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$$ S[\phi]~=~\int_V \mathrm{d}^nx ~{\cal L}, \tag{1}$$

$$ {\cal L}(\phi(x),\partial\phi(x),x). \tag{2}$$

It is implicitly assumed that under a variation the integration region $V$ changes according to the vector field $X^{\mu}$. Then the corresponding infinitesimal variation of the action $S$ takes the form$^2$

$$ \delta S ~\sim~ \int_V \mathrm{d}^n x \left(\epsilon ~ k + j^{\mu} ~ d_{\mu} \epsilon \right) \tag{5}$$

III) Next we assume that the action $S$ has a quasisymmetry$^3$ for $x$-independent infinitesimal $\epsilon$. Then eq. (5) reduces to

$$ \begin{align} 0~\sim~~~~~&\text{(boundary terms)}~\approx~ \delta S \cr ~\stackrel{(5)+(9)+(10)}{\sim}& \int_V \mathrm{d}^n x ~ J^{\mu}~ d_{\mu}\epsilon \cr ~\sim~~~~~& -\int_V \mathrm{d}^n x ~ \epsilon~ d_{\mu} J^{\mu} \end{align}\tag{11}$$

$^3$ A quasisymmetry of a local action $S=\int_V d^dx ~{\cal L}$ means that the infinitesimal change $\delta S\sim 0$ is a boundary term under the quasisymmetry transformation.

$$ S_V[\phi]~=~\int_V \mathrm{d}^nx ~{\cal L}, \tag{1}$$

$$ {\cal L}(\phi(x),\partial\phi(x),x).\tag{2} $$

It is implicitly assumed that under a variation the integration region $V$ changes according to the vector field $X^{\mu}$. Then the corresponding infinitesimal variation of the action $S_V$ takes the form$^2$

$$ \delta S_V ~\sim~ \int_V \mathrm{d}^n x \left(\epsilon ~ k + j^{\mu} ~ d_{\mu} \epsilon \right) \tag{5}$$

III) Next we assume that the action $S_V$ has a quasisymmetry$^3$ for $x$-independent infinitesimal $\epsilon$. Then eq. (5) reduces to

$$ \begin{align} 0~\sim~~~~~&\text{(boundary terms)}~\approx~ \delta S_V \cr ~\stackrel{(5)+(9)+(10)}{\sim}& \int_V \mathrm{d}^n x ~ J^{\mu}~ d_{\mu}\epsilon \cr ~\sim~~~~~& -\int_V \mathrm{d}^n x ~ \epsilon~ d_{\mu} J^{\mu} \end{align}\tag{11}$$

$^3$ A quasisymmetry of a local action $S_V=\int_V d^dx ~{\cal L}$ means that the infinitesimal change $\delta S_V\sim 0$ is a boundary term under the quasisymmetry transformation.

Then the corresponding infinitesimal variation of the action $S$ takes the form$^2$

It is implicitly assumed that under a variation the integration region $V$ changes according to the vector field $X^{\mu}$. Then the corresponding infinitesimal variation of the action $S$ takes the form$^2$

$$\tag{1} S[\phi]~=~\int_V \mathrm{d}^nx ~{\cal L}, $$

$$\tag{2} {\cal L}(\phi(x),\partial\phi(x),x). $$

$$\tag{3} \delta x^{\mu}~=~\epsilon X^{\mu} \qquad\text{and}\qquad \delta\phi^{\alpha}~=~\epsilon Y^{\alpha}$$

$$\tag{4} X^{\mu}(x)\qquad\text{and}\qquad Y^{\alpha}(\phi(x),\partial\phi(x),x).$$

$$\tag{5} \delta S ~\sim~ \int_V \mathrm{d}^n x \left(\epsilon ~ k + j^{\mu} ~ d_{\mu} \epsilon \right) $$

$$\tag{6} k(\phi(x),\partial\phi(x),\partial^2\phi(x),x)$$

$$\tag{7} j^\mu(\phi(x),\partial\phi(x),x).$$

$$\tag{8} 0~\sim~\epsilon\int_V \mathrm{d}^n x~ k. $$

$$\tag{9} k ~=~ d_{\mu}k^{\mu} $$

$$\tag{10} J^{\mu}~:=~j^{\mu}-k^{\mu}.$$

On-shell, after an integration by part, eq. (5) becomes

$$\tag{11} 0~\sim~\text{(boundary terms)}~\approx~ \delta S ~\stackrel{(5)+(9)+(10)}{\sim}~\int_V \mathrm{d}^n x ~ J^{\mu}~ d_{\mu}\epsilon ~\sim~-\int_V \mathrm{d}^n x ~ \epsilon~ d_{\mu} J^{\mu} $$

$$\tag{12} d_{\mu}J^{\mu}~\approx~0, $$

$$ S[\phi]~=~\int_V \mathrm{d}^nx ~{\cal L}, \tag{1}$$

$$ {\cal L}(\phi(x),\partial\phi(x),x). \tag{2}$$

$$ \delta x^{\mu}~=~\epsilon X^{\mu} \qquad\text{and}\qquad \delta\phi^{\alpha}~=~\epsilon Y^{\alpha}\tag{3}$$

$$ X^{\mu}(x)\qquad\text{and}\qquad Y^{\alpha}(\phi(x),\partial\phi(x),x).\tag{4}$$

$$ \delta S ~\sim~ \int_V \mathrm{d}^n x \left(\epsilon ~ k + j^{\mu} ~ d_{\mu} \epsilon \right) \tag{5}$$

$$ k(\phi(x),\partial\phi(x),\partial^2\phi(x),x)\tag{6}$$

$$ j^\mu(\phi(x),\partial\phi(x),x).\tag{7}$$

$$ 0~\sim~\epsilon\int_V \mathrm{d}^n x~ k. \tag{8}$$

$$ k ~=~ d_{\mu}k^{\mu} \tag{9}$$

$$ J^{\mu}~:=~j^{\mu}-k^{\mu}.\tag{10}$$

On-shell, after an integration by parts, eq. (5) becomes

$$ \begin{align} 0~\sim~~~~~&\text{(boundary terms)}~\approx~ \delta S \cr ~\stackrel{(5)+(9)+(10)}{\sim}& \int_V \mathrm{d}^n x ~ J^{\mu}~ d_{\mu}\epsilon \cr ~\sim~~~~~& -\int_V \mathrm{d}^n x ~ \epsilon~ d_{\mu} J^{\mu} \end{align}\tag{11}$$

$$ d_{\mu}J^{\mu}~\approx~0, \tag{12}$$