Solution
= Solution
Along a differentiable curve, the <chain rule> gives
$$
\frac{d}{dx}(F-y'F_{y'})=F_x+y'F_y-y'\frac{dF_{y'}}{dx}=F_x
$$
by the <Euler-Lagrange equation>. Thus when $F_x=0$ the <Beltrami identity> is
$$
\boxed{F-y'F_{y'}=C.}
$$
= Solution
Along a differentiable curve, the <chain rule> gives
$$
\frac{d}{dx}(F-y'F_{y'})=F_x+y'F_y-y'\frac{dF_{y'}}{dx}=F_x
$$
by the <Euler-Lagrange equation>. Thus when $F_x=0$ the <Beltrami identity> is
$$
\boxed{F-y'F_{y'}=C.}
$$