I was practicing conversion of del operators from Cartesian to Curvillinear coordinates, e.g. spherical coordinates
$$\left\{\begin{matrix} x=r sin \theta cos\phi\\ y=rsin\theta sin \phi \\ z=rcos \theta \end{matrix}\right.$$
I first compute the Jacobian matrix and
$$J^TJ=\begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \phi} & \frac{\partial x}{\partial \theta}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \phi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \phi} & \frac{\partial z}{\partial \theta} \end{pmatrix}\begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r} \\ \frac{\partial x}{\partial \phi} & \frac{\partial y}{\partial \phi} & \frac{\partial z}{\partial \phi} \\ \frac{\partial x}{\partial \theta} & \frac{\partial y}{\partial \theta} & \frac{\partial z}{\partial \theta} \end{pmatrix}=\begin{pmatrix} \left(\frac{\partial x}{\partial r}\right)^2 + \left(\frac{\partial x}{\partial \phi}\right)^2 + \left(\frac{\partial x}{\partial \theta}\right)^2 & \frac{\partial x}{\partial r}\frac{\partial y}{\partial r}+ \frac{\partial x}{\partial \phi}\frac{\partial y}{\partial \phi}+\frac{\partial x}{\partial \theta}\frac{\partial y}{\partial \theta}& \frac{\partial x}{\partial r}\frac{\partial z}{\partial r}+ \frac{\partial x}{\partial \phi}\frac{\partial z}{\partial \phi}+\frac{\partial x}{\partial \theta}\frac{\partial z}{\partial \theta}\\ \frac{\partial x}{\partial r}\frac{\partial y}{\partial r}+ \frac{\partial x}{\partial \phi}\frac{\partial y}{\partial \phi}+\frac{\partial x}{\partial \theta}\frac{\partial y}{\partial \theta} & \left(\frac{\partial y}{\partial r}\right)^2 + \left(\frac{\partial y}{\partial \phi}\right)^2 + \left(\frac{\partial y}{\partial \theta}\right)^2 & \frac{\partial y}{\partial r}\frac{\partial z}{\partial r}+ \frac{\partial y}{\partial \phi}\frac{\partial z}{\partial \phi}+\frac{\partial y}{\partial \theta}\frac{\partial z}{\partial \theta}\\ \frac{\partial x}{\partial r}\frac{\partial z}{\partial r}+ \frac{\partial x}{\partial \phi}\frac{\partial z}{\partial \phi}+\frac{\partial x}{\partial \theta}\frac{\partial z}{\partial \theta} & \frac{\partial y}{\partial r}\frac{\partial z}{\partial r}+ \frac{\partial y}{\partial \phi}\frac{\partial z}{\partial \phi}+\frac{\partial y}{\partial \theta}\frac{\partial z}{\partial \theta} & \left(\frac{\partial z}{\partial r}\right)^2 + \left(\frac{\partial z}{\partial \phi}\right)^2 + \left(\frac{\partial z}{\partial \theta}\right)^2 \end{pmatrix}$$ $$=\begin{pmatrix} sin^2\theta cos^2\phi+ r^2sin^2\theta sin^2\phi+r^2cos^2\theta sin^2\phi& sin^2\theta cos\phi sin\phi - r^2sin^2\theta sin\phi cos\phi + r^2cos^2\theta cos\phi sin\phi& sin\theta cos\theta cos\phi + 0 - r^2cos\theta sin\theta cos\phi\\ sin^2\theta cos\phi sin\phi - r^2sin^2\theta sin\phi cos\phi + r^2cos^2\theta cos\phi sin\phi & sin^2\theta sin^2\phi+r^2sin^2\theta cos^2\phi+r^2cos^2\theta sin^2\phi & sin\theta cos\theta sin\phi + 0 - r^2cos\theta sin\theta sin\phi \\ sin\theta cos\theta cos\phi + 0 - r^2cos\theta sin\theta cos\phi & sin\theta cos\theta sin\phi + 0 - r^2cos\theta sin\theta sin\phi & cos^2\theta+0+r^2sin^2\theta \end{pmatrix}$$
$$=\begin{pmatrix} sin^2\theta cos^2\phi+ r^2sin^2\phi& sin^2\theta cos\phi sin\phi - r^2sin^2\theta sin\phi cos\phi + r^2cos^2\theta cos\phi sin\phi& sin\theta cos\theta cos\phi - r^2cos\theta sin\theta cos\phi\\ sin^2\theta cos\phi sin\phi - r^2sin^2\theta sin\phi cos\phi + r^2cos^2\theta cos\phi sin\phi & sin^2\theta sin^2\phi+r^2sin^2\theta cos^2\phi+r^2cos^2\theta sin^2\phi & sin\theta cos\theta sin\phi - r^2cos\theta sin\theta sin\phi \\ sin\theta cos\theta cos\phi - r^2cos\theta sin\theta cos\phi & sin\theta cos\theta sin\phi - r^2cos\theta sin\theta sin\phi & cos^2\theta+r^2sin^2\theta \end{pmatrix}$$
But according to this and this link, the inner product matrix G should look like this
$$G=\begin{pmatrix} 1 & 0 & 0\\ 0 & r^2 & 0 \\ 0 & 0 & r^2sin^2\theta \end{pmatrix}$$
Therefore
Why am I missing some r terms in the Jacobian matrix, thus preventing me from getting the correct G?
