Divergence ($\nabla \cdot$)

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,

For the Levi-Civita connection of a Riemannian metric, the divergence of $X$ is ${\mathrm{div}}_{g}X=\mathrm{tr}(Y\mapsto {\nabla }_{Y}X)$. In coordinates it equals $(\det g{)}^{-1/2}{\partial }_i((\det g{)}^{1/2}{X}^i)$. On an oriented manifold it satisfies ${\mathcal{L}}_X{\omega }_g=({\mathrm{div}}_{g}X){\omega }_g$. The codifferential on a one-form satisfies $\delta \theta =-{\mathrm{div}}_g({\theta }^{♯})$, where $♯$ is the musical isomorphism; this follows by integration by parts or the Hodge star formula.

For a vector field directed radially from a fixed axis and depending only on distance $\rho$ from that axis, cylindrical coordinates give $\nabla \cdot [f(\rho ){\mathbf{e}}_{\rho }]=f\prime +f/\rho$. Zero divergence gives $f=A/\rho$ on any positive radial interval. Unless $A=0$, this field is singular on the axis, so applications of the divergence theorem must use a domain avoiding it or account separately for that singularity.

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.

For $\mathbf{u}=u_r{\mathbf{e}}_r+u_{\theta }{\mathbf{e}}_{\theta }$,

For a scalar field $\varphi$ and vector field $F$,

A total divergence has the form $\nabla \cdot \mathbf{F}$. The divergence theorem converts its volume integral into a boundary flux, which vanishes when the field has suitable decay or boundary conditions.

For a twice continuously differentiable vector field, commuting partial derivatives and antisymmetry of the Levi-Civita symbol give