Question

16. Let Ag, Ay, A, denote the constant components of a vector in Cartesian coordinates. Using the transformation laws (1.2.42) and (1.2.47) to find the contravariant and covariant components of this vector upon changing to (a) cylindrical coordinates (r, 0, 2). (b) spherical coordinates (p, 0,6) and (c) Parabolic cylindrical coordinates.

1.2.41:

1.2.47:

also below is the final answer for part b, it will help in the finding a and c:

Answer 1

(a) In cylindrical coordinates (8,0,2), r = rcoco x+y=12 y = rsmo tamo = h . 7=7 :. Ā = dj agent and h(x) = media) - where x=x, x²= 4 x²= z; A = Ax , Az = Ay co z'= r, izo, = 2. Az = Az că = ai + Az Ox+ As you to S = Ax 3 x + Ay 24 h + A2 0 - Ã = ax coso + Ay smo, R = ax ax + Ay on lo + A2 2 2 : Āz = - YAr sin 0 + ray coso and Āg = 1+ az tag Āg = Az egyes - AZ e 1. .: = A x Dr J + Ay on any + Az on ne P = Axi caso & dy smo - Ax got dy do +2220 :: Āze - Ao sino + Ay como

and Taman - Axon + Ay taz X2 3 =Az. ☺ in parabolic Cylindrical coordinades (0,52) x=62 y =(7203) ., 7- ZE E di = a; axed , TC) = za where x'=x, x² = 4, 2²=2. ž=0, 22. zz. is Āi = A x y + Ayam +A2 ET Axtar Ayet - Az = ax ax +y ou je +Az Z ESAR + Thy .. and Āg -.42. = AX Do lo + Ay 30 do +t2 g at e Ax - Ay Az = ax detay de 2 + 20 = Ő Ax + 5 Ay. Az = Az k