[!]. This exercise is used in Section 15.3. If

f, g are O-forms and w, TJ are r-forms, define

(f,g). (w,TJ) = fw + gTJ·

(a) Prove that if f : R3 ----> R is C1 and gradf := (fx, fy, fz), then the exterior derivative of the o-form W = f can be written in the form dw = ( grad 1) . (dx, dy, dz).

(b) Prove that if F = (P, Q, R) : R3 ----> R3 is cl, then the exterior derivative of the I-form W = P dx + Q dy + R dz can be written in the form dw = (curl F) . (dydz,dzdx,dxdy).

and the exterior derivative of the 2-form TJ = P dy dz + Q dz dx + R dx dy can be written in the form dTJ = (div F) dx dy dz.