Derivative of a map into a level set
= Derivative of a map into a level set
{title2=$DH_{F(x)}DF_x=0$}
If a <differentiable map> $F$ takes values in a level set of a differentiable scalar function $H$, the <chain rule> gives $DH_{F(x)}DF_x=0$. 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.