The exterior derivative of differential forms inR^n and the associated operations on vector fields inR^2 andR^3.

(a) Compute dw when w is the 2-form W = x1x3x4 dx1 / dx2 - el-22-23+4 dx1 / dx4. Note that dw is a 3-form in R4. Don't forget to simplify your answer so it looks like dw = F123 dx1 A dx2 A dx3 + F124 dx1 A dx2 A dx4 + F134 dx1 A dx3 A dx4 + F234 dx2 / dx3 A dx4, for some smooth functions Fijk : R4 -> R. (b) Let F : R2 - R2 be a vector field with scalar curl equal to x. Which of the diagrams below better represents F?\f(c) Let F : R2 -> R? be the same vector field from (b). Based on the answer you gave in (b), select the option below that best describes the divergence of F. Justify your answer. '0 = 4 . A (!) (ii) V . F = 1. .I - = 4 . A ( !) (iv ) V . F = x. (v ) V . F = -x. of = H . A ( !) (vii) V . F = -y