CHAPTER 6
CURVILINEAR COORDINATE SYSTEMS
6.2. Curvilinear Coordinate Systems
6.2-1. Curvilinear Coordinates
6.2-2. Coordinate Surfaces and Coordinate Line
6.2-3. Elements of Distance and Volume
6.3. Representation of Vectors in Terms of Components
6.3-1. Vector Components and Local Base Vectors
6.3-2. Representation of Vectors in Terms of Physical Components
6.3-3. Representation of Vectors in Terms of Contravariant and Covariant Components
6.3-4. Derivation of Vector Relations in Terms of Curvilinear Components
6.4. Orthogonal Coordinate Systems.Vector Relations in Terms of Orthogonal Components
6.4-3. Line Integrals, Surface Integrals, and Volume Integrals
6.5. Formulas Relating to Special Orthogonal Coordinate Systems
6.6. Related Topics, References, and Bibliography
6.6-2. References and Bibliography
6.1-1. Chapter 6 deals with the description (representation) of scalar and vector functions of position (see also Secs.5.4-1 to 5.7-3) in terms of curvilinear coordinates (Sec.6.2-1). Vectors will be represented by components along, or perpendicular to, the coordinate lines at each point (Secs.6.3-1 to 6.3-3). The use of curvilinear coordinates simplifies many problems; one may, for instance, choose a curvilinear coordinate system such that a function under consideration is constant on a coordinate surface (Secs.6.4-3 and 10.4-lc).
In accordance with the requirements of many physical applications, Chap.6 is mainly concerned with orthogonal coordinate systems (Secs.6.4-1 to 6.5-1). The representation of vector relations in terms of non-orthogonal components is treated more elaborately in Chap. 16 in the context of tensor analysis.
6.2. CURVILINEAR COORDINATE SYSTEMS
6.2-1.Curvilinear Coordinates. A curvilinear coordinate system defined over a region V of three-dimensional Euclidean space labels each point (x, y, z) with an ordered set of three real numbers x1, x2, x3.* The
FIG.6.2-1. Illustrating coordinate line elements, coordinate surface elements, and volume element for a curvilinear coordinate system.
curvilinear coordinates x1, x2, x3 of the point (x, y, z) ≡ (xl, x2, x3) are related to the right-handed rectangular cartesian coordinates x, y, z (Sec.3.1-4) by three continuously differentiable transformation equations
where the functions (1) are single-valued and ∂(x1, x2, x3)/∂(x, y, z) ≠ 0 throughout V (admissible transformation, see also Secs.4.5-6 and 16.1-2).
* The indices 1, 2, 3 of x1, x2, x3 are superscripts, not exponents; see also Secs.16.1-2 and 16.1-3.
The x1, x2, x3 coordinate system is cartesian (Sec.3.1-2; not in general rectangular) if and only if the transformation equations (1) are linear equations.
6.2-2. Coordinate Surfaces and Coordinate Lines. The condition xi= xi(x, y, z) = constant defines acoordinate surface. Coordinate surfaces corresponding to different values of the same coordinate xi do not intersect in V. Two coordinate surfaces corresponding to different coordinates xi, xj intersect on the coordinate line corresponding to the third coordinate xk. Each point (x, y, z) ≡ (x1, x2,xz) of V is represented as the point of intersection of three coordinate surfaces, or of three coordinate lines.
6.2-3. Elements of Distance and Volume (Fig.6.2-1; see also Secs.6.4-3 and 17.3-3). (a)In terms of curvilinear coordinates x1, x2, x3, the element of distance ds between two adjacent points (x, y, z) ≡ (x1, x2, x3) and (x + dx, y + dy, z + dz) ≡ (x1 + dx1, x2 + dx2, + x3 + dx 3) is given by the quadratic differential form
The functions gik(x1, x2, x3) are the components of the metric tensor (Sec.16.7-1).
(b)The six coordinate surfaces associated with the points (x1, x2, x3) and (x1 + dxl, x2 + dx2, x3 + dx3) bound the parallelepipedal volume element
(see also Secs.5.4-7, 6.4-3c, and 16.10-10). 
is taken to be positive if the directed increments dx1, dx2, dxz, which define the positive directions on the coordinate lines, form a right-handed system (Sec.3.1-3). The curvilinear coordinate system is then right-handed throughout the region V; otherwise the coordinate system is left-handed throughout V (see also Sec. 16.7-1).
