General curvilinear coordinates pdf file

I have two lat and lon matrices, each 622x810 and they are of a curvilinear projection. Understanding components and vector derivative in general curvilinear coordinates. Classical electromagnetism in a nutshell 2012, princeton. This type of motion is more complex than rectilinear straightline motion. Plotting data on curvilinear coord projection matlab.

This section shows how to calculate these derivatives in other coordinate systems. In your past math and physics classes, you have encountered other coordinate systems such as cylindri. Curvilinear coordinates are a coordinate system where the coordinate lines may be curved. Methods of curvilinear tensor analysis are used to develop the most commonly used equations. 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. The standard cartesian coordinates for the same space are as usual x, y, z.

This is useful because some problems do not fit ideally into cartesian x, y, z 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. Treating bodies as particles is, of course, an idealization which involves an approximation. That is, substituting values for q 1, q2, x1, y, z1, x2, y2, and z2 into the equation will give a. This file is licensed under the creative commons attributionshare alike 2. Cylindrical coordinates a and spherical coordinates b. General dynamic formations consider a follower, which must retain distance p and q from the leader along curvilinear coordinates. We denote the curvilinear coordinates by u 1, u 2, u 3. Two coordinate systems cylindrical polar coordinates.

A set of coordinates u ux,y,z, v vx,y,z and w wx,y,z where the direc. The focus of this study was restricted to the derivation and application of orthogonal threedimensional coordinate systems. With these specific examples in mind, lets go back the the general case, and see where all the. For example in lecture 15 we met spherical polar and cylindrical polar coordinates.

Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. I previously just used vectors of lon and lat to plot my data with surfacem see below but i am not sure how to do this with lat and lon as a matrix of curvilinear coordinates. To illustrate the derivation of scale factors and base vectors, consider the 0 quanti ties in cylindrical coordinates. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.

Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates. Curvilinear coordinate systems are general ways of. Div, grad and curl in orthogonal curvilinear coordinates. Let ul, u2 u3 represent the three coordinates in a general, curvilinear system. More general coordinate systems, called curvilinear coordinate. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. Curvilinear coordinate systems are general ways of locating points in eu clidean space using coordinate functions that are invertible functions of the usual x. Curvilinear motion is defined as motion that occurs when a particle travels along a curved path. We will be mainly interested to find out gen eral expressions for the gradient, the divergence and the curl of. If x, y, zare the cartesian coordinates, the curvilinear ones, u, v, w, can be expressed as smooth functions of x, y, z, according to. A curvilinear coordinate system expresses rectangular coordinates x, y, z in terms of the generalized coordinates u u u by holding and u 3 constant, we form a family of u surfaces.

Curvilinear coordinates in matlab download free open. Their relationship with the more familiar equations in orthogonal coordinates is shown. 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. The equation u j c j, where c j is a constant, represents a. General dynamic formations for nonholonomic systems. Principles and applications of the general theory of.

Generalized coordinates, lagranges equations, and constraints. In the limit, this reduces to the general geometrically conservative expression for the divergence in the curvilinear coordinates. Curvilinear analysis in a euclidean space mechanical engineering. An introduction to curvilinear coordinates, infinitesimal displacement vector, and multiple coordinate systems. The relation between cartesian and cylindrical coordinates is given by. Programs and utility functions employing the symbolic math toolbox are provided to analyze vectors and tensors in general curvilinear coordinates. Problems in curvilinear coordinates lecture notes by dr k m udayanandan cylindrical coordinates 1. Generalized coordinates are usually selected to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of lagranges equations of motion. Their utility arises in problems with obvious geometric symmetries such as cylindrical or spherical symmetry.

Orthogonal curvilinear coordinates particle in cell. Turns out we can and this is the motivation for working in general orthogonal curvilinear coordinates. But despite this, the axes always remain orthogonal. Example in general curvilinear coordinates wiley online library. Cartesian to curvilinear coordinate forward and backward. The difference between a general curvilinear system and the cartesian one is that the axes orientation and scaling changes with the spatial position. Orthogonal curvilinear coordinates the results shown in section 28. However, it can also occur that a useful set of generalized coordinates may be dependent, which means that they are related by one or more constraint equations. Pdf mathematical physics lessons gradient, divergence. Rekach, manual on solutions of elasticity problems, vyshaya shkola, moscow. 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. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. For cartesian coordinates the normalized basis vectors are e. In order to do so it is convenient to start from the expression for the function di erential.

Cartesian coordinates we will start by studying the motion of a particle. Curvilinear coordinates file exchange matlab central. Threedimensional curvilinear motion describes the most general case of motion for a particle. The main application is the subject of curvilinear coordinates. We think of a particle as a body which has mass, but has negligible dimensions. The curved path can be in two dimensions in a plane, or in three dimensions. Orthogonal curvilinear coordinates and div, grad, curl. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame of reference defined by the unit vectors e1,e2,e3. 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. These functions transform cartesian coordinates to channel fitted curvilinear coordinates with respect to a given curve.

Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Mast20009 vector calculus, semester 2, 2016 tutorial 12 questions general curvilinear coordinates for each point p with cartesian coordinates x, y, z, associate a unique set of curvilinear coordinates u 1, u 2, u 3 where x f 1 u 1, u 2, u 3, y f 2 u 1, u 2, u 3, and z f. 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. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. Mathematical physics lessons gradient, divergence and curl in curvilinear coordinates article pdf available april 2007 with 10,025 reads how we measure reads. It is not unusual to encounter a problem that is not conveniently posed in one of the common coordinate systems i. Differentials squared divergence in general orthogonal. Orthogonal curvilinear coordinates 1 definitions university of. The conventional choice of coordinates is shown in fig. The new user interface for automatic computation of curvilinear coordinates is a very practical addition to version 4. A coordinate system is orthogonal if the three families of coordinate surfaces are mutually perpendicular.

The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems. Finally, the practical problems of evaluating the many geometric coefficients which appear in these equations are examined by considering numerically generated. Small changes or variations in the rectangular coordinates. 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. Construct the basis vectors for cylindrical coordinates, fe. Equations of elasticity in general orthogonal curvilinear coordinates. These are two important examples of what are called curvilinear coordinates. In cartesian coordinates, if fi,, x ypxy qxy j, the 2ddivergence is defined as div pq x y f. Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. It uses splines to parameterize the curve to its arclength.

The kinetic energy, t, may be expressed in terms of either r. Gradient, divergence and curl in curvilinear coordinates. The following matlab project contains the source code and matlab examples used for curvilinear coordinates. However, other coordinate systems can be used to better describe some physical situations. Such coordinates qare called generalized coordinates. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. Most of the material presented in this chapter is taken from anupam, g.