ReviewB - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2004 Review B: Coordinate Systems B.1 Cartesian Coordinates ..................................................................................... B-2 B.1.1 Infinitesimal Line Element ......................................................................... B-4 B.1.2 Infinitesimal Area Element ......................................................................... B-4 B.1.3 Infinitesimal Volume Element .................................................................... B-5 B.2 Cylindrical Coordinates .................................................................................. B-5 B.2.1 Infinitesimal Line Element ......................................................................... B-9 B.2.2 Infinitesimal Area Element ......................................................................... B-9 B.2.3 Infinitesimal Volume Element .................................................................. B-10 B.3 Spherical Coordinates ................................................................................... B-11 B.3.1 Infinitesimal Line Element ....................................................................... B-13 B.3.2 Infinitesimal Area Element ....................................................................... B-14 B.3.3 Infinitesimal Volume Element .................................................................. B-14 B-1
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Coordinate Systems B.1 Cartesian Coordinates A coordinate system consists of four basic elements: (1) Choice of origin (2) Choice of axes (3) Choice of positive direction for each axis (4) Choice of unit vectors for each axis We illustrate these elements below using Cartesian coordinates. (1) Choice of Origin Choose an origin O . If you are given an object, then your choice of origin may coincide with a special point in the body. For example, you may choose the mid-point of a straight piece of wire. (2) Choice of Axis Now we shall choose a set of axes. The simplest set of axes are known as the Cartesian axes, -axis, -axis, and the -axis. Once again, we adapt our choices to the physical object. For example, we select the -axis so that the wire lies on the -axis, as shown in Figure B.1.1: x y z x x Figure B.1.1 A wire lying along the x -axis of Cartesian coordinates. B-2
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Then each point in space our P S can be assigned a triplet of values ( x P , y P , z P ) , the Cartesian coordinates of the point . The ranges of these values are: −∞ P < x P < +∞ , −∞ < y P < +∞ , −∞ < z P < +∞ . The collection of points that have the same the coordinate P y is called a level surface. Suppose we ask what collection of points in our space S have the same value of P y y = . This is the set of points { } ( , , ) such that P y P S x y z S y y = = P y S . This set is a plane, the - x z plane (Figure B.1.2), called a level set for constant P y . Thus, the -coordinate of any point actually describes a plane of points perpendicular to the -axis. y y Figure B.1.2 Level surface set for constant value P y . (3) Choice of Positive Direction Our third choice is an assignment of positive direction for each coordinate axis. We shall denote this choice by the symbol + along the positive axis. Conventionally, Cartesian coordinates are drawn with the - x y plane corresponding to the plane of the paper. The horizontal direction from left to right is taken as the positive -axis, and the vertical direction from bottom to top is taken as the positive -axis. In physics problems we are free to choose our axes and positive directions any way that we decide best fits a given problem. Problems that are very difficult using the conventional choices may turn out to be much easier to solve by making a thoughtful choice of axes. The endpoints of the wire now have the coordinates and x y ( / 2,0,0) a ( / 2,0,0) a .
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