Whether $df(X)=\langle\operatorname{grad}f,X\rangle$?

Question

$(M,g)$ is Riemann manifold, $X$ is vector field, $f$ is function on $M$. $\langle\cdot,\cdot\rangle$ is inner.

Whether $df(X)=\langle\operatorname{grad}f,X\rangle$ ?

I only know $df(X)=X(f)$.

Answer 1

If $f$ is a smooth real-valued function on a smooth manifold $M$, then $df$ is a one-form defined as $df(X) = Xf$. If $M$ is equipped with a Riemmanian metric $g$, then the gradient of $f$, denoted $\operatorname{grad} f$, is the unique vector field satisfying $df(X) = g(\operatorname{grad} f, X)$; note the vector field $\operatorname{grad} f$ depends on $g$, but the one-form $df$ does not.

In summary, you should think of the equation $df(X) = g(\operatorname{grad} f, X)$ as a definition of $\operatorname{grad} f$, not of $df$.