Question

15 Total Question 3 Let F: R3R3 be any C2 vector field. 3(a). Prove that the divergence of the curl of F is zero. /4 marks 3(b). For F as defined above, a misguided professor claims that for any closed curve C, F dr 0 because of the argument: (x F)ds F-dr div (eurl F) dV X 0-APO by using Stokes' theorem, the divergence theorem, and then part (a) for an appropriately chosen surface S and volume W. Carefully explain all the errors in this argument /3 marks F.dr 0 for all closed curves C 3(c). If it is true that the line integral show that the line integral over a non-closed curve depends only on the beginning- and end-points of the curve /4 marks 3(d). Suppose that F is given by F(u, v, w) ue, + we, +ve. with respect to an orthogonal curvilinear coordinate system (u, v, w) with scale factors h w, h u and h v. Determine div F. |4 marks
please give some explanation to each step

Answer 1

3.a) Divergence of curl of F is 0.

Let defined F is vector in term of x,y,z.

F= Fi+Fyj+Fk

Curl of vector F,

H

OF OF V X az

Divergence of (curl of vector F)

dXA) (V x F) (i k

ӘF. ӘF, - j ӘF, k д д + k ОF, У E (оxF). дл у т. 7 ду ду дх ду де дх дх дх

ОF, ОF ОF, ОF, ОF. ду д ӘF, д д E o (оxF) - дх У ду де ду дх де де дх

ОF. ӘF, ОF. ӘF, ӘF, OFT E о (уxF) - дудх дхду дгду, дгдх дхде дуд:

V (V x F) - 0