I know the gradient transforms covariantly under a change of basis, so I've been struggling with this, because I always see $\nabla f$ written as $f^i$ for a real-valued function $f$.
I know that using the metric isomorphism $g^{ij} f_j = f^i$ one typically associates to $df$ the gradient vector field (since $df(v) = \langle \nabla f, v \rangle$ where $\langle -,- \rangle$ is the metric-induced inner product).
However, I don't know how to reconcile the fact that the gradient transforms in a covariant fashion with the "contravariant" notation $f^i$ that we typically give vector fields.
Would someone please clear this up for me?