Refer to Secs.6.4-3, 17.3-3, and 17.4-2 for the description of vector path elements and surface elements in terms of curvilinear coordinates.
6.3. REPRESENTATION OF VECTORS
IN TERMS OF COMPONENTS
6.3-1.Vector Components and Local Base Vectors. In Chap. 5, a vector function of position was described in terms of right-handed rectangular cartesian components by reference to local base vectors i, j, k; the magnitude and direction of each cartesian base vector are the same at every point (x, y, z) (Secs.5.2-2 and 5.4-1). If the vector function F(r) is to be described in terms of curvilinear coordinates x1,x2,x3, it is useful to employ local base vectors directed along, or perpendicular to, the coordinate lines at each point (x1, x2, x3). Such base vectors are themselves vector functions of position. The following sections describe three different systems of local base vectors associated with each given curvilinear coordinate system.
6.3-2.Representation of Vectors in Terms of Physical Components. Given a curvilinear coordinate system defined as in Secs. 6.2-1, the unit vectors u1(x1, x2, 33), u2(x1, x2, x3), u3(x1, x2,x3) respectively directed along the positive x1, x2, x3 coordinate lines (6.2-3b) at each point (x1, x2, x3) ≡ (r) may be used as local base vectors (see also Sec. Sec. 16.8-3). Every vector F(r) ≡ F(x1, x2, x3) can be uniquely represented in the form
at each point (r) = (x1, x2, x3).
Refer to Table 6.3-1 for the transformation equations relating the functions
(physical components of F in the coordinate directions, see also Sec. 16.8-3) and the local base vectors u1{xl, x2, x3), u2(x1, x2, x3), u3(x1x2, x3) to their respective rectangular cartesian counterparts Fx, Fy, Fz and i, j,
k. Note that the functions
are the direction cosines of ui (i.e., of the ith-coordinate line) with respect to the x, y, z axes (see also Fig.6.3-1).
6.3-3.Representation of Vectors in Terms of Contravariant and Covariant Components. For any curvilinear coordinate system defined as in Sec.6.2-1, it is possible to introduce the system of local base vectors
directed along the coordinate lines, and the system of local base vectors
directed perpendicularly to the coordinate surfaces. Each vector F(r) can then be represented in the respective forms
at each point (r) = (x1, x2, x3). The magnitudes as well as the directions of the local base vectors ei ei and change from point to point, unless x1, x2, x3 are cartesian coordinates (Sec.16.6-la).
FIG. 6.3-1. Representation of a vector F in terms of unit vectors u1, u2, u3 and physical components 
The contravariant components Fi ≡ Fi(x1 x2, x3) ≡ F • ei and the covariant components Fi ≡ Fi(x1 x2, x3) ≡ F• ei(i= 1, 2, 3) in the scheme of measurements (Sec. 16.1-4) associated with each system of coordinates xl, x2, x3, and the associated base vectors ei and ei may also be defined directly by the transformations relating them to the rectangular cartesian components Fx, Fy, Fz and base vectors i, j, k (Table 6.3-1). Note (see also Secs.16.6-1 and 16.8-4)
Table 6.3-1. Transformations Relating Base Vectors and Vector Components Associated with Different Local Reference Systems
(a) Relations between i, j, k and the Local Base Vectors ui, ei, ei Associated with a Curvilinear Coordinate System
(b) Relations between Fx, Fy, Fz and the Physical components 
i, Contravariant Components Fi and Convariant Components Fi Associated with a Curvilinear Coordinate system
(c) Relations between Local Base Vectors and Vector Components Associated with Two Curvilinear Coordinate Systems (barred base vectors and vector components are associated with the 
system; see also sec. 16.2-1).
In the special case of a rectangular cartesian coordinate system, x1 ≡ x, x2 ≡ y, x3 ≡ z, and
A main advantage of the contravariant and covariant representation of vectors is the relative simplicity of the transformation equations relating contravariant (or covariant) vector components associated with different coordinate systems (Table 6.3-1; see also Sec. 6.3-4).
