Field variation (${\delta }_0\varphi (x)=\varphi \prime (x)-\varphi (x)$)

A fixed-coordinate field variation compares two fields at the same spacetime point. If a transformation also changes coordinates by $\delta x^{\mu }$, its total variation and fixed-coordinate variation differ by $\delta x^{\mu }{\partial }_{\mu }\varphi$. Keeping these separate is essential when applying the Noether current for a spacetime symmetry.