Download englishus transcript pdf the following content is provided under a creative commons license. Vector calculus in curvilinear coordinates gri ths. Of course in cartesian coordinates, the distance between two. Systems of orthogonal curvilinear coordinates are defined as systems for which locally. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar. Polar coordinates lecture 24 integration and curvilinear. For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. Applications general curvilinear motion rectangular components of kinematic vectors group problem solving 2. Note the positon of the subscripts and superscripts in this expression. Change of variables as we saw in lecture 16, in e2 we can use the polar coordinates system.
Two coordinate systems cylindrical polar coordinates and spherical polar. An introduction to differential geometry philippe g. It follows that the jacobian of a coordinate transformation is the reciprocal of the jacobian of its inverse. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Cartesian coordinates we will start by studying the motion of a particle. In this lecture a general method to express any variable and expression in an arbitrary curvilinear. Curvilinear coordinates, namely polar coordinates in two dimensions, and cylindrical and spherical coordinates in three dimensions, are used to simplify problems with cylindrical or spherical symmetry. Central force lecture 25 integration and curvilinear. Generalized curvilinear coordinates for a general 3d orthogonal coordinate system fq. 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. Given a nonlinear transformation between cartesian coordinates xi,i 1. We focus on the orthogonal case, which includes cartesian, cylindrical and spherical coordinates. Treating bodies as particles is, of course, an idealization which involves an approximation. Curvilinear coordinates trigonometry chegg tutors youtube. 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. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates. View notes lecture notes 8 from math 115 at university of pennsylvania. We learn about double and triple integrals, and line integrals and surface integrals. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. Applications the path of motion of each plane in this formation can be tracked with radar and their x, y, and z coordinates relative to a point on earth recorded as a function of time. Here we shall derive the appropriate relations for the latter using vector technique. Note that there is a class of magnetic fields known as potential fields that. Curvilinear coordinates fujiun jiang october 11, 2010 i.
Gradient, divergence and curl in curvilinear coordinates. Being able to change all variables and expression involved in a given problem, when a di erent coordinate system is chosen, is one of. General orthogonal curvilinear coordinates the two sets of polar coordinates above have a. We think of a particle as a body which has mass, but has negligible dimensions. The di erential measures and vector operators for any orthogonal coordinates. Here, the curvilinear coordinates 12 3, are the familiar rz. Vector differential operators in orthogonal curvilinear coordinates, e. You may also be familiar with the use of the symbols r,q for polar coordinates. With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. Mar 17, 2016 curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. This section shows how to calculate these derivatives in other coordinate systems. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame. Consider the transformation from the rectangular cartesian. Examples of curvilinear coordinates include the wellknown cylindrical and spherical coordinates figure 1.
Pdf tensor analysis also known as tensor algebra or tensor calculus is. The purpose of this lecture is to show you a very important use of polar coordinates, and thats to consider newtons equation, when you have a central force, many sort of engineering problems when youre pulling an object into some point and causing it to rotate. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. 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. Your support will help mit opencourseware continue to offer high quality educational resources for free. So, that means whenever you have a geometric problem, and you are not told how to choose your coordinates, its probably wiser to try to center things on the z axis. In this section a general discussion of orthogo nal curvilinear systems is given first, and then the relationships for cylindrical and spher ical coordinates are derived as special cases. Cylindrical and secure coordinates are set up so that the z axis plays a special role. Div, grad and curl in orthogonal curvilinear coordinates.
Coordinate system as a part of reference system to describe satellite motion it is necessary to have a welldefined coordinate system. Most of the students in this course are beginning graduate students in engineering coming from a variety of backgrounds. The di erential measures and vector operators for any orthogonal coordinates system are easily constructed once the scale factors are determined. Thats where these coordinates are the best adapted. It is the central mathematical element of more complex issue reference system. Pdf tensor analysis and curvilinear coordinates researchgate. The coordinates of a point p are given by r, the distance from p to o, and. The particle is located by the radial distance r from a fixed point and by an angular measurement. Learning the basics of curvilinear analysis is an essential first step to reading much of the. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. Lacava, classical electrodynamics, undergraduate lecture notes in. Chapter 1 pages 3854, also appendix a of gri ths scale factors h 1.
A special case of these are orthogonal curvilinear coordinates. This means that one can convert a point given in a cartesian. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. Expressions for the gradient, divergence and curl in terms of curvilinear coordinates. Differential area a vector, ds, that is normal to the two coordinates describing the scalar area ds there are three different differential areas. Thisclassisusuallyreferredtoasstatics,butwellbecoveringsomeextra. Lecture notes on classical mechanics a work in progress. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. The main application is the subject of curvilinear coordinates. These are lecture notes for ame 60611 mathematical methods i, the. Problems in curvilinear coordinates lecture notes by dr k m udayanandan cylindrical coordinates 1.
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