Page 169 - Schaum's Outline of Theory and Problems of Advanced Calculus

160 VECTORS [CHAP. 7

11. r ðr AÞ¼ 0. The divergence of the curl of A is zero.
2
12. r ðr AÞ¼ rðr AÞ r A

VECTOR INTERPRETATION OF JACOBIANS,
ORTHOGONAL CURVILINEAR COORDINATES
The transformation equations
x ¼ f ðu 1 ; u 2 ; u 3 Þ; y ¼ gðu 1 ; u 2 ; u 3 Þ; z ¼ hðu 1 ; u 2 ; u 3 Þ ð15Þ
[where we assume that f ; g; h are continuous, have continuous partial derivatives, and have a single-
valued inverse] establish a one-to-one correspondence between points in an xyz and u 1 u 2 u 3 rectangular
coordinate system. In vector notation the transformation (17) can be written
r ¼ xi þ yj þ zk ¼ f ðu 1 ; u 2 ; u 3 Þi þ gðu 1 ; u 2 ; u 3 Þj þ hðu 1 ; u 2 ; u 3 Þk ð16Þ
A point P in Fig. 7-12 can then be defined not only by rectangular coordinates ðx; y; zÞ but by coordinates
ðu 1 ; u 2 ; u 3 Þ as well. We call ðu 1 ; u 2 ; u 3 Þ the curvilinear coordinates of the point.

Fig. 7-12
If u 2 and u 3 are constant, then as u 1 varies, r describes a curve which we call the u 1 coordinate curve.
Similarly, we define the u 2 and u 3 coordinate curves through P.
From (16), we have
@r @r @r
du 3
dr ¼ du 1 þ du 2 þ ð17Þ
@u 1 @u 2 @u 3
@r @r @r
The collection of vectors ; ; is a basis for the vector structure associated with the curvilinear
@x @y @z
system. If the curvilinear system is orthogonal, then so is this set; however, in general, the vectors are
not unit vectors. he differential form for arc length may be written
2 2 2 2
3
ds ¼ g 11 ðdu 1 Þ þ g 22 ðdu 2 Þ þ g 33 ðdu Þ
where
@r @r @r @r @r @r
; ;
g 11 ¼ g 22 ¼ g 33 ¼
@x @x @y @y @z @z
The vector @r=@u 1 is tangent to the u 1 coordinate curve at P. If e 1 is a unit vector at P in this
direction, we can write @r=@u 1 ¼ h 1 e 1 where h 1 ¼j@r=@u 1 j. Similarly we can write @r=@u 2 ¼ h 2 e 2 and
@r=@u 3 ¼ h 3 e 3 , where h 2 ¼j@r=@u 2 j and h 3 ¼j@r=@u 3 j respectively. Then (17) can be written