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While formulating this question, I arrived at a likely interpretation provided in an answer to my own question below. My problem appears to be one of inexperience in working with ambient coordinates, and since this is not for class but of my own interest I'm still posting this question in the hope that someone might care to verify my interpretation. I hope this is acceptable :-)

Let $M$ be a smooth submanifold of $\mathbb{R}^{3N}$ with a smooth (Lagrangian) function $L:TM\to\mathbb{R}$, let $\gamma$ be a smooth curve $I=[a,b]\to M$, and let $S$ be the action functional defined by integrating $L$ over derivatives of smooth paths (such as $\gamma$).

A variation of $\gamma$ is a smooth map $\widetilde{\gamma}=(-\epsilon,\epsilon)\times I\to M$ extending $\gamma$, or more precisely, coinciding with $\gamma$ at $0\times I$. This yields a family $s\in(-\epsilon,\epsilon)\mapsto\gamma_s$ of paths in $M$, which in turn yields a real function $S(\gamma_s)$ defined on $(-\epsilon,\epsilon)$. In this book, section 1.2.3 page 5, the first variation of $S$ at $\gamma$, defined to be $$\delta S=\left.\frac{\partial}{\partial s}\right|_{s=0}S(\gamma_s),$$ is stated to commute with the integral, yielding the expression $$\delta S=\int_a^b\delta L(\gamma,\dot{\gamma})dt,$$ with $\delta L$ defined as $$DL(\gamma,\dot{\gamma})\cdot(\delta\gamma,\delta\dot{\gamma})$$ where $$\delta\gamma=\left.\frac{\partial}{\partial s}\right|_{s=0}\gamma_s$$ and $$\delta\dot{\gamma}=\left.\frac{\partial}{\partial s}\right|_{s=0}\dot{\gamma}_s.$$

My problem is one of type-checking: I cannot figure out what the symbols $\delta\gamma$ and $\delta\dot{\gamma}$ mean. The partial operator acts on functions!

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While formulating this question I arrived at the following possible interpretation: $(\gamma_s(t),\dot{\gamma}_s(t))$ is a smooth map defined on $(-\epsilon,\epsilon)\times I$ with values in $TM\subset T\mathbb{R}^{3N}=\mathbb{R}^{3N}\times\mathbb{R}^{3N}$, so we get components of $\gamma_s(t)$ and $\dot{\gamma}_s(t)$ respectively, which are real functions on $(-\epsilon,\epsilon)\times I$, and the symbols are meant to mean component-wise partial derivation of these with respect to $s$.

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