CHAPTER 6 CURVILINEAR COORDINATES u  If is independent of ; we have etc r QUESTIONS Q1. Determine the unit vectors in the directions of the following three vectors and test whether they form an orthogonal set 3iˆ 2   2  2iˆ  4kˆ. kˆ, iˆ kˆ, ˆj ˆj ˆj Ans. Yes, an orthogonal set  hu = 1, hv = 2V, h = 2u. Ans. Q3. A vector field is given in cylindrical coordinates by F  (b)  F Determine (a) F  (2 cos 2 cos 2 1) (2sin 2 sin)kˆ Ans. (a)   (b)   , . Fr F and F Hint: You determine , , in terms of eˆr eˆ eˆ iˆ, and kˆ. Use Cramer’s rule to ˆj  Ans.   where  2r sin  sin  cos sin cos sin   3r sin cos cos Fr r 2 2r sin cos sin  cos2  sin  3r sin  cos F  r 2  sin sin  cos cos F r 2 a. Q5. Find the element of arc length on a sphere of radius   Ans.  dS a 2 2
CHAPTER 6 CURVILINEAR COORDINATES If u is independent of  ; we have r etc QUESTIONS Q1. Determine the unit vectors in the directions of the following three vectors and test whether they form an orthogonal set 3iˆ 2 ˆj  kˆ, iˆ  2 ˆj  kˆ, 2iˆ ˆj  4kˆ. Ans. Yes, an orthogonal set  Ans. hu = 1, hv = 2V, h = 2u. Q3. A vector field F is given in cylindrical coordinates by  Determine (a)  F (b)  F Ans. (a) (2 cos  2 cos 2  1) (b) (2sin 2  sin)kˆ  Fr , F and F . Hint: You determine eˆr , eˆ , eˆ in terms of iˆ, ˆj and kˆ. Use Cramer’s rule to  Ans.   where Fr  2r sin 2  sin  cos r sin cos sin   3r sin cos cos F  2r sin cos sin  r cos2  sin  3r sin 2  cos F  sin sin 2  r cos cos Q5. Find the element of arc length on a sphere of radius a. Ans. dS 2  a 2  