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GENERAL CURVILINEAR SYSTEMS 49
X Figure 3
1 Y
The Position Vector in -2herical Coordinates
The position vector, shown in Figure 3.7, is expressed in the spherical system as
- r = (F. C,)$ + (F - &)CO + (F. @+)@+. (3.14)
Because F is always perpendicular to 20 and $4, Equation 3.14 simplifies to
3.4 GENERAL CURVILINEAR SYSTEMS
Although the most common, cylindrical and spherical coordinate systems are just two examples of the larger family of curvilinear systems. A system is classified as curvilinear if it has orthonormal, but not necessarily constant, basis vectors. Other more esoteric curvilinear systems include the toroidal, hyperbolic, and elliptical systems. Instead of individually working out the vector operations of the previous chapter for each of these systems, we present a general approach that can tackle any curvilinear geometry.
The coordinates (41, q 2 , q 3 ) and corresponding basis vectors ($1, q 2 , q 3 ) will be used to represent any generic curvilinear system, as shown in Figure 3.8. Because these
Coordinates, Basis Vectors, and Scale Factors
Y
X Figure 3.8 Curvilinear Coordinates and Basis Vectors
50 CURVILINEAR COORDINATE SYSTEMS
basis vectors are functions of position, we should always be careful to draw them emanating from a particular point, as we mentioned earlier in this chapter. In both the cylindrical and spherical coordinate systems, a set of equations existed which related these coordinates to a “standard” set of Cartesian coordinates. For the general case, we write these equations as
xi = xi(q19q27q3) (3.16)
qi = qi(xl,x2,x3), (3.17)
where the subscript notation has crept in to keep things concise. In both these equa-
tions, the subscript i takes on the values (1,2,3). The variables xi always represent
Cartesian coordinates, while the qi are general curvilinear coordinates. An expression for qi, the unit basis vector associated with the coordinate qi. can be constructed by increasing qi. watching how the position vector changes, and then normalizing:
where hi = ldF/dqiI. This equation is a little confusing, because there actually is
no sum over the i index on the RHS, even though it appears twice. This is subtly
implied by the notation, because there is an i subscript on the LHS. The hi, which are sometimes called scale factors, force the basis vectors to have unit length. They
can be written in terms of the curvilinear coordinates. To see this, write the position
vector in terms of its Cartesian components, which in turn are written as functions of
the curvilinear coordinates:
(3.19)
Therefore,
and
hi = 121 = d m.
The physical interpretation of the scale factors is quite simple. For a change dql of the coordinate 41, the position vector changes by a distance of ldql hl I. Therefore, using Equation 3.18, the displacement vector can be written in the curvilinear system as
