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414 CHAPTER 12 Vector Integral Calculus

12.10 Curvilinear Coordinates

Thus far, we have done vector algebra and calculus in rectangular coordinates. For some set-
tings, other coordinate systems may be more convenient. Spherical coordinates are natural when
dealing with spherical surfaces, cylindrical coordinates for cylinders, and sometimes we invent
systems to deal with other settings we may encounter.
Begin with the usual rectangular coordinate system with axes labeled x, y and z. Suppose
we have some other coordinate system with coordinates labeled q 1 ,q 2 and q 3 . We assume that
the two systems are related by equations
x = x(q 1 ,q 2 ,q 3 ), y = y(q 1 ,q 2 ,q 3 ), z = z(q 1 ,q 2 ,q 3 ). (12.11)

We also assume that these equations are invertible and can be solved for
q 1 = q 2 (x, y, z),q 2 = q 2 (x, y, z),q 3 = q 3 (x, y, z).

In this way we can convert the coordinates of points back and forth from one system to the other.
Finally, we assume that each point in 3-space has exactly one set of coordinates (q 1 ,q 2 ,q 3 ),asit
does in rectangular coordinates. We call (q 1 ,q 2 ,q 3 ) a system of curvilinear coordinates.

EXAMPLE 12.28 Cylindrical Coordinates
As shown in Figure 12.27, a point P having rectangular coordinates (x, y, z) can be specified
uniquely by a triple (r,θ, z), where (r,θ) are polar coordinates of the point (x, y) in the plane,
and z is the same in both rectangular and cylindrical coordinates (the distance from the x, y-plane
to the point).
These coordinate systems are related by

x =r cos(θ), y =r sin(θ), z = z
with 0 ≤ θ< 2π, r ≥ 0 and z any real number. With some care in using the inverse function
tangent function, these equations can be inverted to write
y

2
2
r = x + y ,θ = arctan , z = z.
x
z
P

z
y
θ
r

x
FIGURE 12.27 Cylindrical coordi-
nates of Example 12.28.

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October 14, 2010 14:53 THM/NEIL Page-414 27410_12_ch12_p367-424