I am trying to get a better sense of pullbacks/tensors geometrically by looking at the problem of finding coordinate expressions for the Euclidean metric in $\mathbb{R}^3 $ and spherical coordinates. We have from definition that in rectangular coordinates $g=dx\otimes dx +dy\otimes dy + dz\otimes dz$. We can do this via pullbacks, which is messy albeit mechanical, or from using geometry.
We should get the the spherical coordinate expression $g'= \rho^2(d\phi\otimes d\phi+\sin\phi^2d\theta\otimes d\theta)+ d\rho \otimes d\rho. $
My questions are in regard to the geometric perspective. First, how does one go about computing these from the geometry? Secondly, just from looking at $g'$, is there a nice geometric takeaway that I am missing?
To do this geometrically you only need to look at the partials with respect to each coordinate $\rho,\phi,\theta $ given the change of coordinates $$f:(\rho,\phi,\theta)\rightarrow (\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)$$
Since the spherical coordinates are orthogonal all of the inner products of the partials will be zero, except for those inner products taken with the same coordinate. If you calculate all of these out you get exactly $g'$ with respect to each coordinate.