Curvilinear Coordinate Systems 147(b) Verify that, in cylindrical coordinates,w-i^ViS+g. ("•«> r dr \ dr ) r^2 d6^2 dz^2 '
and use the result to provide an alternative proof that V^2 lnr = 0 in R^2.
(c)c Verify that, in spherical coordinates,
(^2) \<9r dr J r (^2) sin (^2) <pd9 (^2) r (^2) sin<p d<p \ dip
and use the result to provide an alternative proof that V^2 r_1 =0 in R^3.
Next, we derive the formula for the CURL using (3.1.26). Consider a
continuously differentiable vector field F written in the Q coordinates as
F(qi,q2,qs) = Fi(q 1 ,q 2 ,q 3 )q 1 + F 2 (q 1 ,q 2 ,q 3 ) q 2 + F 3 (q 1 ,q 2 ,q 3 ) q 3 - Let us
compute curlF(P) • q1: where the point P has Q-coordinates (c\, c 2 ,c 3 ).
For a sufficiently small a > 0, consider a rectangle spanned by the vectors
ar'k(0), k = 2,3, so that P is at the center of the rectangle. The vertices
of the rectangle have the Q coordinates (ci,C2 ± (a/2),c 3 ± (a/2)) and
the area of this rectangle is approximately a^2 h 2 h 3 ; draw a picture and see
equations (3.1.35). The line integral of F along the two sides parallel to q 2
is approximately
a[h 2 (ci,c 2 ,c 3 - (a/2))F 2 (ci,C2,c 3 - (a/2))
- h 2 (ci, c 2 , c 3 + (a/2))F 2 [cu c 2 , c 3 + (a/2))J
^ n2d(h 2 F 2 )
with the quality of approximation improving as a —> 0. The line integral
over the remaining two sides is approximately a^2 d(h 3 F 3 )/dq 2. Dividing by
the area of the rectangle and passing to the limit a —> 0, we find that
curlF(P)iir/m - 1 • fd(h^3 F^3 ) d(h^2 F^2 )\ u
9l = -^ - ^ - ftiThe other two components, curl F(P) • q 2 and curlF(P) • q 3 are computed
similarly, and then we get the representation of the curl as a symbolic