Note on curvilinear coordinates physics libretexts. Base and unit vectors in orthogonal curvilinear coordinates. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Gradient, divergence and curl in curvilinear coordinates.
By using this reference surface, the coordinates of any. Note on transformation to general curvilinear coordinates. Pdf tensor analysis and curvilinear coordinates researchgate. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. We will begin with the case that the coordinates are orthogonal.
We shall say nothing further about nonorthogonal coordinate systems, for these find no application in conventional hydrodynamic problems. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. Tensor analysis and curvilinear coordinates xmission. Fourier series expansion in a nonorthogonal system of.
With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. Strain gradient theory in orthogonal curvilinear coordinates. A tensor operations in orthogonal curvilinear coordinate. I am beginning to study the mathematics of curvilinear coordinates and all textbooks and web sites do not have realistic examples of non othogonal systems. Pdf strain gradient theory in orthogonal curvilinear. Generalization of the second order vector potential formulation for arbitrary non orthogonal curvilinear coordinates systems from the covariant form of maxwells equations 401. Curvilinear coordinates are a coordinate system used for euclidean space where the coordinate lines think. Rans equations in curvilinear coordinates to begin with, the reynoldsaveraged navierstokes rans equations are presented in the familiar vector and cartesian tensor forms. A quick overview of curvilinear coordinates 1 introduction curvilinear coordinate systems are general ways of locating points in euclidean space using coordinate functions that are invertible functions of the usual x icartesian coordinates. An orthogonal system is one in which the coordinates arc mutually perpendicular. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. We need to know all the lengths and all the angles between the. The subscript i indicates a covariant quantity and the superscript i indicates a contravariant quantity.
Generalization of the second order vector potential formulation for arbitrary nonorthogonal curvilinear coordinates systems from the covariant form of. Generalization of the second order vector potential formulation for arbitrary nonorthogonal curvilinear coordinates systems from the covariant form of maxwells equations. Generalization of the second order vector potential. We therefore propose to examine the identity in a general curvilinear coordinate system,not necessarily orthogonal for which we assume that. 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. 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.
In a non orthogonal coordinate system, there are two sets of basis vectors. Below is a summary of the main aspects of two of the most important systems, cylindrical and spherical polar coordinates. In this section we derive the expressions of various vector concepts in an orthogonal curvilinear coordinate system. Examples of orthogonal coordinate systems include the. Piercey univserity of arizona november 21, 2007 abstract this paper will explore how to perform di. Since in these systems lines of constant components e. Pdf orthogonal curvilinear coordinates mohamed sobhy. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. Curvilinear coordinate an overview sciencedirect topics. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates. Most of the coordinate systems we are interested in are orthogonal.
What are some examples of non orthoganal curvilinear coordinates so that i can practice on. To ensure that vector components can be treated in a simple way, a curvilinear coordinate system should reduce to a rectangular system when a sufficiently small region is considered, and it must not reduce to a net of obliquely inclined lines. Generalized curvilinear coordinates in hybrid and electromagnetic codes daniel w. Here, the curvilinear coordinates 12 3, are the familiar rz. Div, grad and curl in orthogonal curvilinear coordinates. Gradient, divergence, laplacian, and curl in non euclidean coordinate systems math 225 supplement to colleys text, section 3. Curvilinear coordinates georgia institute of technology. Lecture notes on mathematical methods mihir sen joseph m. The laplacian in curvilinear coordinates the full story. For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. Grad, div, and curl in orthogonal curvilinear coordinate systems. In this note, we derive two formulas for the divergence and curl operators in a general coordinates system, whether orthogonal or not. Curvilinear analysis in a euclidean space mechanical engineering. The use of an orthogonal basis makes vector manipulations simpler than for non orthogonal.
In addition, nbfc grid makes the application of boundary conditions more complex. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. An introduction to curvilinear orthogonal coordinates overview. A tensor operations in orthogonal curvilinear coordinate systems a. Expression of strain tensor in orthogonal curvilinear coordinates 49 d surface nested in a cartesian coordinate system. Gradient, divergence, laplacian, and curl in noneuclidean. You can think of curvilinear coordinates as a generalized coordinate system, free of the straight line restraints that you see with a euclidean system. Hypersonic threedimensional nonequilibrium boundary. Conclusion the formulae by ward and pendry 7, 9 for the maxwells curl equations in curvilinear non orthogonal coordinates are amended to allow for the marked di. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. Curvilinear coordinates fujiun jiang october 11, 2010 i. This cylindrical system is itself a special case of curvilinear coordinates in that the base vectors are always orthogonal to each other. Following this theorem, we can separate the given vector.
Swift geophysical institute, university of alaska, fairbanks, alaska, 997757320, usa. Nonorthogonal systems are hard to work with and they are of little or no practical use. A point or vector can be represented in any curvilinear coordinate system, which may be orthogonal or nonorthogonal. Curvilinear coordinates and vector calculus 3 3 3 3 2 2 2 1 1 1 h u h u h. Note on transformation to general curvilinear coordinates for maxwells. Powers department of aerospace and mechanical engineering university of notre dame notre dame, indiana 465565637. The laplacian in curvilinear coordinates the full story peter haggstrom. 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. Effectiveness of grid orthogonality on body fitted. The graphics look ratty in windows adobe pdf viewers when not scaled up. However, other coordinate systems can be used to better describe some physical situations. Pdf generalization of the second order vector potential.
It may also be described by a cylindrical coordinate system, which is a non cartesian coordinate system. Each term in the transport equations is examined to see whether it. For example, the threedimensional cartesian coordinates x, 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. Expression of strain tensor in orthogonal curvilinear.
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