6.3-4. Derivation of Vector Relations in Terms of Curvilinear Components. In principle, every vector relation given, as in Chap. 5, in terms of rectangular cartesian components can be expressed in terms of curvilinear components with the aid of Table 6.3-1. If the relations in question involve differentiation and/or integration, note that the base vectors associated with curvilinear-coordinate systems are functions of position.
Many practically important problems permit the use of orthogonal coordinate systems (Sec. 6.4-1). In this case, the formulas of Secs.6.4-1 to 6.4-3 and Tables 6.4-1 to 6.5-11 yield comparatively simple expressions for many vector relations directly in terms of physical components. When these special methods do not apply, it is usually best to employ contravariant and covariant vector components rather than physical components; the formulation of vector analysis in terms of contravariant and covariant components is treated in detail in Chap. 16 as part of the more general subject of tensor analysis.
In particular, one uses the formulas of Secs. 16.8-1 to 16.8-4 for computing scalar and vector products, and the relatively straightforward method of covariant differentiation (Secs.16.10-1 to 16.10-8) yields expressions for differential invariants like ∇Φ, ∇ • F, and ∇ X F.
6.4. ORTHOGONAL COORDINATE SYSTEMS. VECTOR RELATIONS IN TERMS OF ORTHOGONAL COMPONENTS
6.4-1.Orthogonal Coordinates. An orthogonal coordinate system is a system of curvilinear coordinates x1, x2, x3 (Sec. 6.2-1) chosen so that the functions gik{xl, x2, x3) satisfy the relations
at each point (xl, x2, x3). The coordinate lines, and thus also the local base vectors u1 u2, u3, of an orthogonal coordinate system are perpendicular to each other at each point; each coordinate line is perpendicular to all coordinate surfaces corresponding to constant values of the coordinate in question.
6.4-2.Vector Relations, (a) The formulas of Table 6.4-1 express the most important vector relations in terms of orthogonal coordinates and components. The appropriate functions gii = gii{xl, x2, x3) for each specific orthogonal coordinate system are obtained from (Eq.6.2-2) or from Tables 6.5-1 to 6.5-11.
Table 6.4-1. Vector Formulas Expressed in Terms of Physical Components for Orthogonal Coordinate Systems (Sec. 6.4-1) Plus and minus signs refer, respectively, to right-handed and left-handed orthogonal coordinate systems. The appropriate functions gii{xl, x2, x3) ≡ |ei|2 are obtained from (6.2-2) or from Tables Eq. 6.5-1 to 6.5-11
(a) Scalar and Vector Products
(b) Differential Invariantas* (g ≡ g11g22g33)
(b)For every orthogonal coordinate system,
where the plus and minus signs refer, respectively, to right-handed and left-handed orthogonal coordinate systems.
NOTE : Expressions for ∇Φ, ∇ • F, and ∇ X F may be derived directly from the definitions of Sec. 5.5-1 with the aid of the volume element shown in FIG.6.2-1.
6.4-3. Line Integrals, Surface Integrals, and Volume Integrals (see also FIG.6.2-1.). (a) Given a rectifiable curve C (Sec. 5.4-4), the vector path element dv has the physical components 
in each curvilinear coordinate system (6.2-1), i.e.
The appropriate expression (6) for dr must be substituted in each line integral defined in Sec.5.4-5. Thus, for orthogonal coordinates x1, x2, x3
Note that for every curvilinear coordinate system
(see also Sec. 17.2-1); for orthogonal coordinates xl, x2, x3,
(b) The description of a surface S and of the vector surface element dh in terms of surface coordinates is discussed in Secs. 5.4-6 and 17.3-1. In particular, for orthogonal coordinates x1, x2, x3 chosen so that S is a portion of an xk coordinate surface (Sec6.2-2), the space coordinates xi and xj are (orthogonal) surface coordinates on S, and
The relative simplicity of the expressions (5.4-13) and (5.4-14) for surface integrals resulting from the use of (Eq.8) is often the main reason for introducing a curvilinear coordinate system (see also Sec.10.4-lc). The sign of the square root in (Eq.8) determines the direction of the positive normal (Sec.17.3-2) and is taken to be positive for right-handed coordinate systems.
