Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. See orthogonal one person multiple careers pdf coordinates for the important special case, skew coordinates for a. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. If you are dealing with a flat, euclidean, geometry, im not sure that you need curvilinear coordinates. With values of the cartesian coordinates known in the field as functions of the curvilinear coordinates, the network of intersecting lines formed by contours surfaces in 3d on which a curvilinear coordinate is constant, i. Orthogonal curvilinear coordinate systems suppose that the cartesian coordinates x,yz are expressed in terms of the new coordinates x12 3,xx by the equations x xx x x 12 3, yyxxx 12 3, zzxxx 12 3, where it is assumed that the correspondence is unique and that the inverse mapping exists. Expression of strain tensor in orthogonal curvilinear coordinates. The relation between cartesian and cylindrical coordinates is given by. To find the curvilinear coordinate for three functions of f f 1 x, y, z, f 2 x, y, z, and f 3 x, y, z, set each function to a. General curvilinear coordinate systems sciencedirect. Newtonian molecular dynamics in general curvilinear internal coordinates article in chemical physics letters 35356. In our study of curvilinear coordinates, we will examine this.
Two coordinate systems cylindrical polar coordinates. The name curvilinear coordinates, coined by the french mathematician lame, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. A set of coordinates u ux,y,z, v vx,y,z and w wx,y,z where the direc. Most of the material presented in this chapter is taken from anupam, g.
The curvature kl becomes the instantaneous curvature of the reference agent. Curvilinear coordinates fujiun jiang october 11, 2010 i. Orthogonal curvilinear coordinates and div, grad, curl. Cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. An introduction to curvilinear orthogonal coordinates overview throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar cartesian x,y,z coordinate system. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. These are two important examples of what are called curvilinear coordinates. Strain gradient theory in orthogonal curvilinear coordinates jidong zhao, dorival pedroso centre for geotechnical and materials modelling, university of newcastle, university drive, callaghan, nsw 2308, australia received 11 september 2007. This means that one can convert a point given in a cartesian coordinate system to its curvilinear coordinates and. Curvilinear coordinates are a coordinate system where the coordinate lines may be curved.
In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. This document provides a very nice treatment of coordinate transformation using the index notation. Rekach, manual on solutions of elasticity problems, vyshaya shkola. Threedimensional curvilinear motion describes the most general case of motion for a particle. Curvilinear analysis in a euclidean space mechanical engineering. The cylindrical polar system is related to cartesian coordinates x.
Derivatives of the unit vectors in orthogonal curvilinear coordinate systems. Piercey november 21, 2007 1 introduction on the inside of the cover of my undergraduate electromagnetism textbook ps02, there were cryptic formulae for di. Advanced fluid dynamics 2017 navier stokes equation in. With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and twodimensional body shapes of interest to aerodynamicists. The main application is the subject of curvilinear coordinates. In your past math and physics classes, you have encountered other coordinate systems such as cylindri. Incompressible ns equations in cylindrical polar systems. Divergence in curvilinear coordinates, continued first show that r e3 h1h2 0 problem 1 assume e 1 e 2 e 3 orthogonal coordinate system, and then obviously rx 1 e1 h1 and rx 2 e2 h2, and rx 1 rx 2 e3 h1h2, and next r e 3 h 1h 2 r rx 1 rx 2 the vector relations at the end of chapter 6 help to work out the righthand. Treating bodies as particles is, of course, an idealization which involves an approximation. With these specific examples in mind, lets go back the the general case, and see. These same examples are used to illustrate the general ideas.
Spiegel, schaums outline of vector analysis, chapter 7 and part of chap. However, other coordinate systems can be used to better describe some physical situations. These coordinates may be derived from a set of cartesian coordinates by using a transformation that is locally invertible a onetoone map at each point. The kinetic energy, t, may be expressed in terms of either r. Wellknown examples of curvilinear coordinate systems in threedimensional euclidean space r 3 are cylindrical and spherical polar coordinates. The curved path can be in two dimensions in a plane, or in three dimensions. Gradient, divergence and curl in curvilinear coordinates. This cylindrical system is itself a special case of curvilinear coordinates in that the base vectors are always orthogonal to each other. Such coordinates qare called generalized coordinates. Sekhar chivukula for phy 415810 at michigan state university. Orthogonal, curvilinear coordinates, author unknown.
