 Orbital derivative

The orbital derivative of a differentiable scalar $V$ along an autonomous differential equation $\dot{x}=f(x)$ isThe chain rule identifies it with $dV(x(t))/dt$. It measures change along a trajectory, rather than the size of the gradient alone.

For $V(x,y)=(x^2+y^2)/2$ and the rotation field $f=(-y,x)$, it is $-xy+yx=0$ even though the gradient is nonzero away from the origin. For the contracting field $f=(-x,-y)$, the same function has orbital derivative $-x^2-y^2$. This directional distinction is what makes an orbital derivative useful in a Lyapunov function or a first integral.