First variation of the scalar-field action and the Euler-Lagrange equation

Given: The affine Minkowski model $(A,V,\eta ,C^{+})$ with a fixed inertial chart $x$ and its coordinate partial derivatives ${\mathrm{\partial }}_a$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$), a nonempty open $\mathrm{\Omega }\subseteq A$, numerical constants $c,K>0$, a smooth $V:\mathbb{R}\to \mathbb{R}$, a smooth real scalar field $\phi :\mathrm{\Omega }\to \mathbb{R}$ (C^k tensor fields on affine Minkowski space and their inertial components), and smooth compactly supported real $h:\mathrm{\Omega }\to \mathbb{R}$. In the inertial chart the metric components are the constant matrices ${\eta }_{ab}={\eta }^{ab}=\mathrm{diag}(-1,1,1,1)$, with ${\eta }^{ab}={\eta }^{ba}$ (Inertial frames and inertial charts of the affine Minkowski model). We write $\mathrm{\square }\phi :={\eta }^{ab}{\mathrm{\partial }}_a{\mathrm{\partial }}_b\phi$ and $L(\varphi ):=\frac{1}{2}{\eta }^{ab}{\mathrm{\partial }}_a\varphi {\mathrm{\partial }}_b\varphi +V(\varphi )$ for a smooth scalar function $\varphi$.

Compactly supported differentiation under the integral sign: if $F:(-s_0,s_0)\times \mathrm{\Omega }\to \mathbb{R}$ is continuous with continuous ${\mathrm{\partial }}_{s}F$ and there is a compact $K⋐\mathrm{\Omega }$ with $F(s,z)=0$ for $z\mathrm{\notin }K$ and all $s$, then $s\mapsto {\int }_{\mathrm{\Omega }}F(s,z)d^{4}z$ is $C^1$ on $(-s_0,s_0)$ with derivative ${\int }_{\mathrm{\Omega }}{\mathrm{\partial }}_{s}F(s,z)d^{4}z$. (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(c))

Compact-support integration by parts: if $f:\mathrm{\Omega }\to \mathbb{R}$ is $C^1$ and compactly supported and $g:\mathrm{\Omega }\to \mathbb{R}$ is $C^1$, then for each $a$, ${\int }_{\mathrm{\Omega }}({\mathrm{\partial }}_{a}f)g{d}^{4}x=-{\int }_{\mathrm{\Omega }}f({\mathrm{\partial }}_{a}g)d^{4}x$. (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(b))

Fundamental lemma: if $g:\mathrm{\Omega }\to \mathbb{R}$ is continuous and ${\int }_{\mathrm{\Omega }}g\psi d^{4}x=0$ for every smooth compactly supported $\psi :\mathrm{\Omega }\to \mathbb{R}$, then $g=0$ on $\mathrm{\Omega }$. (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(a))

Chain rule for total derivatives: if $f$ is totally differentiable at $a$ and $g$ at $f(a)$, then $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$. (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$)

Real one-variable sum, scalar multiple and product rules for derivatives. (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$)

If a map is totally differentiable at a point, then all its directional derivatives exist there and equal the total derivative evaluated on the direction; in particular coordinate partial derivatives are the entries of the Jacobian. (A total derivative computes every directional derivative, and its matrix is the Jacobian)

Sums, scalar multiples and products of continuous real-valued maps on a topological space are continuous where defined, so every finite polynomial expression in continuous real-valued maps is continuous; a composite of continuous maps is continuous, and pulling a continuous real-valued function back along the coordinate projection $(s,z)\mapsto z$ preserves continuity, both immediately from the definition of continuity through preimages of open sets and from the descriptions of the product and subspace topologies. Consequently the maps $(s,z)\mapsto {\mathrm{\partial }}_a\phi (z)$, $(s,z)\mapsto {\mathrm{\partial }}_{a}h(z)$, $(s,z)\mapsto h(z)$, $(s,z)\mapsto s$, $(s,z)\mapsto \phi (z)+sh(z)$ and the composites $V(\phi (z)+sh(z))$ and $V^{\mathrm{\prime }}(\phi (z)+sh(z))$ are continuous on $\mathbb{R}\times \mathrm{\Omega }$. (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\bar{A})\subseteq \overline{f(A)}$)