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Let $\varphi: \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth coordinate transform in $\mathbb{R}^3$. Also, let $\mathcal{E} = C(\mathbb{R}^3)$ to be a space of real, scalar-valued functions on $\mathbb{R}^3$. Now, we define $T_\varphi: \mathcal{E} \to \mathcal{E}$ as a non-linear map of the form: $T_\varphi (f) = f \circ \varphi$. I am interested in computing the first variational derivative of $T_\varphi (f)$ w.r.t. f. I realize it should have the form of a linear map from E to itself, but I have been having a hard time to formalize it. Any help will be greatly appreciated. Thanks!

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