This section shows how to calculate these derivatives in other coordinate systems. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. More general coordinate systems, called curvilinear coordinate. Two coordinate systems cylindrical polar coordinates and. Up until now, a rectangular cartesian coordinate system has been used, and a set of orthogonal unit base vectors i e has been employed as. Since in these systems lines of constant components e. For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. In this chapter we will write the general form of the differential.
For example calculations in spherical coordinates result easier for systems with spherical symmetry. Learning the basics of curvilinear analysis is an essential first step to reading much of the. Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems. Shikakhwa department of physics, the university of jordan, amman 11942 jordan and middle east technical university northern cyprus campus,kalkanl, guzelyurt, via mersin 10, turkey n. The di erential measures and vector operators for any orthogonal coordinates system are easily constructed once the scale factors are determined. Orthogonal curvilinear coordinates often it is convenient to use coordinate systems other than rectangular ones. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame. Div, grad and curl in orthogonal curvilinear coordinates. Strain gradient theory in orthogonal curvilinear coordinates. Signals and systems lecture s2 orthogonal functions and fourier series march 17, 2008 todays topics 1.
Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often. Orthogonal curvilinear coordinates the results shown in section 29. Cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. It should be recognized that the derivation can also be accomplished using tensor analysis 1. For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates. Pdf mathematical physics lessons gradient, divergence and. Here we shall derive the appropriate relations for the latter using vector technique.
Orthogonal curvilinear coordinate systems include bipolar cylindrical coordinates, bispherical coordinates, cartesian coordinates, confocal ellipsoidal coordinates, confocal paraboloidal coordinates, conical coordinates, cyclidic coordinates, cylindrical coordinates, ellipsoidal coordinates, elliptic cylindrical coordinates, oblate spheroidal. Expressions for the gradient, divergence and curl in terms of curvilinear coordinates. Gradient, divergence and curl in curvilinear coordinates. For example, the threedimensional cartesian coordinates x, y, z is an orthogonal coordinate system, since its coordinate surfaces x constant, y constant, and z constant are planes that meet at right angles to one another, i. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. A special case of these are orthogonal curvilinear coordinates. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocityvorticity formulations. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the.
Note on curvilinear coordinates physics libretexts. Boundary layer formulations in orthogonal curvilinear. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved.
Orthogonality of a curvilinear coordinate system physics forums. In this short note, general formulations of the toupinmindlin strain gradient theory in orthogonal curvilinear coordinate systems are derived, and are then specified for the cases of. Once an origin has been xed in space and three orthogonal scaled axis are anchored to this origin, any point in space is uniquely determined by three real numbers, its cartesian coordinates. Periodic signals can be represented as a sum of sinusoidal functions. However, as transforming all the way to cartesian coordinates seems to be a minor nightmare i hope to show that uvw is orthogonal in spherical space, which got no proof implies its orthogonal also in cartesian space. You are familiar, for example, with polar coordinates r. The notation and contents are primarily from ogden, naghdi, simmonds, green and zerna, basar and weichert, and ciarlet. Orthogonal curvilinear coordinates and div, grad, curl.
So far we have only used cartesian x,y,z coordinates. Later we generalize the results to the more general setting, orthogonal curvilinear coordinate system and it will be a matter of taking into account the scale factors. Curvilinear coordinates and vector calculus 3 3 3 3 2 2 2 1 1 1 h u h u h. Adjustments need to be made in the calculation of line, surface and volume integrals.
These coordinates may be derived from a set of cartesian coordinates. However, other coordinate systems can be used to better describe some. The focus of this study was restricted to the derivation and application of orthogonal threedimensional coordinate systems. The treatment here is standard, following that in abraham and becker, classical theory of electricity and magnetism. Pdf orthogonal curvilinear coordinates mohamed sobhy.
In your past math and physics classes, you have encountered other coordinate systems such as cylindri. Lautrup december 17, 2004 1 curvilinear coordinates let xi with i 1. For example, in cartesian coordinates where ql x, q2 y, qa z, we find that hi h2 ha i. We shall use ordinary cartesian vector notation x x1. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over windgenerated surface waves. The laplacian in curvilinear coordinates the full story. It will be demonstrated that results in many existing studies can be covered as special cases by our formulations. Signals and systems lecture s2 orthogonal functions and. Aug 07, 2015 which means that i cant avoid finding r.
Curvilinear coordinate an overview sciencedirect topics. A system of curvilinear coordinates u, v, w is used to describe spiral bevel and hypoid gears. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Curvilinear coordinates trigonometry chegg tutors youtube. Pdf strain gradient theory in orthogonal curvilinear. Orthogonality of a curvilinear coordinate system physics. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Orthogonal curvilinear coordinates particle in cell. Orthogonal curvilinear coordinate systems suppose that the cartesian coordinates,x y z. Fourier series take away periodic complex exponentials have properties analogous to vectors in n dimensional spaces. Vector and tensor calculus in threedimensional curvilinear coordinates note. It should be recognized that the derivation can also be accomplished using tensor analysis.
In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. Div, grad, curl, and the laplacian the most common way that the gradient of a function, the divergence of a vector field, and the curl of a vector field are presented is entirely algebraic with barely any indication of what these mean. The laplacian in curvilinear coordinates the full story peter haggstrom. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates.
The coordinates u, v, w used to parameterize these families of pitch, transverse, and axial surfaces are formulated using the cylindroid defined by the input and output axes of rotation. Orthogonal curvilinear coordinates university of notre dame. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates. Hamiltonian for a particle in a magnetic eld on a curved surface in orthogonal curvilinear coordinates m. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. A curvilinear coordinate system expresses rectangular. Nov 26, 2015 in geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. Some useful relations in the algebra of vectors and secondorder tensors in curvilinear coordinates are given in this section. Mar 17, 2016 curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. In orthogonal curvilinear coordinates, since the total differential change in r is. Curvilinear coordinates fujiun jiang october 11, 2010 i. Another reason to learn curvilinear coordinates even if you never explicitly apply the knowledge to any practical problems is that you will develop a far deeper understanding of cartesian tensor analysis. In this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system.
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