5 Curvilinear coordinates
TABLE OF CONTENTS
1. Introduction
2. General coordinate transformations
2.1 Orthogonal curvilinear coordinates
2.2 Unit vectors in curvilinear system
2.3 Representation of a vector
2.4 Element of arc length, surface and volume
3. The gradient
3.1 Equality of two bases
3.2 The divergence
3.3 The curl
4. Specific coordinate systems
4.1 Spherical coordinates
4.2 Cylindrical coordinates
4.3 Other orthogonal systems
LEARNING OBJECTIVES
1. Systems of coordinates other than the Cartesian coordinates, called curvilinear coordinates, are introduced.
2. The idea of orthogonal coordinates is explained.
3. The unit vectors, the Jacobian and the elements arc length, surface and volume are described in terms of these orthogonal curvilinear coordinates.
4. Next expressions for differential operators, the gradient, divergence, curl and Laplacian are obtained.
5. Two most common and important curvilinear coordinates, spherical and cylindrical coordinates, are described in detail. Explicit expressions for Jacobian, the elements of arc length, surface and volume and the various differential operators are obtained.
6. Some other coordinate systems are mentioned in passing.
C u r v i l i n e a r C o o r d i n a t e s
1. Introduction
For the description of physical processes we need a system of coordinates. The most convenient and the most common coordinate system that is employed to measure the position of a particle, the magnitude and direction of vectors and tensors etc., is the Cartesian coordinate system that we have studied so far. However there are many situations where the inherent symmetry of the system makes it difficult, sometimes almost impossible, to work with the Cartesian coordinates. If a system has spherical symmetry, spherical polar coordinates are the natural ones to be used rather than the Cartesian coordinates. For example, the equation for the surface of a sphere in spherical polar coordinates is simply r = constant. In the Cartesian coordinate system the three axes are orthogonal to each other. Though, in principle, it is not necessary to do so, “orthogonality” of the coordinates leads to huge simplification. So we would like the alternative system that we develop and employ should also have this feature of orthogonality built into it. In this module we develop the general theory of these alternative coordinate systems, the general orthogonal curvilinear coordinates, and then consider in detail the special case of the most common such coordinates, the spherical polar and the cylindrical coordinates in somewhat greater detail.
2. General transformation of coordinates
2.1 Orthogonal curvilinear coordinates
2.2 Unit vectors in curvilinear system
Thus at each point P of a curvilinear coordinate system there exist two sets of unit vectors, which are in general distinct from each other. The two sets are identical, if and only if, the curvilinear system is orthogonal. Later on, in fact we will prove that in this case
2.3 Representation of a vector
2.4 Elements of arc length, surface and volume
The square of the arc length is given by
where we have used equation (13) for the unit vectors.
with similar results for the other two.
2.5 The Jacobian
From vector analysis we know that the determinant can also be written as a scalar triple product
3. The gradient
For Cartesian coordinates the three scale factors are each equal to unity and equation (24) reduces to the usual expression for the gradient.3.1 Equality of the two bases
3.2 The divergence
From equation (25)
3.3 The curl
Once again using expression (24) for the gradient, we have
Dealing with the other two expressions in the curl, collecting all the terms together, we have for the curl of a vector field in orthogonal curvilinear coordinates:
3.4 The Laplacian
Finally we will obtain expression for the Laplacian in orthogonal curvilinear coordinates. If Φ is a scalar field, then
4. Specific orthogonal coordinate systems
There is a large number of orthogonal coordinate systems that are of use in various problems of physics. Which system to employ depends upon the symmetry of the problem. For systems with linear symmetry, the best choice obviously is the Cartesian coordinates. Apart from this, the problems studied most commonly have either a spherical or a cylindrical symmetry. We will study these two in detail and mention a few others in passing.
4.1 Spherical coordinates
The range of the three coordinates is
4.2 Cylindrical coordinates
The unit vectors in these directions are
Hence the three scale factors are
The Jacobian of the transformation is
and the volume element is
Finally using equations (24), (31), (32), (33), the various differential operators are
4.3 Other orthogonal systems
The coordinate surfaces in case of Cartesian coordinates are planes. For the spherical and cylindrical coordinates the surfaces are planes, circles or cylinders. The nomenclature depends on the nature of the coordinate surface. Thus we have parabolic cylindrical coordinates, paraboloidal coordinates, elliptic cylindrical coordinates, prolate spheroidal coordinates, oblate spheroidal coordinates , ellipsoidal coordinates and bipolar coordinates, apart from a few others. To give just one example, the parabolic cylindrical coordinates are
There is a simple relationship with the cylindrical coordinates:
The coordinate surfaces in this case are confocal parabolic cylinders. These coordinates find many applications in potential theory. A typical example is the electrostatic field surrounding a flat semi-infinite conducting plate.
SUMMARY
- We introduce the idea of alternative coordinate system to describe physical processes which are useful depending on the symmetry of the system.
- We explain the idea of orthogonal curvilinear coordinates as coordinates in which surfaces perpendicular to the three coordinates being normal to each other.
- We then obtain general expressions for the unit vectors, the Jacobian and the elements of arc length, surface and volume for these orthogonal curvilinear coordinates.
- Next we obtain expressions for differential operators, the gradient, divergence, curl and Laplacian.
- Then we study in detail the two most common and important curvilinear coordinates, spherical and cylindrical coordinates. We obtain explicit expressions for Jacobian, the elements of arc length, surface and volume and the various differential operators for these two systems.
- Finally we mention some other coordinate systems in passing.