= Variational symmetry of a Lagrangian density
A differentiable infinitesimal transformation is a variational symmetry when its change of the <Lagrangian density> is a divergence, $\delta\mathcal L=\varepsilon\partial_\mu K^\mu$. Under suitable <boundary conditions> the <action> is invariant, so exact pointwise invariance is unnecessary. The off-shell variation identity gives the <Noether current for a first-derivative scalar field>. This definition is for a constant transformation parameter; a spacetime-dependent parameter is a different, gauge-symmetry problem.
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