \delta r} & {\delta y \over \delta \theta} & {\delta y \over \delta \phi} \\ {\delta z \over \delta r} & {\delta z \over \delta \theta} & {\delta z \over \delta \phi} \\ \end{vmatrix} \\ {\delta (x,y,z) \over \delta (r, \theta, \phi)} = \begin{vmatrix} sin\theta cos\phi & rcos\theta cos\phi & -rsin\theta sin\phi \\ sin \theta sin\phi & r cos \theta sin\phi & rsin \theta cos \phi \\ cos \theta & -r \;sin \theta & 0 \end{vmatrix}\\ = cos \theta (r^2 sin \theta cos \theta cos^2 \phi + r^2 sin \theta cos \theta sin^2 \phi) + r sin \theta ( r^2 sin^2 \theta cos ^2 \phi + r sin^2 \theta sin^2 \phi) \\ = cos \theta . r^2 sin \theta \; cos\theta + r \; sin \theta.r \; sin^2 \theta \\ = r^2 sin \theta cos^2 \theta + r^2 sin \theta sin^2 \theta \\ =r^2sin \theta (cos^2 \theta + sin^2 \theta) \\ = r^2 \; sin\theta \\ If \; J= {\delta (x,y,z) \over \delta (r, \theta, \phi)} \; then, \\ J^{'} = { \delta (r, \theta, \phi) \over \delta (x,y,z) } \\ Now, \; J.J^{'} = 1 \\ \therefore J^{'} = \frac1J \\ \therefore { \delta (r, \theta, \phi) \over \delta (x,y,z) } = {1 \over r^2 sin \theta }$