M427LDifferentialformPracticeforthethirdexam6. Explain why a 2-form onRnis the same thing as a anti-symmetricn×nmatrix.7. LetF= (F1, F2, . . ., Fn) (take the corresponding column vector) be a vectorfield onRn.Letω=∑Fidxibe the corresponding 1-from.Show thatd(ω)has the same data asA, the anti-symmetric part of the matrixD=D(F), i.e.A= 1/2(D-DT). Explain why forn= 3, exterior derivative of 1-forms is thesame thing as curl of vector fields. Explain why exterior derivative of 2-forms(onR3) is the same thing as divergence.8. Letf:R2→R2be the mapf(r, θ) = (rcos(θ), rsin(θ)). Computef*(dxdy)(here the firstR2has coordinates (r, θ) and the second has (x, y)).9. Compute the exterior derivative of the 2 form
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