Page 437 - Advanced engineering mathematics

12.10 Curvilinear Coordinates 417

A similar, but more complicated, calculation shows that spherical coordinates are orthogonal
curvilinear coordinates.
In rectangular coordinates, the differential element ds of arc length is given by

2
2
2
2
(ds) = (dx) + (dy) + (dz) . (12.12)
We assume that this differential element of arc length is given in terms of the orthogonal
curvilinear coordinates q 1 ,q 2 ,q 3 by the quadratic form

3 3

2 2
(ds) = h dq i dq j .
ij
i=1 j=1
The numbers h ij are called scale factors for the curvilinear coordinate system. We want to
determine these scale factors so that we can compute such quantities as arc length, area,
volume, gradient, divergence and curl in curvilinear coordinates. Begin by differentiating
equations (12.11):

∂x ∂x ∂x
dx = dq 1 + dq 2 + dq 3 ,
∂q 1 ∂q 2 ∂q 3
∂y ∂y ∂y
dy = dq 1 + dq 2 + dq 3 ,
∂q 1 ∂q 2 ∂q 3
∂z ∂z ∂z
dz = dq 1 + dq 2 + dq 3 .
∂q 1 ∂q 2 ∂q 3
Substitute these into equation (12.12). This is a long calculation, but after collecting the
2
2
2
coefficients of (dq 1 ) , (dq 2 ) and (dq 3 ) (the terms in the double sum with i = j), and leav-
ing the cross product terms involving dq i dq j with i = j within the summation notation, we
obtain
2 2 2
" #

∂x ∂y ∂z
2 2
(ds) = + + (dq 1 )
∂q 1 ∂q 1 ∂q 1
" #
2 2 2
∂x ∂y ∂z

+ + + (dq 2 ) 2
∂q 2 ∂q 2 ∂q 2
" #
2 2 2

∂x ∂y ∂z
+ + + (dq 3 ) 2
∂q 3 ∂q 3 ∂q 3
3 3

+ h ij dq i dq j
i=1 j=1, j =i
3 3

2 2 2 2 2 2
=(h 11 ) (dq 1 ) + (h 22 ) (dq 2 ) + (h 33 ) (dq 3 ) + h ij dq i dq j
i=1 j=1, j =i
3 3

= h ij dq i dq j .
i=1 j=1
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October 14, 2010 14:53 THM/NEIL Page-417 27410_12_ch12_p367-424