dotting gradient in spherical coordinates with displacement vector The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \right)$ on the other hand the gradient is supposed to give us: $\nabla f \cdot d\vec r = df$ where $d\vec r$ is the displacement vector if I write $\left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \right) \cdot (dr,d\theta,d\phi)$ it will be wrong because: $df = \frac{\partial f}{\partial r}dr + \frac{\partial f}{\partial \theta}d\theta + \frac{\partial f}{\partial \phi} d\phi$ I realize my displacement vector is wrong and is not $(dr,d\theta,d\phi)$, but on the other hand isn't the displacement vector by definition just composed of the small changes in each coordinate?

$$ \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial \phi} \right) \cdot (dr, r d\theta , r \sin \theta d\phi ) = \frac{\partial f}{\partial r} dr + \frac{\partial f}{\partial \theta} d \theta + \frac{\partial f}{\partial \phi} d \phi = d f $$