On a pseudo-Riemannian manifold $M$ , and in Euclidean space in particular, one can express the gradient operator, the divergence operator, and the curl operator (which makes sense only if $M$ is 3-dimensional) in terms of the exterior derivative. Let $\fM$ denote the ring of smooth functions on $M$ ; let $\vfM$ denote the $\fM$ -module of smooth vector fields, and let $\Omega^1(M)$ denote the $\fM$ -module of smooth 1-forms. The contraction with the metric tensor $g$ and its inverse $g^{-1}$ , respectively, defines the $\fM$ -module isomorphisms$$\flat:\vfM\to\Omega^1(M),\quad \sharp \colon \Omega^1(M)\to\vfM$$ In local coordinates, this isomorphisms is expressed as$$ \left(\ddx{i}\right)^\flat= \sum_j g_{ij} dx^j,\quad \left(dx^j\right)^\sharp = \sum_i g^{ij} \ddx{i}.$$ or as the lowering of an index. To wit, for $V\in\vfM$ , we have

Another natural structure on an $n$ -dimensional Riemannian manifold is the volume form, $\omega \in \Omega^n(M)$ , defined by $$\omega = \sqrt{\det g_{ij}}\, dx^1\wedge\ldots\wedge dx^n.$$ Multiplication by the volume form defines a natural isomorphism between functions and $n$ -forms: $$f\mapsto f\omega,\quad f\in \fM.$$ Contraction with the volume form defines a natural isomorphism between vector fields and $(n-1)$ -forms: $$X\mapsto X\iprod \omega,\quad X\in \vfM,$$ or equivalently $$\ddx{i} \mapsto (-1)^{i+1} \sqrt{\det g_{ij}}\, dx^1\wedge\ldots \wedge \widehat{dx^i}\wedge\ldots \wedge dx^n,$$ where $\widehat{dx^i}$ indicates an omitted factor. The divergence operator, which in tensor notation is expressed as $$\Div X = \nabla_i X^i,\quad X\in \vfM$$ can be defined in a coordinate-free way by the following relation: $$(\Div X)\, \omega = d(X\iprod \omega),\quad X\in \vfM.$$

Finally, on a $3$ -dimensional manifold we may define the curl operator in a coordinate-free fashion by means of the following relation: $$(\curl X)\iprod \omega = d(X^\flat),\quad X\in \vfM.$$