Given an open set $U\in\mathbb{R}^{3}$ and an orthagonal matrix $A\in SO(3)$ I've got a vector field $g\in C^{1}(U,\mathbb{R}^{3})$ with $g^{A}\in C^{1}(A(U),\mathbb{R}^{3})$ defined as $g^{A}:=Ag(A^{-1}x)$. I'm supposed to prove, that given a $C^{1}$-surface $S$ with the norm $v_{S}(x)$ with a $C^{1}$-boundary $\partial S$ that: $\int_{S}g\cdot v_{S}dS=\int_{S^{A}}g^{A}\cdot v_{S^{A}}dS\Rightarrow curl(f^{A})=(curl(f))^{A}\forall f\in C^{1}(U,\mathbb{R}^{3})$.