{"id":266,"date":"2018-11-16T04:39:47","date_gmt":"2018-11-16T04:39:47","guid":{"rendered":"http:\/\/msp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=266"},"modified":"2018-11-16T06:26:22","modified_gmt":"2018-11-16T06:26:22","slug":"curvilinear-coordinates","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp01\/chapter\/curvilinear-coordinates\/","title":{"rendered":"Curvilinear coordinates"},"content":{"raw":"
\r\n\r\nTABLE OF CONTENTS<\/strong>\r\n

1. Introduction<\/p>\r\n

2. General coordinate transformations<\/p>\r\n

2.1 Orthogonal curvilinear coordinates<\/p>\r\n

2.2 Unit vectors in curvilinear system<\/p>\r\n

2.3 Representation of a vector<\/p>\r\n

2.4 Element of arc length, surface and volume<\/p>\r\n

3. The gradient<\/p>\r\n

3.1 Equality of two bases<\/p>\r\n

3.2 The divergence<\/p>\r\n

3.3 The curl<\/p>\r\n

4. Specific coordinate systems<\/p>\r\n

4.1 Spherical coordinates<\/p>\r\n

4.2 Cylindrical coordinates<\/p>\r\n

4.3 Other orthogonal systems<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nLEARNING OBJECTIVES<\/strong>\r\n

1. Systems of coordinates other than the Cartesian coordinates, called curvilinear coordinates, are introduced.<\/p>\r\n

2. The idea of orthogonal coordinates is explained.<\/p>\r\n

3. The unit vectors, the Jacobian and the elements arc length, surface and volume are described in terms of these orthogonal curvilinear coordinates.<\/p>\r\n

4. Next expressions for differential operators, the gradient, divergence, curl and Laplacian are obtained.<\/p>\r\n

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.<\/p>\r\n

6. Some other coordinate systems are mentioned in passing.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

C u r v i l i n e a r\u00a0 C o o r d i n a t e s<\/strong><\/span><\/p>\r\n \r\n\r\n1. Introduction<\/strong><\/span>\r\n

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<\/em> = constant. In the Cartesian coordinate system the three axes are orthogonal to each other. Though, in principle, it is not necessary to do so, \u201corthogonality\u201d 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<\/em>, and then consider in detail the special case of the most common such coordinates, the spherical polar<\/em> and the cylindrical coordinates<\/em> in somewhat greater detail.<\/p>\r\n \r\n\r\n2. General transformation of coordinates<\/strong><\/span>\r\n\r\n\"\"\r\n\r\n2.1 Orthogonal curvilinear coordinates<\/strong>\r\n\r\n\"\"\r\n\r\n2.2 Unit vectors in curvilinear system<\/strong>\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\"\"\r\n\r\n<\/div>\r\n
\r\n

\u00a0 \u00a0 Thus at each point P<\/em> of a curvilinear coordinate system there exist two sets<\/em> 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<\/em>. Later on, in fact we will prove that in this case<\/p>\r\n\"\"\r\n\r\n2.3 Representation of a vector<\/strong>\r\n\r\n\"\"\r\n\r\n2.4 Elements of arc length, surface and volume<\/strong>\r\n\r\n\"\"\r\n

The square of the arc length is given by<\/p>\r\n\"\"\r\n

where we have used equation (13) for the unit vectors.<\/p>\r\n

\"\"\u00a0with similar results for the other two.<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

2.5 The Jacobian<\/strong><\/p>\r\n\"\"\r\n

From vector analysis we know that the determinant can also be written as a scalar triple product<\/p>\r\n\"\"\r\n\r\n3. The gradient<\/strong><\/span>\r\n

\"\"\u00a0For Cartesian coordinates the three scale factors are each equal to unity and equation (24) reduces to the usual expression for the gradient.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

3.1 Equality of the two bases<\/strong><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

3.2 The divergence<\/strong><\/p>\r\n

\"\"From equation (25)<\/span><\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n

3.3 The curl<\/strong><\/p>\r\n\"\"\r\n

Once again using expression (24) for the gradient, we have<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

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:<\/p>\r\n\"\"\r\n\r\n3.4 The Laplacian<\/strong>\r\n

Finally we will obtain expression for the Laplacian in orthogonal curvilinear coordinates. If \u03a6 is a scalar field, then<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n4. Specific orthogonal coordinate systems<\/strong><\/span>\r\n

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.<\/p>\r\n \r\n\r\n4.1 Spherical coordinates<\/strong>\r\n\r\n\"\"\r\n\r\nThe range of the three coordinates is\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n<\/div>\r\n

\r\n

4.2 Cylindrical coordinates<\/strong><\/p>\r\n\"\"\r\n

The unit vectors in these directions are<\/p>\r\n\"\"\r\n\r\nHence the three scale factors are<\/span>\r\n\r\n\"\"\r\n

The Jacobian of the transformation is<\/span><\/p>\r\n\"\"\r\n

and the volume element is<\/span><\/p>\r\n\"\"\r\n

Finally using equations (24), (31), (32), (33), the various differential operators are<\/span><\/p>\r\n\"\"\r\n\r\n \r\n\r\n4.3 Other orthogonal systems<\/span><\/strong>\r\n

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<\/span><\/p>\r\n\"\"\r\n

There is a simple relationship with the cylindrical coordinates:<\/span><\/p>\r\n\"\"\r\n

The coordinate surfaces in this case are <\/span>confocal <\/em><\/a>parabolic <\/a>cylinders<\/em>. These coordinates find many applications in potential theory. A typical example is the electrostatic field surrounding a flat semi-infinite conducting plate.<\/span><\/p>\r\n \r\n\r\nSUMMARY<\/strong>\r\n