First integral of an ordinary differential equation ($I_t+\nabla I\cdot f=0$)

A first integral of $y^{\prime }=f(t,y)$ is a differentiable function $I(t,y)$ whose value is constant along solutions through every time and state in its domain. The chain rule makes this equivalent to the displayed pointwise identity. For $I(y)=y^{T}Sy$ with symmetric $S$, it becomes $y^{T}Sf(t,y)=0$. This identity at the implicit stage is what proves exact Runge-Kutta conservation of quadratic invariants for the implicit midpoint rule. Invariance asserted only on the image of one initial-time flow needs that image to contain the stage under consideration.