partial derivative ∂u∂r .Similarly, the curves r =r (c 1 , v, c 3 )and r =r (c 1 , c 2 , w )are coor-
dinate curves and have the respective tangent vectors ∂r∂v and ∂w∂r .One can calculate
the magnitude of these tangent vectors by defining the scalar magnitudes as
h 1 =hu=|∂r
∂u |, h^2 =hv=|∂r
∂v |, h^3 =hw=|∂r
∂w |. (8 .75)The unit tangent vectors to the coordinate curves are given by the relations
ˆeu=^1
h 1∂r
∂u, ˆev=^1
h 2∂r
∂v, eˆw=^1
h 3∂r
∂w. (8 .76)
The coordinate surfaces and coordinate curves may be formed from the equations
(8.68) and are illustrated in figure 8-15
Figure 8-15. Coordinate curves and surfaces.
Consider the point u=c 1 , v =c 2 , w =c 3 in the curvilinear coordinate system. This
point can be viewed as being created from the intersection of the three surfaces
u=u(x, y, z) = c 1
v=v(x, y, z ) = c 2
w=w(x, y, z) = c 3