Assume that $f^{(j)}:\mathbb{R}^3\rightarrow \mathbb{R}$ for $j=1,2$ are some $C^1$ functions such that $(\nabla f^{(1)}, \nabla f^{(2)})$ are independent over the curve $\gamma = \{ x \in \mathbb{R}^3 | f^{(1)}(x)=0=f^{(2)}(x)\}$. Prove that the function $O :=sgn(det(\nabla f^{(1)} \nabla f^{(2)} u))$ defines orientation for the curve $\gamma$.

I don't know how to approach this kind of question... What is the usual way to prove that some function defines orientation?


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