I am unsure whether the following vector form of the Fundamental Lemma of the Calculus of Variations is correct. This lemma appears to be very useful in deriving the compact form of the Euler-Lagrange equation using generalized coordinates. Please use SVG math renderer to see the formulas.
If $ f(\cdot) \in \mathcal{C}\left([a,b]; \mathbb{R}^n \right) $ and satisfies $$ \forall h(\cdot) \in \mathcal{T} \left( [a, b]; \mathbb{R}^n \right), \quad \int_{a}^{b} f^\top(x) h(x) \, \mathsf{d} x = 0, $$ then $ f(x) \equiv \mathbf{0}_n, $ where $\mathcal{T} \left( [a, b]; \mathbb{R}^n \right) := \left\{ \varphi(\cdot) \in \mathcal{C}^1 \left([a,b]; \mathbb{R}^n \right): \varphi(a) = \varphi(b) = \mathbf{0}_n \right\}. $