Divergence, curl, and local conservation

For a C¹ vector field F in three-dimensional Cartesian coordinates, divergence is the sum of the derivative of each component in its own direction. Consider a tiny rectangular box. The x-directed flux leaving its right face minus that entering its left face is approximately the x derivative of Fx times the box volume. Adding the corresponding y and z contributions gives divergence times volume to leading order. A positive value therefore describes net outward flux locally; it is not a statement that all arrows point outward from one universal origin.

A spatially constant field can transport a great deal of material through a region while having zero divergence, because inflow and outflow balance. Likewise a field can have zero vector value at a point but nonzero divergence there: F=(x,y,z) vanishes at the origin yet has divergence three. Derivatives compare neighboring field values, so inspecting a single arrow cannot determine them. The box argument motivates the local interpretation; exact finite-region identities require integration theorems and their regularity hypotheses.

The component of curl along a unit normal describes circulation density around a small loop oriented by the right-hand rule. In the plane, embed F=(P,Q) as (P,Q,0). The relevant component is Qx−Py. A rigid rotation field (−Ωy,Ωx) has scalar curl 2Ω, not Ω. The factor two reflects contributions from both pairs of opposite sides of a small loop. Reversing loop orientation reverses measured circulation, matching the normal direction used for the curl component.

For sufficiently smooth fields, curl of a gradient is zero and divergence of a curl is zero because the corresponding mixed second partial derivatives cancel. The smoothness matters: standard pointwise identities assume the mixed partials exist and commute, for example under C² regularity. These identities impose necessary conditions for potentials, but their converses can require topology and boundary conditions. In particular, a curl-free field on a region with a hole need not have a globally defined scalar potential, as a loop around the missing region can detect.

Let ρ be a conserved density, J its transport flux, and s a production rate per unit volume. In a fixed control region, rate of stored amount equals production minus outward flux. Under sufficient smoothness, converting the boundary flux to a volume integral gives the local continuity equation. Its minus sign has a physical meaning: net outflow reduces storage. The balance law alone does not determine J; an advective, diffusive, or other constitutive relation must be supplied separately and justified for the application.

For compressible advection, expanding divergence of ρv gives v dotted with the density gradient plus ρ times divergence of velocity. Omitting the latter term assumes an incompressible velocity field and can violate conservation otherwise. For diffusion with spatially varying κ, the expression is divergence of κ times gradient ρ, not simply κ times the Laplacian unless κ is constant. Boundary conditions then control whether total material is conserved in a domain. A correct local equation does not automatically imply a closed system with zero boundary flux.