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FUNDAMENTAL LEMMA OF THE CALCULUS OF VARIATIONS:
If a continuous multivariable function \(f\) on an open set \({\Omega\subset \mathbb {R} ^{d}}\) satisfies the equality
\[\displaystyle\int_\Omega\mathcal{f}(x)\mathcal{g}(x)\ \mathrm{d}x=0\]
for all compactly supported smooth functions \(g\) on \(\Omega\), then \(f\) is identically zero.
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