Total divergence

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.

Adding $dF(q,t)/dt$ to a mechanical Lagrangian changes its action by $F(q_f,t_f)-F(q_i,t_i)$. Fixed endpoint variations therefore leave its bulk Euler-Lagrange equations unchanged. For fields, adding ${\partial }_{\mu }{K}^{\mu }$ changes the action by a boundary flux. The same conclusion holds for compactly supported variations or appropriate fixed boundary data. If $K$ depends on field derivatives, those boundary data must also control their variations. Such terms can still change canonical momenta, boundary observables, and representatives of Noether currents; equivalence of bulk equations does not mean that every boundary-value problem is identical.