16. Consider the same polar and cartesian coordinate systems on a 2-dimensional flat plane, and the vector v described in the previous question. Consider also the scalar function = br² cos sin 0. Show that v¹µÞ has the same numerical value bxy = in both the polar and cartesian coordinate systems. 15. Consider polar coordinates on a 2-dimensional flat plane. The transformation equations between the polar coordinates r, (the primed coordinate system) and cartesian coordinates x, y (the unprimed coordinate system are) x = r cos, and y = r sin. The inverse transformations are r = √√x²+ y² and = tan¹ (2) 4 Эхи a) Find the transformation partials (Reminder: the primed coordinate system is the polar coordinates and the unprimed coordinate system is the cartesian coordinates). b) Consider a vector v whose components are v1 and v = 0. Use these partials to transform the cartesian components of the vector v to polar coor- dinates, and so verify that v 1. c) We have seen that the polar coordinate metric is: I'v = Show that the inverse metric is: g'mv d) Use this to raise the polar-coordinate components of the covector v. You should get the same components as you found in part b.