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416 CHAPTER 12 Vector Integral Calculus

z
∇q (P )
3
0
Half-plane θ = θ 0
Cone φ = φ 0 q = k 2
2
ρ 0 q 1 = k 1

y
θ 0 ∇q (P ) q = k 3 ∇q 2 (P )
0
3
1
0
x
FIGURE 12.30 Coordinate surfaces in curvilinear
FIGURE 12.29 Intersection of coordinate
coordinates.
surfaces in spherical coordinates.
In general curvilinear coordinates, which need not be any of these three systems, we sim-
ilarly specify a point ((q 1 ) 0 ,(q 2 ) 0 ,(q 3 ) 0 ) as the intersection of the three coordinate surfaces
q 1 = (q 1 ) 0 , q 2 = (q 2 ) 0 and q 3 = (q 3 ) 0 (Figure 12.30).
In rectangular coordinates, the coordinate surfaces are planes x = x 0 , y = y 0 , z = z 0 , which
are mutually orthogonal. Similarly, in cylindrical and spherical coordinates, the coordinate sur-
faces are mutually orthogonal, in the sense that their normal vectors are mutually orthogonal at
any point of intersection. Because of this, we refer to these coordinate systems as orthogonal
curvilinear coordinates.

EXAMPLE 12.30
We will verify that cylindrical coordinates are orthogonal curvilinear coordinates. In terms of
rectangular coordinates, cylindrical coordinates are given by

2
2
r = x + y ,
y

θ = arctan
x
z = z,
except at the origin, which is called a singular point of these coordinates. Suppose P 0 is the point
of intersection of the cylinder r = r 0 , the half-plane θ = θ 0 and the half-plane z = z 0 .Toverify
that these surfaces are mutually orthogonal, we will show that their normal vectors are mutually
orthogonal. Compute these normal vectors using the gradient in rectangular coordinates:
1
(xi + yj),
∇r =
x + y 2
2
1
∇θ = (−yi + xj),∇z = k.
x + y 2
2
Now it is routine to verify that
∇r ·∇θ =∇r ·∇z =∇θ ·∇z = 0.

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October 14, 2010 14:53 THM/NEIL Page-416 27410_12_ch12_p367-424