Covariant and contravariant Piola maps and H(div)/H(curl) spaces

Assume Countable Choice. Let $d\ge 2$, take $\widehat{K}={\widehat{S}}_d$ as in Matching affine simplicial meshes and shape regularity, and let $K=F(\widehat{K})$ under an affine cell map $F(\widehat{x})=A\widehat{x}+b$ with $\det A>0$. Weak derivatives use the signed test identity of Weak derivative of a locally integrable function. For $d\ge 2$ define $H(\mathrm{div},K):=\{v\in L^2(K;{\mathbb{R}}^d):\mathrm{div}v\in L^2(K)\},$ where $\mathrm{div}v={\sum }_i{D}_i{v}_i$ weakly, meaning ${\int }_{K}v\cdot \mathrm{\nabla }\varphi =-{\int }_K(\mathrm{div}v)\varphi$ for every $\varphi \in C_c^{\mathrm{\infty }}(K)$. For $d=2$, define the weak scalar curl by ${\int }_K(\mathrm{curl}v)\varphi ={\int }_K(v_1{\mathrm{\partial }}_2\varphi -v_2{\mathrm{\partial }}_1\varphi )$ for every $\varphi \in C_c^{\mathrm{\infty }}(K)$. For $d=3$, define the weak vector curl by ${\int }_K(\mathrm{curl}v)\cdot \psi ={\int }_{K}v\cdot \mathrm{curl}\psi$ for every $\psi \in C_c^{\mathrm{\infty }}(K;{\mathbb{R}}^3)$. In dimensions $d=2,3$ set $H(\mathrm{curl},K):=\{v\in L^2(K;{\mathbb{R}}^d):\mathrm{curl}v\in L^2(K;{\mathbb{R}}^{d_{\ast }})\},$ with $d_{\ast }=1$ for $d=2$ and $d_{\ast }=3$ for $d=3$.

The covariant Piola pushforward of a reference vector field is $({\mathcal{P}}_F^{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}\widehat{v})(F(\widehat{x})):=A^{-T}\widehat{v}(\widehat{x}),$ and the contravariant Piola pushforward is $({\mathcal{P}}_F^{\mathrm{d}\mathrm{i}\mathrm{v}}\widehat{v})(F(\widehat{x})):=\frac{1}{\det A}A\widehat{v}(\widehat{x})\mathrm{.}$ The scalar pullback is $u\mapsto u\circ F$. The exact mapping properties, weak differential identities, and oriented trace-moment transformations are established in the corresponding affine Piola theorem below.