Invariance equation for a graph ($Dh(x)F(x,h(x))=G(x,h(x))$)

For $\dot{x}=F(x,y)$ and $\dot{y}=G(x,y)$, a differentiable graph $y=h(x)$ is invariant precisely when $Dh(x)F(x,h(x))=G(x,h(x))$. This is the chain rule along a trajectory on the graph. The equation determines coefficients in a local power-series approximation of a stable manifold or unstable manifold; checking tangent directions alone does not ensure invariance.