Localized Ten Dimensional Variations Detect Their Coefficients

Let $U\subset {\mathbb{R}}^{10}$ be open, let $W$ be a finite-dimensional real coordinate vector space, let $w:U\to (0,\mathrm{\infty })$ be continuous, and let $A:U\to W$ be continuous. A dual-coordinate test field is a smooth map $Q:U\to W^{\ast }$, where $W^{\ast }$ is the linear dual, with compact support $\mathrm{supp}Q:=\overline{\{x\in U:Q(x)\mathrm{\ne }0\}}\subset U$. For each such $Q$, put $h_Q(x)=w(x)⟨A(x),Q(x)⟩$ on $U$ using the canonical pairing $W\times W^{\ast }\to \mathbb{R}$, and extend $h_Q$ by zero to ${\tilde{h}}_Q$ on ${\mathbb{R}}^{10}$. Since $\mathrm{supp}Q$ is a compact subset of $U$, ${\tilde{h}}_Q$ is continuous with compact support. Define ${\int }_U{h}_Q{d}^{10}x$ to be the Darboux integral of ${\tilde{h}}_Q$ over any closed nondegenerate coordinate rectangle whose interior contains its support; the value is independent of the chosen rectangle, since a common larger rectangle can be subdivided along their faces and ${\tilde{h}}_Q=0$ on the added subrectangles. If

${\int }_{U}w(x)⟨A(x),Q(x)⟩d^{10}x=0$

for every smooth compactly supported test field $Q$ of the dual coordinate type, where $⟨,⟩$ is a nondegenerate finite-dimensional coordinate pairing, then $A=0$ on $U$. This applies to scalar, covector, form, and symmetric covariant two-tensor coefficients in local Euler–Lagrange variations.

Smooth manifolds and their smooth charts

A smooth tensor field

The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$

A smooth bump between concentric Euclidean balls