Problem in deducing gradient in spherical coordinates.

Question

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the gradient is $\dfrac{\partial}{\partial r}\cdot\widehat{r} + \dfrac{1}{r}\dfrac{\partial}{\partial\theta}\cdot\widehat{\theta} + \dfrac{1}{r\sin\theta}\dfrac{\partial}{\partial \phi}\cdot\widehat{\phi}$. Can anyone show me the deduction please? I am new to this & came across it when was studying Schroedinger's equation in spherical coordinate form.

Answer 1

Following this answer, https://math.stackexchange.com/a/587298/169296, you can use : $$ df = \frac{\partial f}{ \partial r} dr + \frac{\partial f}{ \partial \theta} d \theta + \frac{\partial f}{ \partial \phi} d\phi = \vec{\nabla f }\cdot \vec{dr}$$ Since $\vec{e_r} , \vec{e_\theta} $ and $ \vec{e_\phi}$ is a set of basis vectors you can suppose that $\vec{\nabla f } = \alpha \vec{e_r} + \beta \vec{e_\theta} + \gamma \vec{e_\phi}$.

Then you will find your answer by identifying the coefficients using your expression of $\vec{dr}$.