Question

9. Using Generalized Curvilinear Coördinates, if the scalar S and the vector Ä are both functions of position, prove the following relations: (a) 7 (SA)=s(8. A)+A.IS; (b) (x(SA)=s(0xĂ)+()xA; (c) Ex(s)=0; (a) 8x(0xA)=(. A)-VA. Most of these identities are independent of the specific coördinate system that one employs; however, care must be taken with the operator VA, since in Cartesian Coördinates, it is defined as G?A=i(v?4)+1(v24,)+R(v?4.), but in the Cartesian case, the unit vectors are independent of the coordinate variables, which is not true of most other coordinate systems, so one must generalize the definition of this operator for any curvilinear coordinate system.

Answer 1

(a) 8 (s) (b) Cox (SA))- ai (sau) = (ais) Ai t sai Ai (vs). A + s vẢ (is). Å + Si tijk Di (saj) tijk (138) Aj + S ajaj) Eijk@;s) A; + Stijk Di Aj = (lºs)x X) +(slöxă), → vvs) - Ús) xả + slová) por (0) - & Eijk 0; (@s); Fijk bilajs) Ejik 2; (dis) {idj are dummy indices} --Eijk 9; (35) Eijk --Ejik 3:03 = 3; :. [ox (os))n = fijk (235) - Eijk ai (aj s) = a 2 Eijk diej s)

60 Eijk di Ijs = d) û x(ös) (öxă), & Eijk di Aj (ox (oxă) = L epka ap (BxA), Opko {tijk:) Eska Eijk aplin ;) Edpk Eijk ophiaj) - boj-jp) po 43) - Sci Spj apli aj) – Saj opiopeiaj) ajlód A; - di (aingi Od Oj aj – oloi Ad) ² AL fa D A 5(8. A)a - 2 AL 2 : 8x ( 6xÅ に (VA) -