I have learned divergence, gradient and rotational in vector analysis of $\mathbb R^3$. However, when I read Riemannian Geometry, there are only definitions about divergence and gradient. So I have an idea to generalize the conception of Rotational.
In Do Carmo's book Differential Forms and Applications, rotational is defined as below: $$X\rightarrow\omega\rightarrow d\omega\rightarrow*(d\omega)$$ Let $\omega$ denote the differential one-form obtained from $X$ by the canonical isomorphism induced by the inner product $<,>$ and $d$ be exterior differential, $*$ be Hodge star operation.
My questions are:
(1) Can this definition be generalized for a Riemannian Manifold?
I think yes. But does it still mean rotational?
(2) Is there another way to define the rotational?
Although there is a definition above, I still feel it is complex or strange with reference to physics, compared with that in $\mathbb R^3$.