Vector analysis university of colorado colorado springs. Download englishus transcript pdf the following content is provided under a creative commons license. This means that one can convert a point given in a cartesian. We can express the location of p in polar coordinates as r ru r.
Curvilinear coordinates fujiun jiang october 11, 2010 i. Two commonlyused sets of orthogonal curvilinear coordinates are cylindrical polar. With riemanns insight, it grew to represent infinite space, and curvilinear coordinate systems were utilized in physical applications. Let x x1,x2,x3 be the cartesian coordinates of a point m with respect to a frame. We focus on the orthogonal case, which includes cartesian, cylindrical and spherical coordinates. Polar coordinates lecture 24 integration and curvilinear. Curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved. Mar 17, 2016 curvilinear coordinates utilize a coordinate system where the coordinate lines, or axes, can be curved.
Problems in curvilinear coordinates lecture notes by dr k m udayanandan cylindrical coordinates 1. Thats where these coordinates are the best adapted. We learn about double and triple integrals, and line integrals and surface integrals. Here we shall derive the appropriate relations for the latter using vector technique. Gradient, divergence and curl in curvilinear coordinates. Your support will help mit opencourseware continue to offer high quality educational resources for free.
Examples of curvilinear coordinates include the wellknown cylindrical and spherical coordinates figure 1. We think of a particle as a body which has mass, but has negligible dimensions. 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. Learning the basics of curvilinear analysis is an essential first step to reading much of the.
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. Of course in cartesian coordinates, the distance between two. Generalized curvilinear coordinates for a general 3d orthogonal coordinate system fq. Lecture notes on classical mechanics a work in progress. The di erential measures and vector operators for any orthogonal coordinates. Central force lecture 25 integration and curvilinear. This cylindrical system is itself a special case of curvilinear coordinates in that the base vectors are always orthogonal to each other. Note the positon of the subscripts and superscripts in this expression. Systems of orthogonal curvilinear coordinates are defined as systems for which locally. Lacava, classical electrodynamics, undergraduate lecture notes in. You may also be familiar with the use of the symbols r,q for polar coordinates. An example of a curvilinear system is the commonlyused cylindrical coordinate system, shown in fig. In your past math and physics classes, you have encountered other coordinate systems such as cylindri.
Orthogonal curvilinear coordinates university of notre dame. Two coordinate systems cylindrical polar coordinates and spherical polar. Applications general curvilinear motion rectangular components of kinematic vectors group problem solving 2. Cylindrical and secure coordinates are set up so that the z axis plays a special role. General orthogonal curvilinear coordinates the two sets of polar coordinates above have a. 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. Expressions for the gradient, divergence and curl in terms of curvilinear coordinates. Lecture 1 introduction, units, linear algebra 0introduction welcometoengineeringmechanicsi. Cartesian coordinates we will start by studying the motion of a particle. This section shows how to calculate these derivatives in other coordinate systems.
Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. These are lecture notes for ame 60611 mathematical methods i, the. This is useful because some problems do not fit ideally into cartesian x, y, z coordinates. Consider the transformation from the rectangular cartesian. Curvilinear coordinates trigonometry chegg tutors youtube. In this lecture a general method to express any variable and expression in an arbitrary curvilinear. Div, grad and curl in orthogonal curvilinear coordinates. The main application is the subject of curvilinear coordinates. Treating bodies as particles is, of course, an idealization which involves an approximation. 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. View notes lecture notes 8 from math 115 at university of pennsylvania. 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.
Differential area a vector, ds, that is normal to the two coordinates describing the scalar area ds there are three different differential areas. Orthogonal curvilinear coordinates, in particular, were used in solving select partial differential equations, including the laplace and helmholtz equations. 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. Most of the students in this course are beginning graduate students in engineering coming from a variety of backgrounds. Problems with a particular symmetry, such as cylindrical or spherical, are best attacked using coordinate systems that take full advantage of that symmetry. The particle is located by the radial distance r from a fixed point and by an angular measurement. In geometry, curvilinear coordinates are a coordinate system for euclidean space in which the coordinate lines may be curved. The coordinates of a point p are given by r, the distance from p to o, and. It should be recognized that the derivation can also be accomplished using tensor analysis. Change of variables as we saw in lecture 16, in e2 we can use the polar coordinates system. For example in lecture 15 we met spherical polar and cylindrical polar coordinates. Here, the curvilinear coordinates 12 3, are the familiar rz. Note on curvilinear coordinates physics libretexts.
The di erential measures and vector operators for any orthogonal coordinates system are easily constructed once the scale factors are determined. It follows that the jacobian of a coordinate transformation is the reciprocal of the jacobian of its inverse. For example, the schrodinger equation for the hydrogen atom is best solved using spherical polar coordinates. 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. 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. Vector calculus in curvilinear coordinates gri ths. Chapter 1 pages 3854, also appendix a of gri ths scale factors h 1. Vector differential operators in orthogonal curvilinear coordinates, e. Pdf tensor analysis also known as tensor algebra or tensor calculus is. A special case of these are orthogonal curvilinear coordinates. Pdf tensor analysis and curvilinear coordinates researchgate. Thisclassisusuallyreferredtoasstatics,butwellbecoveringsomeextra. Note that there is a class of magnetic fields known as potential fields that.
186 1406 1410 849 162 935 147 345 588 158 1416 863 177 1442 1046 7 1011 162 350 143 163 819 681 1462 1358 94 1224 925 1374 52 239 1400 1166 1382 1391 82 1259 1057 813 197 1133 1284