Derivative of a map into a level set ($D{H}_{F(x)}D{F}_x=0$)

If a differentiable map $F$ takes values in a level set of a differentiable scalar function $H$, the chain rule gives $D{H}_{F(x)}D{F}_x=0$. At a regular point, the image of the derivative lies in the kernel of the nonzero covector $DH$: the tangent space of the level set. For a map into the unit circle this forces rank at most one.