You are familiar, for example, with polar coordinates r. Expression of strain tensor in orthogonal curvilinear coordinates 49 d surface nested in a cartesian coordinate system. Curvilinear coordinates and vector calculus 3 3 3 3 2 2 2 1 1 1 h u h u h. Just as in cartesian coordinates, the distance element.
Cartesian coordinate system is length based, since dx, dy, dz are all lengths. Base vectors in a cartesian left and in a generalized orthogonal curvilinear coordinate system right a. Another application of such examples is for a highly directional high frequency. 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. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. Orthogonal coordinate systems cartesian, cylindrical, and. Numerical relativityin singular curvilinear coordinate systems ian ruchlin,1 zachariah b. In this generic orthogonal curved coordinate system three coordinate surfaces meet at each point pin space. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Cylindrical and spherical coordinates are just two examples of general orthogonal curvilinear coordinates. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. With these specific examples in mind, lets go back the the general case, and see where all the.
In this project we are exploring the possibility of combining data from. Nonorthogonal systems are hard to work with and they are of little or no practical use. Orthogonal curvilinear coordinates the results shown in section 29. However, other coordinate systems can be used to better describe some physical situations. Principles and applications of the general theory of. We will begin with the case that the coordinates are orthogonal. The laplacian in curvilinear coordinates the full story. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. The same vector transformed into the curvilinear coordinate system. This general idea has been applied previously to twodimensional regions. This allows us to write the divergence as an inner product of. Baumgarte3 1department of mathematics, west virginia university, morgantown, west virginia 26506, usa 2center for gravitational waves and. In the above equations we have used spatial derivatives x and y written in terms of physical distance rather than with derivatives in the curvilinear coordinate l and.
A tensor operations in orthogonal curvilinear coordinate. Calculus with curvilinear coordinates problems and. In orthogonal curvilinear coordinates, the vector derivatives. For such cases, it is natural to derive and solve mode parabolic equations in curvilinear coordinates. In general, the basis vectors are neither unit vectors nor mutually orthogonal. Description of the northogonal curvilinear coordinate. 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.
Pdf strain gradient theory in orthogonal curvilinear. A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and twodimensional body shapes of interest to aerodynamicists. 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. These parallel and reciprocal bases coincide for curvilinear orthogonal coordinates. Vector analysis can be performed in different coordinate systems, an optimal system considers the symmetry of the problem in order to reduce calculatory. However, other coordinate systems can be used to better describe some. The reason to prefer orthogonal coordinates instead of general curvilinear coordinates is simplicity. For orthogonal curvilinear coordinates, the component aiis obtained by taking the scalar product of awith the ith curvilinear basis vector ei ai eiar nb aimust be expressed in terms of uinot x. As previously mentioned, in cartesian coordinates the his are equal to one. Expression of strain tensor in orthogonal curvilinear. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. Although this report assumes a newtonian fluid model, the viscous stress components are listed separately so that, given the proper expressions for the viscous stress com. A random and an orthonormal vector basis in threedimensional space.
Using the theory of orthogonal curvilinear coordinates, the spheroidal equations of relative atmospheric motion are derived from the vector equation of absolute motion. The hamiltonians of both spin zero and spin onehalf particles, with soc present in the latter, are constructed. Most practicing engineers can get along just fine without ever having to learn the theory behind general curvilinear coordinates. Many of the steps presented take subtle advantage of the orthogonal nature of these systems.
Pdf lesson for the module mathematical physics ii, dept of physics, university of. In this approach, the decoupling of the transverse dynamics from the hamiltonian in the limit q 3. Strain gradient theory in orthogonal curvilinear coordinates article in international journal of solids and structures 451112. Classical electromagnetism in a nutshell 2012, princeton. The focus of this study was restricted to the derivation and application of orthogonal threedimensional coordinate systems. What would be the right way to derive the for example spherical. Implementation of an atmosphereocean general circulation. The other is an algebraic method that relies on being able to determine the effects of the del operator on a vector written in curvilinear coordinates. Recent patchbased amr implementations for cartesian, cylindrical, and spherical geometries were also demonstrated in 25. Aug 07, 2015 which means that i cant avoid finding r.
There are two main reasons for using curvilinear coordinates for space. Looking again at the result in equation 6, we learn that. Here we shall derive the appropriate relations for the latter using vector technique. Div, grad and curl in orthogonal curvilinear coordinates the treatment here is standard, following that in abraham and becker, classical theory of electricity and magnetism.
This completes the general results for orthogonal curvilinear coordinates. It presents the material in arbitrary orthogonal coordinates and discusses parametrization methods and topics such as potential theory and integral theorems. Examples of orthogonal coordinate systems include the. The use of an orthogonal basis makes vector manipulations simpler than for non orthogonal. Pseudodifferential adiabatic mode parabolic equations in. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. Orthogonality of a curvilinear coordinate system physics forums.