(c)The volume element dV appearing in the expressions (5.4-16) and (5.4-17) for volume integrals is given by (Eq 6.2-3.), or
is positive wherever the coordinate system is right-handed (Sec.6.2-36).
6.5.FORMULAS RELATING TO SPECIAL
ORTHOGONAL COORDINATE SYSTEMS
6.5-1.Introduction. Tables 6.5-1 to 6.5-11 present formulas relating to a number of special orthogonal coordinate systems. In particular, the functions gii(xl, x2, x3) = |ei|2 are tabulated for each coordinate system and may be substituted in the relations of Table 6.4-1 and of Secs.6.4-3 and 16.10-1 to 16.10-8 to yield additional formulas.
NOTE: A given family of coordinate surfaces x1 = x1(x, y, z) = constant, x2 = x2{x, y, z) = constant, x3 = x3{x, y, z) = constant is necessarily common to all systems of coordinates x̄1 = xl{xl), x̄2 = x2(x2), x̄3 = x3(x3). The particular systems described in Tables 6.5-1 to 6.5-11 are representative.
6.6.RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY
6.6-1.Related Topics. The following topics related to the study of vector relations in terms of curvilinear coordinates are treated in other chapters of this handbook:
Solid analytic geometry Chap. 3
Elementary vector analysis Chap. 5
Coordinate transformations Chaps. 3,14
Contravariant and covariant vectors Chap. 16
Transformation of base vectors Chap. 16
Differential invariants Chap. 16
Differential geometry Chap. 17
Potential theory Chap. 15
6.6-2. References and Bibliography (see also Sec.5.8-2).
6.1.Courant, R., and D. Hilbert: Methods of Mathematical Physics, vol. I, Wiley, New York, 1953.
6.2.Kellogg, O. D.: Foundations of Potential Theory, Springer, Berlin, 1929.
6.3.MacMillan, W. D.: Theory of the Potential, McGraw-Hill, New York, 1930.
6.4.Madelung, E.: Die mathematischen Hilfsmittel des Physikers, 7th ed., Springer, Berlin, 1964.
6.5.Magnus, W., and F. Oberhettinger: Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954; 3d ed., Springer, Berlin, 1966.
6.6.Margenau, H., and G. M. Murphy: The Mathematics of Physics and Chemistry, Van Nostrand, Princeton, N.J., 1943.
6.7.Stratton, J. A.: Electromagnetic Theory, Chap. I, McGraw-Hill, New York, 1941.
6.8.Whittaker, E. T., and G. N. Watson: A Course of Modern Analysis, Cambridge, New York, 1927.
Table 6.5-1. Vector Formulas in Terms of Spherical and (Circular) Cylindrical Coordinates
(see also Figs. 2.1-2 and 3.1-1 b)
The formulas for cylindrical coordinates also apply to polar coordinates in the xy plane (Sec. 2.1-8)
* To find the Laplacian of a vector, ues 
or use Eq. (6.4-5) and Table 16.10-1.
Table 6.5-2. General Ellipsoidal Coordinates λ, μ, ν or u, υ, ω
(a) Coordinate Surfaces (solve each equation to obtain λ, μ, ν in terms of x, y, z)
(b) Transformation to Ellipsoidal Coordinates
(c) Alternative System (see Ref. 6.4 for other alternative systems). Introduce u, v, w by
so that (Sec. 21.6-2)
with 
(d) 
(e) 
Table 6.5-3. Prolate Spheroidal Coordinates σ, Ƭ,φ or u, υ, φ (z axis is axis of revolution; see also FIG.6.2-1.)
(a) Coordinate Surfaces (solve each equation to obtain σ, Ƭ,φ in terms of x, y, z)
All spheroids and hyperboloids have common foci (0, 0, α), (0, 0, -α); 2ασ and 2ατ are, respectively, the sum and difference of the focal radii of the point (x, y).
