Please help me with 2.3 thanks!

Definition 2.1: The gradient operator V in two dimensional rectangular coordinates is a a - dit V= ax' dy ax ay Exercise 2.2 (25 pts): This exercise has three parts. (a) Use the chain rule to write the partial derivatives a, and ay in terms of a and Hint: draw a chain diagram for an arbitrary differentiable function of two variables F(r, 0) where r and 0 are each parameterized by both x and y.) (b) Use the transformation equations r = vx2 + y2 and 0 = arctany/x to find or Or de and 20 . (c) Use parts (a) and (b) with the fact that i = cos Our - sin Que j = sin Our + cos Que, to express V in only polar coordinates (i.e. there should only be or , a6, ur, ue, and r and 0 in your answer). Now, a quicker way to find the two-dimensional gradient in polar coordinates! Let F(r) be a function in polar coordinates; i.e. F(r) = F(r, 0). The small change in moving from the point r with coordinates (r, 0) to the point r + dr with coordinates (r + dr, 0 + do) is given by: dF = OF -dr OF ar Exercise 2.3 (30 pts) This exercise has three parts. (a) Observe that dF = dr . VF. Find a formula for dr in terms of dr, do, ur, and ue, the standard unit basis vectors in polar coordinates from above. (b) Suppose that VF = our + Bue for some unknown functions o, B that we must find. Write dF = dr . VF in terms of dr, d, a and B. (Hint: why does up . ur = ue . up = 1?) (c) Solve for the unknown functions a and B in terms of a, , 20, r and 0. Conclude that the gradient operator in polar coordinates is given by V = our + , aqu