c and d only

(a) Consider the 1-forms @1 = ydx + xdy + zdz and @2 = xdx + zdy + ydz in R'. Compute their wedge product, @1 A @2. Note that @1 A @2 is a 2-form on R, so you want to put it in the form W1 A @2 = A(x, y, z) dy A dz + B(x, y, z) dz A dx + C(x, y, z) dx A dy. (b) Now recall that 1-forms on R' are represented by vector fields. For example, @1 is represented by the vector field F = (y, x, z), and @2 is represented by the vector field G = (x, Z, y). Compute the cross product F X G.(c) Finally, recall that 2-forms on R are also represented by vector fields: we say that the 2-form A dy A dz + B dz A dx + C dx A dy is represented by the vector field (A, B, C). Now compare your answers in (a) and (b), and fill in the blanks (with the name of the suitable operation): upon identifying 1-forms and 2-forms on R with vector fields, we see that the product of 1-forms is represented by the product of vector fields. (Note that for this identification to be correct, we really need to write the 2-form in that specific form with dy A dz, dz A dx and dx A dy!) (d) For the final activity, let's just make sure you know how the exterior derivative works. Compute dw when w is the 2-form W = X1X3x4 dx1 A dx3 + e2xi+*2+x3-*4 dx2 A dx4. Note that dw is a 3-form in R*, so your answer should looks like do = A dx1 A dx2 A dx3 + Bdx1 A dx2 A dx4 + C dx] A dx3 A dx4 + D dx2 A dx3 A dx4, for suitable functions A, B, C and D (some of them possibly equal to 0)