(b) Transformation to Elliptic cylindrical Coordinates
(c) Alternative System (removes ambiguities)
(d) 
(e) 
Table 6.5-4. Oblate Spheroidal Coordinates σ, Ƭ, φ or u, ν, φ (z axis is axis of revolution; see also FIG.6.5.1)
(a) Coordinate Surfaces (solve each equation to obtain σ, Ƭ,φ in terms of x, y, z)
(b) Transformation to Paraboloidal Coordinates
(c) Alternative System (removes ambiguities)
(d) 
(e) 
FIG.6.5.1. An orthogonal system of confocal ellipses and hyperbolas with foci F, F′. Such curves define an elliptic coordinate system in their plane and will generate coordinate surfaces of
1. A prolate spheroidal coordinate system if the figure is rotated about the axis P′OP (Table 6.5-3)
2. An oblate spheroidal coordinate system if the figure is rotated about the axis Q′OQ (Table 6.5-4)
3. An elliptic cylindrical coordinate system if the figure is translated at right angles to the plane of the paper (Table 6.5-5)
FIG. 6.5-2.An orthogonal system of confocal parabolas with focus F. Such curves define a parabolic coordinate system in their plane and will generate coordinate surfaces of
1. A parabolic coordinate system if the figure is rotated about the axis P′OP (Table 6.5-8)
2. A parabolic cylindrical coordinate system if the figure is translated at right angles to the plane of the paper (Table 6.5-9)
Table 6.5-5. Elliptic Cylindrical Coordinates σ, Ƭ, z or u, υ, z (also used as confocal elliptic coordinates in xy plane; see also Fig.6.5-1)
(a) Coordinate Surfaces (solve each equation to obtain σ, Ƭ, z in terms of x, y, z)
(b) Transformation to Prolate Spheroidal Coordinates
(c) Alternative System (removes ambiguities)
(d) 
(e) 
Table 6.5-6. Conical Coordinates γ, υ, ω
(a) Coordinate Surfaces (solve each equation to obtain r, v, w in terms of x, y, z)
(b) Transformation to Prolate Spheroidal Coordinates
(c) 
Table 6.5-7. Paraboloidal Coordinates λ, μ, ν
(a) Coordinate Surfaces (solve each equation to obtain λ, μ, ν in terms of x, y, z)
(b) Transformation to Prolate Spheroidal Coordinates
(c) 
Table 6.5-8. Parabolic Coordinates σ, γ, φ (z axis is axis of revolution; see also Fig.6.5-2)
(a) Coordinate Surfaces (solve each equation to obtain σ, γ, φ in terms of x, y, z)
(b) Transformation to Prolate Spheroidal Coordinates
(c) 
(d) 
Table 6.5-9. Parabolic Cylindrical Coordinates σ, γ, z (also used as parabolic coordinates in xy plane; see alsoFig. 6.5-2)
(a) Coordinate Surfaces (solve each equation to obtain σ, γ, z in terms of x, y, z)
(b) Transformation to Prolate Spheroidal Coordinates
(c) 
(d) 
FIG. 6.5-3. A family of circles through two poles A, B, and the family of circles orthogonal to those of the first family. Such curves define a bipolar coordinate system in their plane and will generate coordinate surfaces of
1. A bipolar coordinate system if the figure is translated at right angles to the plane of the paper (Table 6.5-10)
2. A toroidal coordinate system if the figure is rotated about the axis Q′OQ (Table 6.5-11)
Table 6.5-10. Bipolar Coordinates σ, Ƭ, z (also used as bipolar coordinates in xy plane; see also Fig.6.5-3)
(a) Coordinate Surfaces
For any point (x, y) in the xy plane, σ is the angle subtended at (x, y) by the two poles (-α, 0) and (α, 0). er is the ratio of the polar radii to (x, y).
(b) Transformation
(c) 
(d) 
Table 6.5-11. Toroidal Coordinates σ, Ƭ, φ (z axis is axis of revolution; see also Fig.6.5-3)
(a) Coordinate Surfaces
(b) Transformation
(c) 
(d) 
* To obtain ∇2F, use ∇2F ≡∇(∇ • F) – ∇ X (∇ X F), or refer to Sec. 16.10-7. To obtain (F • ∇)G, refer to Table 5.5-1 or to Sec.16.10-7.