This section shows how to calculate these derivatives in other coordinate systems. General dynamic formations for nonholonomic systems. The common differential operators including the vector laplacian are expressed in arbitrary curvilinear coordinates. Suppose now we take an infinitesimally small cube with edges parallel to the local curvilinear coordinate directions, and therefore with faces satisfying u i constant, i 1, 2, 3 for the three pairs of faces. If you are dealing with a noneuclidean geometry, then almost by definition there is no linear coordinate system that will work everywhere ove. Ntis hc a03bp a01 cscl 200 cjnclas 6334 0 1001 17 august 1987 nasa. For cartesian coordinates the normalized basis vectors are e. Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of cartesian coordinates, which. The key to deriving expressions for curvilinear coordinates is to consider the arc.
The presentation here closely follows that in hildebrand 1976. Let ul, u2 u3 represent the three coordinates in a general, curvilinear. Incompressible ns equations in orthogonal curvilinear coordinate systems. Newtonian molecular dynamics in general curvilinear internal. Generalized coordinates, lagranges equations, and constraints. To relate them to cartesian coordinates we usually assume that the zaxis is the polar axis. Small changes or variations in the rectangular coordinates. This file is licensed under the creative commons attributionshare alike 2. The particle is located by the radial distance r from a fixed point and by an angular measurement. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates.
Construct the basis vectors for cylindrical coordinates, fe. The new user interface for automatic computation of curvilinear coordinates is a very practical addition to version 4. The standard cartesian coordinates for the same space are as usual x, y, z. Div, grad and curl in orthogonal curvilinear coordinates. Principles and applications of the general theory of. It will be demonstrated that results in many existing studies can be covered as special cases by our formulations. Please contact me if you have the link, or if you would like my copy of this document. Any good book on vector or tensor analysis should provide you with a first principles derivation for some specific curvilinear coordinate systems spherical and cylindrical systems for example. We denote the curvilinear coordinates by u 1, u 2, u 3. For the more general case, you need to understand that your definition for the dot product of two vectors is valid only for cartesian coordinates. Cartesian coordinates we will start by studying the motion of a particle. 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.
Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Readers who are already wellversed in functional analysis will probably find more rigorous manuscripts such as 14 more suitable. Curvilinear motion is defined as motion that occurs when a particle travels along a curved path. Classical electromagnetism in a nutshell 2012, princeton. If you are completely new to the subject of general curvilinear coordinates or if you seek guidance on the basic machinery associated with nonorthonormal.
May 28, 20 curvilinear coordinates are a coordinate system where the coordinate lines may be curved. Pdf tensor analysis and curvilinear coordinates researchgate. Orthogonal curvilinear coordinates particle in cell. In this lecture a general method to express any variable and expression in an arbitrary curvilinear coordinate system will be introduced and explained. Let ul, u2 u3 represent the three coordinates in a general, curvilinear system, and let e. In this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system.
Question bank in curvilinear coordinates 1 obtain expression for area and volume element in spherical polar coordinates. In general, curvilinear coordinates allow the natural basis vectors hi not all mutually perpendicular to each other, and not required to be of unit length. Such body shapes include spheres, ellipses, spherically capped cones, flatfaced cylinders with rounded corners, circular disks, and planetary probe vehicles. The use of an orthogonal basis makes vector manipulations simpler than for nonorthogonal. More general coordinate systems, called curvilinear coordinate. So far we have only used cartesian x,y,z coordinates. In order to do so it is convenient to start from the expression for the function di erential.
For example in lecture 15 we met spherical polar and cylindrical polar coordinates. To find the curvilinear coordinate for three functions of f f 1 x, y, z, f 2 x, y, z, and f 3 x, y, z, set each function to a constant u 1, u 2, and u 3, which defines each function as a. 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. Strain gradient theory in orthogonal curvilinear coordinates. The governing equations were derived using the most basic coordinate system, i. This type of motion is more complex than rectilinear straightline motion. Given a nonlinear transformation between cartesian coordinates xi,i 1. The equation u j c j, where c j is a constant, represents a. Nasa technical memorandum 03 boundarylayer equations in generalized curvilinear coordinates argyris g. Orthogonal curvilinear coordinates the results shown in section 28.
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