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My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...

Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient. Show that $u\in H^1(\Omega), E(u)\in L^r(\...