Page 170 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 7] VECTORS 161

dr ¼ h 1 du 1 e 1 þ h 2 du 2 e 2 þ h 3 du 3 e 3 ð18Þ
The quantities h 1 ; h 2 ; h 3 are sometimes caleld scale factors.
If e 1 ; e 2 ; e 3 are mutually perpendicular at any point P, the curvilinear coordinates are called
orthogonal. Since the basis elements are unit vectors as well as orthogonal this is an orthonormal
basis. In such case the element of arc length ds is given by
2 2 2 2 2 2 2
ds ¼ dr dr ¼ h 1 du 1 þ h 2 du 2 þ h 3 du 3 ð19Þ
and corresponds to the square of the length of the diagonal in the above parallelepiped.
Also, in the case of othogonal coordinates, referred to the orthonormal basis e 1 ; e 2 ; e 3 , the volume of
the parallelepiped is given by
dV ¼jg jk jdu 1 du 2 du 3 ¼jðh 1 du 1 e 1 Þ ðh 2 du 2 e 2 Þ ðh 3 du 3 e 3 Þj ¼ h 1 h 2 h 3 du 1 du 2 du 3 ð20Þ
which can be written as

@r @r
@r @ðx; y; zÞ

dV ¼ du 1 du 2 du 3 ¼ du 1 du 2 du 3 ð21Þ

@u 1 @u 2 @u 3 @ðu 1 ; u 2 ; u 3 Þ
where @ðx; y; zÞ=@ðu 1 ; u 2 ; u 3 Þ is the Jacobian of the transformation.
It is clear that when the Jacobian vanishes there is no parallelepiped and explains geometrically the
significance of the vanishing of a Jacobian as treated in Chapter 6.
Note: The further significance of the Jacobian vanishing is that the transformation degenerates at the
point.

GRADIENT DIVERGENCE, CURL, AND LAPLACIAN IN ORTHOGONAL
CURVILINEAR COORDINATES
If is a scalar function and A ¼ A 1 e 1 þ A 2 e 2 þ A 3 e 3 a vector function of orthogonal curvilinear
coordinates u 1 ; u 2 ; u 3 ,we have the following results.
1 @ 1 @ 1 @
e 3
1: r ¼ grad ¼ e 1 þ e 2 þ
h 1 @u 1 h 2 @u 2 h 3 @u 3
1 @ @ @
2: r A ¼ div A ¼ ðh 2 ; h 3 A 1 Þþ ðh 3 h 1 A 2 Þþ ðh 1 h 2 A 3 Þ
h 1 h 2 h 3 @u 1 @u 2 @u 3

h 1 e 1 h 2 e 2 h 3 e 3

1 @ @ @

3: r A ¼ curl A ¼

@u 1 @u 2 @u 3
h 1 h 2 h 3

h 1 A 1 h 2 A 2 h 3 A 3

1 @ h 2 h 3 @ @ h 3 h 1 @ @ h 1 h 2 @
2
4: r ¼ Laplacian of ¼ þ þ
h 1 h 2 h 3 @u 1 h 1 @u 1 @u 2 h 2 @u 2 @u 3 h 3 @u 3
These reduce to the usual expressions in rectangular coordinates if we replace ðu 1 ; u 2 ; u 3 Þ by ðx; y; zÞ,
in which case e 1 ; e 2 ; and e 3 are replaced by i, j, and k and h 1 ¼ h 2 ¼ h 3 ¼ 1.
SPECIAL CURVILINEAR COORDINATES
1. Cylindrical Coordinates ( ; ; z). See Fig. 7-13.
Transformation equations:

x ¼ cos ; y ¼ sin ; z ¼ z