We will begin with the case that the coordinates are orthogonal. The approximate method of london and seban for predicting the solidification rate and for melting in or around slabs, cylinders, and spheres has been. This is a standard way of setting up structural deformation problems. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates.
In your past math and physics classes, you have encountered other coordinate systems such as cylindri. In an earlier communication dealing with flow in curved pipes a numerical procedure capable of calculating variable property flows in arbitrary curvilinear orthogonal coordinates was reported. The treatment here is standard, following that in abraham and becker, classical theory of electricity and magnetism. Such body shapes include spheres, ellipses, spherically capped cones, flatfaced cylinders with rounded corners, circular disks, and planetary probe vehicles. It should be recognized that the derivation can also be accomplished using tensor analysis. The focus of this study was restricted to the derivation and application of orthogonal threedimensional coordinate systems. This cylindrical system is itself a special case of curvilinear coordinates in that the base vectors are always orthogonal to each other. 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. Pdf orthogonal curvilinear coordinate systems corresponding to. Most of the material presented in this chapter is taken from anupam, g. Jun 21, 2018 the coordinates are initially orthogonal. The last vector is clearly orthogonal to the other ones, but the first two arent orthogonal as far as i can tell, which means theres something im missing. Now lets look at the change of the position vector r, in our new coordinate system, when we change the coordinates.
Gradient, divergence and curl in curvilinear coordinates. Pdf orthogonal curvilinear coordinates mohamed sobhy. As previously mentioned, in cartesian coordinates the his are equal to one. Wellknown examples of curvilinear coordinate systems in threedimensional euclidean space r 3 are cylindrical and spherical polar coordinates. We can equally introduce cylindrical polar coordinates which we will use here as the prime example for curvilinear coordinate systems. Apr 15, 2014 2 runmetric computes and prints metric tensor properties for a general curvilinear coordinate system specified in a user defined function such as sphr. Cartesian to curvilinear coordinate forward and backward. Convective and diffusive base vector variation source terms appearing in the differential momentum equation were tabulated but their derivation was not. This allows us to write the divergence as an inner product of.
Cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. We only look at orthogonal coordinate systems, so that locally the three axes such as r. The cartesian orthogonal coordinate system is very intuitive and easy to handle. Stress and strain tensors, the equilibrium equation in terms of the stress tensor and transformation laws for the tensors, in orthogonal curvilinear coordinates, are presented. An orthogonal system is one in which the coordinates arc mutually perpendicular. The name curvilinear coordinates, coined by the french mathematician lame, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. However, when the tire membrane deforms, the material coordinates become nonorthogonal, and the displacements of the material points are expressed functions of the original material coordinates. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Vector calculus and continuum conservation equations in curvilinear orthogonal coordinates. Most of the coordinate systems we are interested in are orthogonal, i. For this and other differential equation problems, then, we need to find the expressions for differential operators in terms of the appropriate coordinates. However, when the tire membrane deforms, the material coordinates become non orthogonal, and the displacements of the material points are expressed functions of the original material coordinates. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig.
In curvilinear coordinates, the basis vectors also depend on positions, so every time you differentiate a vector field, you need to make sure. Here, the curvilinear coordinates 12 3, are the familiar rz. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. Expression of strain tensor in orthogonal curvilinear coordinates. Curvilinear coordinates fujiun jiang october 11, 2010 i. Aug 07, 2015 the last vector is clearly orthogonal to the other ones, but the first two arent orthogonal as far as i can tell, which means theres something im missing. Orthogonal curvilinear coordinates unit vectors and scale factors suppose the point phas position r ru 1. Principles and applications of the general theory of. Curvilinear coordinates synonyms, curvilinear coordinates pronunciation, curvilinear coordinates translation, english dictionary definition of curvilinear coordinates. Div, grad and curl in orthogonal curvilinear coordinates. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame of reference defined by. The components of the tensor are defined as covariant components of secondorder tensors, which are identical to stress and strain components.
Operators in arbitrary orthogonal cuvilinear coordinate systems. So far we have only used cartesian x,y,z coordinates. Coordinate systems and vector elds coordinate systems a point in coordinate space r is often represented as r x. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Then, the orthogonal curvilinear coordinates are introduced as a general coordinate to describe the physical quantities included in the rmm.
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. Jan 22, 2016 these functions transform cartesian coordinates to channel fitted curvilinear coordinates with respect to a given curve. Although cartesian orthogonal coordinates are very intuitive. This means that one can convert a point given in a cartesian coordinate system to its. Nonorthogonal systems are hard to work with and they are of little or no practical use. Pdf mathematical physics lessons gradient, divergence and.
Classical electromagnetism in a nutshell 2012, princeton. Examples of nonorthogonal curvilinear coordinates physics. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Maybe i got the idea wrong, but at the moment i cant put my finger on it. This is bad notation since it suggests that the divergence can be written as an inner product which is. A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and twodimensional body shapes of interest to aerodynamicists. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. 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. Theory of orthogonal curvilinear coordinate systems was very popular among. 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. Apr 02, 2017 an introduction to curvilinear coordinates, infinitesimal displacement vector, and multiple coordinate systems. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. Although the coordinate curves of different orthogonal curvilinear coordinates occ are different, their partial reference. Please can someone give me a formal definition of curvilinear coordinates, preferably with as source.
This cylindrical system is itself a special case of curvilinear coordinates in that the. Nov 26, 2015 in geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. See orthogonal one person multiple careers pdf coordinates for the important special case, skew coordinates for a. It uses splines to parameterize the curve to its arclength. Orthogonal curvilinear coordinates 569ated by converting its components but not the unit dyads to spherical coordinates, andintegrating each over the two spherical angles see section a. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. Curvilinear coordinates file exchange matlab central. Then, the orthogonal curvilinear coordinates are introduced as a general coordinate to describe the physical quantities included.
An introduction to curvilinear coordinates, infinitesimal displacement vector, and multiple coordinate systems. The once that i have found dont seem to be very formal. We denote the curvilinear coordinates by u 1, u 2, u 3. Here we shall derive the appropriate relations for the latter using vector technique. A curvilinear coordinate system expresses rectangular. 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. Pdf reduced micromorphic model in orthogonal curvilinear. If the coordinate surfaces intersect at right angles i.
Orthogonality of a curvilinear coordinate system physics forums. In this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system. These coordinates may be derived from a set of cartesian coordinates. Orthogonal curvilinear coordinates and div, grad, curl. Orthogonal curvilinear coordinates pdf orthogonal curvilinear coordinates pdf download. With each coordinate is associated a scale factor hu, hv or hw respectively where hu x. Orthogonal curvilinear coordinates 1 definitions university of. Vector calculus and continuum conservation equa tions in. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame.
These are two important examples of what are called curvilinear coordinates. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. The equation u j c j, where c j is a constant, represents a. These functions transform cartesian coordinates to channel fitted curvilinear coordinates with respect to a given curve.
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