Curvilinear Coordinate Systems 143(ci,C2 + t,C3), t€R;- The qz coordinate curve with q\= c\, q 2 = c 2 , is the collection of points
(ci,c 2 ,c 3 + t), iel.
Since, by assumption, Xk = Xk(qi,q2,q3), the equations of these curves in
cartesian coordinates are
ri(t) = xi(ci +i,c 2 ,c 3 )l + X2(ci +t,c 2 ,c 3 ) j + X3(ci +t,c 2 ,c 3 )k
r2{t) = Xi(<=i, c2 +1, c 3 ) i + X2(ci ,c 2 + t, c 3 ) 3 + Xz{ci ,c 2 + t, c 3 ) K
^3(*) = Xl (Cl, C 2 , C 3 + i) Z + X2(Cl, C 2 , C 3 + t) J + X3(Cl, C 2 , C 3 + 0 K
(3.1.35)We define the functions hk, k = 1,2,3 so that the tangent vectors are
r'fc(0) = ^fc«fc. (3.1.36)By (3.1.35),
hk(P) = ||r'fc(0)|| =
NS(fe). <-->where the partial derivatives are evaluated at (c\, c 2 , c 3 ), the Q coordinates
of the point P. The orthogonality condition means that
7^(0) • r'n(0) = J2^^=0,m^n. (3.1.38)
The functions hk defined in (3.1.36) will play the central role in many
computations to follow. We will assume that, away from the special points,
the functions hk = hk(qi,q 2 ,qs) have continuous partial derivatives of every
order.
EXERCISE 3.1.26? (a) Verify the relations (3.1.37) and (3.1.38). (b) Verify
that cylindrical and spherical coordinates are orthogonal. Hint: a picture is
helpful.
Our next goal is to study ARC LENGTH in Q coordinates. Let C be a
curve represented in cartesian coordinates X by the vector function r(t) —
xi(t)t--x 2 (t)3 + xs(t) ii. We rewrite the expression (1.3.12) on page 29 for
the line element as (ds/dt)^2 = J2k=i(dxk/dt)^2. Since Xk = Xk{qi,q2,qz),
we have xk(t) = Xk(qi(t),q 2 (t),q 3 (t)), where qk(t) = qk(xi(t),x 2 (t),x 3 (t)),
k = 1,2,3 By the chain rule, dxk/dt = Tlli=i(dXk/dqm)(dqm/dt), and