CHAPTER 6 CURVILINEAR COORDINATES Arc Lengths ds and Volume Element dv Q h3du3eˆ3 u3 d r u2 O h1du1e1 h2du2e2 u1 Fig( A) dS a) Element of arc dS o Q Element of arc from to given by   2 2 2 2 2 2 2 2 2 2 and Example 6.4 Find the square of the element of arc length in cylindrical coordinates and determine the corresponding scale factors.    r xiˆ yˆj zkˆ We have   cos   sin   iˆ zkˆ ˆj    sin d 2 sin  cos  cos2 d     dd dz 2 2 2 2 2 2 2 2 1, , 1 and hU1  hU2  hU3  are the scale factors.
CHAPTER 6 CURVILINEAR COORDINATES Arc Lengths ds and Volume Element dv Q u3 h3du3eˆ3 d r u2 h1du1e1 O h2du2e2 u1 Fig( A) a) Element of arc dS Element of arc dS from o to Q given by   2 2 2 2 2 and 2 2 2 2 2 Example 6.4 Find the square of the element of arc length in cylindrical coordinates and determine the corresponding scale factors. We have r  xiˆ  yˆj  zkˆ   cos iˆ   sin  ˆj  zkˆ     sin 2 d 2  2 sin  cos dd   2 cos2 d 2  dz 2 2 2 2 and hU1  1, hU2  , hU3  1 are the scale factors.