I have this problem of determining the integral curve of the vector field $\vec\varphi(x,y,z)=(y,-x,1)$, this problem comes with a hint : show that $z'=\big(-\arctan(\frac{y}{x}) \big)'$.
My understanding is that I can solve the system $$\begin{cases}\dfrac{dx}{dt}=y\\ \dfrac{dy}{dt}=-x\\ \dfrac{dz}{dt}=1 \end{cases}$$ and so the integral curves are $x(t)=c_1\cos t + c_2 \sin t$, $y(t)=c_1\cos t - c_2 \sin t$ and $z(t)=t+c_3$.
My question is related to the hint, I want to understand how to use the hint to solve this problem.