Consider the vector field $X=-x_2\frac{\partial}{\partial x_1}+x_1\frac{\partial}{\partial x_2}$. I want to calculate the flow of this vector field in $\mathbb R^3$.
As far I understand the calculate the flow we need to solve $X(\gamma(t))=\frac d{dt}\gamma(t)$. Let $\gamma(t)=(x(t),y(t),z(t))$. Then $X(\gamma(t))=(-y(t),x(t),0)$ and $\frac d{dt}\gamma(t)=(x'(t),y'(t),z'(t))$. This is satisfied by $\gamma(cos(t),sin(t),0)$.
Is this calculation correct, it seem I should do something about initial conditions?