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Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in $\ …

If $F(f) = G(f,f')$ for some $G\in C^\infty(\mathbb{R}^2)$ then I think you can establish that $$ F'(f)u = (\partial_1 G)(f,f')u + (\partial_2 G)(f,f')u' $$ holds in much the same way as if you were d …

Not an answer, but too long for a comment. The general idea with this stuff seems to be to pair your family $\mathscr{D}$ of admissible domains with a(ny) suitable normed-space $\mathscr{V}$ of vector …

The classical material derivative $D\varphi/Dt$ of a test function $\varphi \in C_c^\infty(\mathbb{R}_+\times D)$ is obtained by setting $$ \dfrac{D\varphi}{Dt}(x) := \dfrac{\partial\tilde\varphi}{\pa …

I think $X:=S^2$ being a compact Riemannian manifold already gives you quite a lot, without any extra structure. The key seems to be the existence of geodesic normal coordinates, and in particular the …