where Although an$D$orientation runs is over required for the usual notion of Hodge star as given above, we may take it as valued inopen submanifold?pseudoforms s containing to point show that the orientation (or even orientability) of$\mathrm{<span><del\; class="diffmod">\; x</del><ins\; class="diffmod">\; M</ins></span>}$ x M is irrelevant (since the Hodge star is applied twice, returning us to untwisted forms, and with since smooth a boundary bounding hypersurface has a natural ‘outwards’ pseudoorientation). However, the metric, which is hidden in the volume form and in the “dot product”, is relevant.$\partial D$ and $\vec{n}$ is the unit vector of outer normal to the hypersurface $S$. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and balls with decreasing radius in this particular coordinate chart.

Example

If $(M,g)$ is the Cartesian space $\mathbb{R}^n$ endowed with the canonical Euclidean metric, then the divergence of a vector field $X^i \partial_i$ is

Remarks

The divergence was first developed in quaternion analysis, where its opposite appeared most naturally, called the convergence $con(X) = - div(X)$. In many applications of the divergence to the successor field, classical vector analysis, the metric is irrelevant and we may use differential forms instead: we translate a vector field $X$ into the $(n-1)$-form $\star_g g(X)$ and a scalar field $f$ into the $n$-form $\star_g f$, so that the divergence is simply the de Rham differential, and simply use the differential forms from the start.

Related concepts

• nabla

• gradient

  • gradient flow

  • symplectic gradient

• Hamiltonian flow

• curl

• divergence