The vector grad u(x, y, z )is a vector normal to the surface u=c 1 .A unit normal
to the u=c 1 surface has the form
Eu= grad u
|grad u|.
Similarly, the vectors
Ev= grad v
|grad v|
, and Ew= grad w
|grad w|
are unit normal vectors to the surfaces v=c 2 and w=c 3.
The unit tangent vectors ˆeu, ˆev, ˆew and the unit normal vectors Eu, Ev, Ew
are identical if and only if gij = 0 for i=j;for this case, the curvilinear coordinate
system is called an orthogonal coordinate system.
Example 8-14. Consider the identity transformation between (x, y, z ) and
(u, v, w ).We have u=x, v =y, and w=z. The position vector is
r (x, y, z) = xˆe 1 +yˆe 2 +zˆe 3 ,
and in this rectangular coordinate system, the element of arc length squared is given
by ds^2 =dx^2 +dy^2 +dz^2 .In this space the metric components are
gij =
1 0 0
0 1 0
0 0 1
,
and the coordinate system is orthogonal.
Figure 8-16. Cartesian coordinate system.
