Please show detailed work. Thank you.

(a) Consider the 1-forms wi = ydx + xdy + zdz and w2 = xda + zdy + ydz in R' . Compute their wedge product, w1 A w2. Note that wj A w2 is a 2-form on R', so you want to put it in the form W1 Aw2 = 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, w1 is represented by the vector field F = (y, x, z), and w2 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 + Bdz 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 / dz, dz / dx and dax A dy!)(d) For the nal activity, let's just make sure you know how the exterior derivative works. In class, we learned how to differentiate a 1-form to onbtain a 2-form. We can mimic that denition so that we can now differentiate a 2form and obtain a 3form! Compute do; when w is the 2form Note that do; is a 3form in R4, so your answer should looks like do: : A til-ml .. 713:3 l B (1:121 37:34 l (1' (is. for suitable functions A: B: C? and D (some of them possibly equal to (J)