In the Calculus of Variations book by Gelfand and Fomin it says to consider the transformation $$x^{*} = \Phi(x,y,y')$$ $$y^{*} = \Psi(x,y,y').$$ Here it seems that $y'$ is the derivative of $y$ with respect to $x$.
This doesn't make any sense to me. Normally when we change coordinates we have that $(x,y)$ is a fixed coordinate system and we change to say $(x^{*},y^{*})$ where $x^* = f(x,y), y^* = g(x,y)$. Should I consider the variable $y$ inside $\Phi$ to be a function. If I just consider (x,y) as fixed basis then $y'$ makes no sense.