I was reading some physics when I read this particular paragraph:
In spherical coordinates, a general change in $f$ is given by $d f=$ $(\partial f / \partial r) d r+(\partial f / \partial \theta) d \theta+(\partial f / \partial \phi) d \phi$. However, the interval $d$ s takes a more involved form compared with the Cartesian $d$ s. It is $$ d \mathbf{s}=(d r, r d \theta, r \sin \theta d \phi) \equiv d r \hat{\mathbf{r}}+r d \theta \hat{\boldsymbol{\theta}}+r \sin \theta d \phi \hat{\boldsymbol{\phi}} $$
I understand that along $\hat \theta $, $ds=rd\theta$ and also the preceding argument about $dr$ but I really don't get the $rsin\theta \space d\phi\space \hat\phi$ part. I have read some other answers about gradient in spherical coordinates but most of it went over my head. I am in high school and I have learnt some multivariable calculus from Khan academy. Please can you explain this in a geometric way? Also, please tell me if there are more online resources where I can learn this?
Please help me.