Say I want to parametrize the plane. I can use cartesian (x,y) or polar (r,theta). But I cannot use x and r, because if I draw a circle and a vertical line, there are two points of intersection. I guess x and theta will do, because theta specifies a ray that will intersect any vertical line. Except when theta=pi/2 :-) In thermodynamics we have a lot of variables and maybe three dimensions (for the ideal gas) connected through the equation of state. How can I know which combinations can parametrize the space and which can't ?