Exercise 8.5.8 (Polar coordinates): Define a mapping (, ) cos(), sin(). a) Show that is continuously differentiable (for all (, ) R2). b) Compute (0, ) for all . c) Show that if 0, then (, ) is invertible, therefore an inverse of exists locally as long as 0. d) Show that : R2 R2 is onto, and for each point (, ) R2, the set 1(, ) is infinite. e) Show that : R2 R2 is an open map, despite not satisfying the condition of the inverse function theorem