Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. however, orthogonal curvilinear coordinate system pdf in other curvilinear coordinate systems, such as cylindrical and spherical coordinate systems, some differential changes are not length based, such as d θ, dφ. however, other coordinate systems can be used to better describe some physical situations. , the rotation gradient tensor and the deviatoric part of the symmetric couple stress tensor, and the classical strain and stress tensors are derived for. we denote the curvilinear coordinates by ( u 1, u 2, u 3). certainly the most common is the cartesian or orthogonal curvilinear coordinate system pdf rectangular coordinate system ( xyz).
most simulation codes, such as gtc [ 102, 103], xgc [ 104, 105] and gem [ 106– 108] adopted curvilinear coordinate systems. this asymmetric orthogonal curvilinear coordinate system has coordinates s i ∈ r ( i = 1, 2, 3) with s i in ( a 0, a 1), ( a 1, a 2) or( a 2, a 3), respectively, where a 0 < a 1 < a 2 < a 3 are given numbers. turn your pdf or hard copy worksheet into an editable digital worksheet! in this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system. 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. we only look at orthogonal coordinate systems, so that locally the three axes ( such as r, θ, ϕ ) are a mutually perpendicular set. this is indeed correct. of the orthogonal coordinate systems, there are several that are in common use for the description of the physical world. 1 deﬁnition of a vector a vector, v, in three- dimensional space is represented in the most general form as the summation of three components, v1, v2 and v3, aligned with three “ base” vectors, as follows: v = v1g1 + v2g2 + v3g3 = 3 ∑ i= 1 vig i ( b. transformation of the connection components. change of curvilinear coordinates.
- duration: 13: 14. we realize that the gradient operator in curvilinear coordinates can in general be written as ~ ñf = 3 å j= 1 ~ e j 1 h j ¶ f ¶ a j ( 23) where h j = ¶ ~ x ¶ aj are scaling factors in the respective coordinate system ( for example in cylindrical coordinates they are given in eq. amy almada recommended for you. the emission current density j is a function of x 2. e, cartesian coordinates: x i j k x y. a curvilinear coordinate system expresses rectangular. 1 are the de nitions of the coordinate functions.
orthogonal curvilinear coordinate system is given by dv = ( h1du 1e1 ) ⋅ ( h2du 2e2 ) × ( h3du 3e3) = h1h2h3du 1du 2du 3 ( 7) since e1 ⋅ e2 × e3 = 1. in this case, the three vectors e 1, e 2, e 3 are mutually orthogonal at every point i. this is also readily veriﬁed in cartesian coordinates. 1 orthogonal curvilinear coordinate system pdf tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update: maple code is available upon request. nal curvilinear systems is given first, and then the relationships for cylindrical and spher ical coordinates are derived as special cases. example: incompressible n- s equations in cylindrical polar systems the governing equations were derived using the most basic coordinate system, i. 5- cyclide coordinates. cylindrical and spherical coordinate systems in r3 are examples of or- thogonal curvilinear coordinate systems in r3. stress in these two coordinate systems. thus, we need a conversion factor to convert ( mapping) a non- length based differential change ( d θ, dφ, etc.
note that while the de nition of the cylindrical co- ordinate system is rather standard, the de nition of the spherical coordinate system varies from book to book. the base vectors are also mutually perpendicular, and the ordering is “ right- handed. ” below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. the focus of this study was restricted to the derivation and application of orthogonal three- dimensional coordinate systems. now let’ s look at the change of the position vector ~ r, in our new coordinate system, when we change the coordinates.
well- known examples of curvilinear coordinate systems in three- dimensional euclidean space ( r 3 ) are cylindrical and spherical polar coordinates. many of the steps pre- sented take subtle advantage of the. a large subclass of interesting coordinate systems are orthogonal, which means that gab = j~ a ¢ j~ b = 0 ( a 6= b) ( 6) in that case it is better to write = ha~ ea ( 7) where ha is a scale factor and ~ ea is a unit vector. 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. orthogonal curvilinear coordinate system is given by dv = ( h1du 1e1 ) ⋅ ( h2du 2e2 ) × ( h3du 3e3) = h1h2h3du 1du 2du 3 ( 7) since e1 ⋅ e2 × e3 = 1. derivatives of the unit vectors in orthogonal curvilinear coordinate systems 4.
2 have been given in terms of the familiar cartesian ( x, y, z) co- ordinate system. operators in terms of the appropriate coordinates. 6 for any scalar function φ, we can express its gradient in orthogonal curvilinear. plasmas, for which cylindrical or toroidal coordinate systems are more convenient. nate system which conforms to the surface of the duct or body which shapes the flow. curvilinear coordinates in geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. these coordinates may be derived from a set of. there are always m degrees of freedom ( where m d 2 in 2- d and m d 3 in 3- d) in choosing the mapping functions [ 8]. nonorthogonal systems are hard to work with and they are of little or no practical use.
since in these systems lines of constant compo- nents ( e. the name curvilinear coordinates, coined by the french mathematician lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. a curvilinear coordinate system is called orthogonal if the coordinate curves are everywhere orthogonal. this report utilizes the methods of tensor analysis to transform the basic equations from their cartesian forms to expressions in ten orthogonal curvilinear coordinate systems. for example, the three- dimensional cartesian coordinates ( x, orthogonal curvilinear coordinate system pdf 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. let x 1, x 2 be a curvilinear orthogonal coordinate system in meridian plane z, r associated with the emitter surface x 1 = 0 ( figure 21). a point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal.
orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. div, grad and curl in orthogonal curvilinear coordinates. remark: an example of a curvilinear coordinate system which is not orthogonal is provided by the system of elliptical cylindrical coordinates ( see tutuorial 9. these vectors form a local basis in each point ~ ea ¢ ~ eb = – ab x a ~ ea~ ea = ˆ! coordinate system. this coordinate system is described by coordinate surfaces s i = const which are ﬁve compact cyclides. of these, the rectangular cartesian coordinate system is the most popular choice. orthogonal curvilinear coordinates. , constant r) are curved, we refer to such coordinate systems as “ orthogonal curvilinear coor- dinates. ) into a change in length dl as shown below.
e, cartesian coordinates: xi= + + xˆˆyzj. operator in orthogonal curvilinear coordinates ( 1) gradient in orthogonal curvilinear coordinates fig. the presentation here closely follows that in hildebrand ( 1976). if the curvilinear coordinates are not orthogonal, the more general metric coeﬃcients are required. r = aˆcos i+ bˆsin j+ zk ( a 6= b) in the following we shall only consider orthogonal systems arc length the arc length dsis the length of the in nitesimal vector dr: - ( ds) 2 = drdr:.
the derivation process is outlined, and the final. in your past math and physics classes, you have encountered other coordinate systems such as cylindri-. the precise de nitions used here are. let ( ul, u2' u3) represent the three coordinates in a general, curvilinear system, and let e. orthogonal curvilinear coordinates the results shown in section 28.
incompressible n- s equations in orthogonal curvilinear coordinate systems 5. for example by setting off- diagonal components to zero, the coordinate system becomes orthogonal. divergence in orthogonal curvilinear coordinate system. most of the coordinate systems we are interested in are orthogonal, i. both systems to be studied are orthogonal. cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. the formulations for the modified couple stress theory ( mcst) are consistently derived in general orthogonal curvilinear coordinate systems. orthogonal curvilinear coordinate systems in what follows, we adopt much of the notation and verbage of morse and feshbach ( 1953). e 1 • e 2 = e 2 • e 3 = e 3 • e 1 = 0. in particular, the expressions for the rotation vector, higher- order strain, and stress tensors, i. onal curvilinear coordinates as they are used in electromagnetics.
in the present study, we extend the structure- preserving geometric pic algorithm to arbitrary curvilinear orthogonal coordinate systems, in. for an orthogonal curvilinear coordinate system, off- diagonal elements of the metric tensor must be zero. 1, the base vectors are unit vectors. the vast majority of engineerin g applications use on e of the coordinate systems illustrated in fig. the trajectories are determined by the differential equation. vectorial and tensorial ﬁelds in curvilinear coordinates. probably the second most common and of paramount importance for astronomy is the system of spherical or polar coordinates ( r, θ, φ). , are perpendicular. ds2 = x3 i= 1 x3 j= 1 g ijdx idx i the g ij is the metric tensor. the idea of generating coordinate meshes by numerically solving a set of partial differential equations. a set of coordinates u = u( x, y, z), v = v( x, y, z) and w = w( x, y, z) where the direc-.
introduction to curvilinear coordinates b. curvilinear coordinates for an vector displacement ds~ ds~ = ^ e 1h 1dx 1 + ^ e 2h orthogonal curvilinear coordinate system pdf 2dx 2 + ^ e 3h 3dx 3 back to our example of cylindrical coordiantes, ^ e 1 = ^ e r, ^ e 2 = ^ e, and ^ e 3 = ^ e z, and ds~ = ^ e rdr + ^ e rd + ^ e zdz these are orthogonal systems, but it would not have to be! such coordinate systems come equipped with a set of functions, called the lam´ e coeﬃcients. for example, the schrödinger equation for the hydrogen atom is best solved using spherical polar coordinates.
1 writing coulomb’ s law in various coordinate systems the following examples illustrate the advantages and disadvantages of dif- ferent choices of coordinate systems for writing coulomb’ s law, the formula for the force on a point charge q1 caused by another point charge q2. thus we can write ds2 = ( h 1 dqh 2 dqh 3 dq3) 2: ( 20) the hi’ s are called scale factors, and are 1 for cartesian coordinates. an orthogonal system is one in which the coordinates arc mutually perpendicular. 1 the concept of orthogonal curvilinear coordinates the cartesian orthogonal coordinate system is very intuitive and easy to handle. the standard cartesian coordinates for the same space are as usual ( x, y, z). for all three systems in fig. 1 a generalized coordinate system consists of a threefold family of surfaces whose equations in terms of cartesian coordinates are x 1 ( x, y, z) = constant, x 2 ( x, y, z) = constant, x 3 ( x, y, z) = constant. diﬀerentiation of tensor ﬁelds in curvilinear coordinates. the problem of generating orthogonal or non- orthogonal curvilinear coordinate systems in arbitrary domains is a problem of current interest in many branches of physics and engineering, and particularly in fluid mechanics and aerodynamics.