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