Signals and systems lecture s2 orthogonal functions and fourier series march 17, 2008 todays topics 1. The common differential operators including the vector laplacian are expressed in arbitrary curvilinear coordinates. Signals and systems lecture s2 orthogonal functions and. So far we have only used cartesian x,y,z coordinates. 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 di erential measures and vector operators for any orthogonal coordinates system are easily constructed once the scale factors are determined. For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates. Curvilinear coordinate an overview sciencedirect topics. 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 the results shown in section 29. In this chapter we will write the general form of the differential. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocityvorticity formulations. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame. 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.
Orthogonal curvilinear coordinates particle in cell. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates. 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. These coordinates may be derived from a set of cartesian coordinates. Pdf strain gradient theory in orthogonal curvilinear. Periodic signals can be represented as a sum of sinusoidal functions. Adjustments need to be made in the calculation of line, surface and volume integrals. Two coordinate systems cylindrical polar coordinates and. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Boundary layer formulations in orthogonal curvilinear.
Fourier series take away periodic complex exponentials have properties analogous to vectors in n dimensional spaces. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. Some useful relations in the algebra of vectors and secondorder tensors in curvilinear coordinates are given in this section. Orthogonality of a curvilinear coordinate system physics forums. A curvilinear coordinate system expresses rectangular. You are familiar, for example, with polar coordinates r.
In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the. Vector and tensor calculus in threedimensional curvilinear coordinates note. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. Note on curvilinear coordinates physics libretexts. Hamiltonian for a particle in a magnetic eld on a curved surface in orthogonal curvilinear coordinates m. 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. Orthogonal curvilinear coordinates university of notre dame. In your past math and physics classes, you have encountered other coordinate systems such as cylindri. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. 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.
Orthogonal curvilinear coordinate systems suppose that the cartesian coordinates,x y z. Strain gradient theory in orthogonal curvilinear coordinates. Mar 17, 2016 curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often. Learning the basics of curvilinear analysis is an essential first step to. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. The laplacian in curvilinear coordinates the full story peter haggstrom. It should be recognized that the derivation can also be accomplished using tensor analysis 1. The treatment here is standard, following that in abraham and becker, classical theory of electricity and magnetism. Lautrup december 17, 2004 1 curvilinear coordinates let xi with i 1. Pdf orthogonal curvilinear coordinates mohamed sobhy. In orthogonal curvilinear coordinates, since the total differential change in r is.
It should be recognized that the derivation can also be accomplished using tensor analysis. In this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system. For example, in cartesian coordinates where ql x, q2 y, qa z, we find that hi h2 ha i. For example calculations in spherical coordinates result easier for systems with spherical symmetry. We will begin with the case that the coordinates are orthogonal. 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. We shall use ordinary cartesian vector notation x x1. However, other coordinate systems can be used to better describe some. 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. A system of curvilinear coordinates u, v, w is used to describe spiral bevel and hypoid gears. Orthogonality of a curvilinear coordinate system physics. Curvilinear coordinates fujiun jiang october 11, 2010 i. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Expressions for the gradient, divergence and curl in terms of curvilinear coordinates.
Gradient, divergence and curl in curvilinear coordinates. Curvilinear coordinates and vector calculus 3 3 3 3 2 2 2 1 1 1 h u h u h. Cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. This section shows how to calculate these derivatives in other coordinate systems. 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.
Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems. Aug 07, 2015 which means that i cant avoid finding r. More general coordinate systems, called curvilinear coordinate. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. Div, grad and curl in orthogonal curvilinear coordinates. Since in these systems lines of constant components e. Pdf mathematical physics lessons gradient, divergence and. The main application is the subject of curvilinear coordinates.
The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. The focus of this study was restricted to the derivation and application of orthogonal threedimensional coordinate systems. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates. Learning the basics of curvilinear analysis is an essential first step to reading much of the. 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. The laplacian in curvilinear coordinates the full story. Curvilinear coordinates trigonometry chegg tutors youtube. 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.
For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. Here we shall derive the appropriate relations for the latter using vector technique. A special case of these are orthogonal curvilinear coordinates. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. The notation and contents are primarily from ogden, naghdi, simmonds, green and zerna, basar and weichert, and ciarlet. Orthogonal curvilinear coordinates often it is convenient to use coordinate systems other than rectangular ones. Orthogonal curvilinear coordinates and div, grad, curl. 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. It will be demonstrated that results in many existing studies can be covered as special cases by our formulations. 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.
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