Pages 5 by markus antoni this book presents problems and solutions in calculus with curvilinear coordinates. For cartesian coordinates the normalized basis vectors are e. The display coordinate system determines the system in which geometry items are listed or displayed. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocityvorticity formulations. Combining balance laws and stressstrain relations leads to. Orthogonal curvilinear coordinates particle in cell. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar. Hybrid blockamr in cartesian and curvilinear coordinates. Expression of strain tensor in orthogonal curvilinear coordinates. 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. However, in other curvilinear coordinate systems, such as cylindrical and spherical coordinate systems, some differential changes are not length based, such as d. Two coordinate systems cylindrical polar coordinates and. Most of the material presented in this chapter is taken from anupam, g.
Know the four equivalent tests conservative forces. The resultant matrix is called the jacobian matrix. By using this reference surface, the coordinates of any. Strain gradient theory in orthogonal curvilinear coordinates. Notes for elementary help with curvilinear coordinates and. Orthogonal curvilinear coordinates olc is a formalism that emphasizes the similarities of these three coordinate systems rather than their di. 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 spherical coordinates can be translated to cartesian coordinates and vice versa. The laplacian in curvilinear coordinates the full story peter haggstrom. Orthogonal curvilinear coordinates often it is convenient to use coordinate systems other than rectangular ones. In your past math and physics classes, you have encountered other coordinate systems such as cylindri. It should be recognized that the derivation can also be accomplished using tensor analysis 1. Vector calculus and continuum conservation equa tions in.
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. Derivation of the equations of atmospheric motion in. When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates. The treatment here is standard, following that in abraham and becker, classical theory of electricity and magnetism. Applied elasticity in engineering materials technology. 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. Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found. Description of the n orthogonal curvilinear coordinate systems and hamiltonian integrable systems of hydrodynamic type. Here, the curvilinear coordinates 12 3, are the familiar rz. Note on curvilinear coordinates physics libretexts.
In this article we derive the vector operators such as gradient, divergence, laplacian, and curl for a general orthogonal curvilinear coordinate system. So far we have only used cartesian x,y,z coordinates. 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 special case of these are orthogonal curvilinear coordinates. For orthogonal coordinates the metric tensor isdiagonal, g12 5 g21 5 0, so that orthogonal coordinates can be seen as a simplifying subset of general curvilinear coordinates. Newest curvilinearcoordinates questions mathematics.
Pdf mathematical physics lessons gradient, divergence and. In the present study, we combine the aforementioned motivations and derive pseudodifferential, mpes pdmpes in orthogonal curvilinear coordinates. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. I will do the problem algebraically because it gives further insight into general curvilinear coordinates. One feature of orthogonal coordinate systems is that the expression for. The hybrid blockamr strategy is made compatible with general curvilinear orthogonal coordinate systems, encompassing 2d polar as well as 3d cylindrical and spherical grids. An orthogonal system is one in which the coordinates arc mutually perpendicular. Such body shapes include spheres, ellipses, spherically capped cones, flatfaced cylinders with rounded corners, circular disks, and planetary probe vehicles.
With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar coordinates. Orthogonal curvilinear coordinates and div, grad, curl. For example in lecture 15 we met spherical polar and cylindrical polar coordinates.
The cartesian orthogonal coordinate system is very intuitive and easy to handle. The nodal coordinate system defines the degree of freedom directions at each node and the orientation of nodal results data. It will be demonstrated that results in many existing studies can be covered as special cases by our formulations. Combining these two results in accordance with the vector identity previ. This is bad notation since it suggests that the divergence can be written as an inner product which is. This book provides many exercises and solutions in calculus with curvilinear coordinates, helpful for lecturers in engineering mathematics. Curvilinear coordinates fujiun jiang october 11, 2010 i. A set of coordinates u ux,y,z, v vx,y,z and w wx,y,z where the directions at any point indicated by u, v and w are orthogonal perpendicular to each other is referred to as a set of orthogonal curvilinear coordinates. In order to express di erential operators, like the gradient or the divergence, in curvilinear coordinates it is convenient to start from the in nitesimal increment in cartesian coordinates, 1. Examples of transversal isotropy are found in hexagonal crystals chp, zn. It should be recognized that the derivation can also be accomplished using tensor analysis.
The conventional choice of coordinates is shown in fig. Div, grad and curl in orthogonal curvilinear coordinates. The derivation process is outlined, and the final results are tabulated for each of the coordinate systems. Let b 1, b 2, b 3 be an arbitrary basis for threedimensional euclidean space. Pdf orthogonal curvilinear coordinates mohamed sobhy. Gradient, divergence and curl in curvilinear coordinates. A curvilinear coordinate system expresses rectangular. Differential operators on curvilinear coordinates new, consider an orthogonal curvilinear coordinate described as a aiei, ei 1 hi.
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