Suppose I have a function $f(r, \theta) = r\sin\theta$ with $r$ being in meters and $\theta$ being in radians. The gradient of $f$ is $\nabla f = \frac{\partial f}{\partial r}\hat{r} + \frac{\partial f}{\partial \theta} \hat{\theta} = (\sin\theta) \hat{r} + (r\cos\theta)\hat{\theta}$.
Would I be correct to say that, once evaluated, the unit of the first component of $\nabla f$ is $dimensionless$ and the unit of the second component of $\nabla f$ is the $meter$?
If I am correct, the conclusion would contradict this answer. But, if I am wrong, the conclusion would contradict $[\frac{\partial a}{\partial b}] = [\frac{a}{b}]$, taken from this answer.
To be more succinct: What is resultant unit of the function $f$ after the del operator is applied?
