= Noether current for a first-derivative scalar field
{c}
{title2=$j^\mu=\mathcal L_{\partial_\mu\phi}X-K^\mu$}
For $\delta\phi=\varepsilon X$ and a <variational symmetry of a Lagrangian density>, write $\Pi^\mu=\partial\mathcal L/\partial(\partial_\mu\phi)$ and $\mathcal E=\mathcal L_\phi-\partial_\mu\Pi^\mu$. The product rule gives $\delta\mathcal L/\varepsilon=\mathcal E X+\partial_\mu(\Pi^\mu X)$. Therefore $j^\mu=\Pi^\mu X-K^\mu$ obeys $\partial_\mu j^\mu=-\mathcal E X$, so it is a <conserved current> <on shell>. Spatial integration gives a conserved <Noether charge> only when the boundary flux vanishes. The formula extends to several components by summation.
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