Coordinate Systems Are Used to Describe Positions of Particles Or Points at Which Quantities Are to Be Defined Or Measured
COORDINATE SYSTEMS
Concepts of primary interest: The line element Coordinate directions Area and volume elements Sample calculations: Coordinate direction derivatives Velocity and acceleration in polar coordinates Application examples: Velocity and acceleration in spherical coordinates **** add solid angle Tools of the Trade Changing a vector Area Elements: dA = dr12× dr *** TO Add *****
Appendix I – The Gradient and Line Integrals
Coordinate systems are used to describe positions of particles or points at which quantities are to be defined or measured. They are often used as references for specifying directions. The coordinate system or reference frame is used extensively in describing the physical problem or situation, but it is not a part of the problem. No physical result can depend on the choice of coordinates. The coordinate system is a passive aid to the observer, and it may be chosen or adjusted to suit the purposes of the observer. Problem statements may use a coordinate system as a convenience, but no physical problem comes with axes glued to it. We add them to facilitate the description of the problem. Once the coordinates have been chosen for a problem and the description has been started, further changes are usually not advised as a complicated transformation scheme is often required to translate information stated relative to one set of coordinates into a form suitable for use in another set of coordinates. A system of coordinates for three dimensions assigns an ordered triplet of numbers [(x, y, z) or
(q1,q2,q3)] to each point in space. Three such coordinate systems are commonly used by undergraduate physics majors: Cartesian, cylindrical and spherical. A common characteristic of these systems is that
Contact: [email protected] they are locally orthonormal coordinate systems. This phrase means that each coordinate system specifies three mutually perpendicular (orthogonal and unity normalized) directions at every point in space. An infinitesimal displacement along one coordinate direction is independent of small displacements along the other coordinate directions because their directions are mutually perpendicular. For example, in Cartesian coordinates, a displacement in the x direction does not change the y or z coordinate. Each system is to be discussed in a parallel fashion to emphasize their common features and their distinguishing characteristics.
Rene Descartes (1596-1650): French scientific philosopher who developed a theory known as the mechanical philosophy. This philosophy was highly influential until superseded by Newton’s methodology. Descartes was the first to make a graph, allowing a geometric interpretation of a mathematical function and giving his
name to Cartesian coordinates. Eric W. Weisstein @ scienceworld.wolfram.com/biography/Descartes.html
Cartesian Coordinates To understand a coordinate system, you must know its relation to the Cartesian coordinate system, the representation of the position vector, the shapes of the constant coordinate surfaces, the three independent coordinate directions, and the line element represented as d or dr . For this reason, the
Cartesian system is studied first. The relations between the coordinates of a Cartesian system and those of a second Cartesian system with the same origin and axes directions are: x' = x, y' = y, and z' = z. (A more interesting set of transformations is used to relate one set of Cartesian coordinates to another Cartesian set with a different origin or orientation. That problem is studied in a second semester course in mechanics.) Constant Coordinate Surfaces: The constant coordinate surfaces are planes parallel to the plane defined by the other two axes. For example, x = a is a plane parallel to the y-z plane that is perpendicular to the x axis at the point (a, 0, 0). The point (a, b, c) is located at the intersection of the planes x = a, y = b, and z =c. You should sketch some constant coordinate planes illustrating the intersections of pairs and triplets of such planes. Coordinate Orbits: We define a coordinate orbit as the locus of points mapped as one coordinate
2/11/2008 Physics Handout Series.Tank: Basic Coordinate Systems 2 runs through its full range in the positive sense while the other coordinates are held fixed. An x-orbit is an infinite line parallel to the x-axis that passes through the x = 0 plane at (0, y, z). Position Vector: The position vector for a point P is the displacement from the origin to that point. ˆˆˆ The Cartesian position vector corresponding to the point P = (xP, yP, zP) is rxiyjzPP=+ P + Pk. Coordinate Directions: One can find the coordinate directions by examining the change in position due to a small positive variation in one coordinate while the other coordinates are kept fixed. Imagine the Cartesian coordinate axes and a point (x, y, z) hanging in otherwise empty space. Increase each coordinate in turn by a small positive increment to visualize each of the independent coordinate ˆˆˆ directions { xˆˆ,,yzˆ }(also known as:{ijk,, }; { eeeˆˆˆx ,,yz} or { eeeˆˆˆ123,,}). The xˆ direction is the direction a point is displaced if its x coordinate is given a small positive increment while its y and z coordinates are held fixed.