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 they are locally orthonormal coordinate systems. This phrase means that each coordinate system specifies three mutually CS 1 Coordinate Systems 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 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. The CS 2 Coordinate Systems ˆˆˆ Cartesian position vector corresponding to the point P = (xP, yP, zP) is rxiyjzkPP P P. 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 directions { xˆˆ,,yzˆ }(also known as:{ ˆˆˆ 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. x dxiˆˆ yj zkˆˆ xi ˆˆ yj zk rx(,,)(,,) dxyz rxyz xˆ Limit iˆ dx0 rx(,,)(,,) dxyz rxyz dx FORGET the equation! It is the picture that you need. Imagine the axes and point hanging out in space. In your mind, move the point from (x, y, z) to (x + x, y, z). In what direction did the point move? Line Element: The next vital quantity is the line element which is found as the displacement from the point (x, y, z) to the point (x + dx, y + dy, z + dz) at which each of the coordinates has been given an infinitesimal increment. d dr rx(, dxydyz ,)(,,) dz rxyz dxiˆˆ dyj dzkˆ Area and volume elements are built up from the mutually orthogonal components of the line element. For an area element with its normal in the x direction, x is fixed, and dAx = dy dz. The area element is just the product of the two perpendicular components of the line element. All three components of the differential of area are summarized as: ˆˆˆˆ ˆ ˆ dA dAxy i dA j dA z k dy dzi dz dx j dx dy k (There are other notations for dA such as dS, and dr2 .) Note that the direction of an area element is defined to be one of its normal directions. For a closed surface, the convention is to choose the outward directed normal. For the area element dA , the convention is that one takes the cross product of the each pair of the dir4ected components of the line element in right hand rule order. dAdxiˆˆ dy j dy ˆj dz kˆˆ dz k dxi ˆ dy dziˆˆ dz dx j dxdy kˆ Finally, we get the volume element by computing the product of the three orthogonal components of the line element. A volume element can be swept out by taking a small area element and moving it a small distance in the direction of its normal, or it can be computed as the triple vector product of the line elements component CS 3 Coordinate Systems vectors. ˆˆ ˆ ˆ ˆˆ dV dxi dAxy i dy j dA j dz k dA z k dx dy dz dV dxiˆˆ() dy j dz kˆ dV = dx dy dz z ˆ kˆ dz k dyˆ j dA = dy dz x dz kˆ ˆj iˆ ˆ dyˆ j dyˆ j dxi ˆ y dxi dV = dx dy dz x dAz = dx dy Everything is constructed from the components of the line element. Note the coordinate cube (volume element) has a small coordinate corner at (x,y,z) and a large coordinate corner at (x + dx, y + dy, z + dz). The components of the line element {,dxiˆˆ dy j , dz kˆ } are drawn from the small coordinate corner and highlighted. Then, the remaining 9 edges are added. Volumes and areas are easy because the components of the line element are mutually perpendicular. ˆ Exercise: Consider an area element dAy j . Compute the volume that is swept out by the area as it is given each of the following displacements: dxiˆˆ,and dy j dz kˆ . Prepare sketches. Direction Cosines: A general direction is expressed as: eieijejkekˆˆ(ˆˆˆ ) ( ˆ ) ˆ (ˆˆ ˆ ) (cos ) iˆ (cos ) ˆj (cos ) k ˆ where cos, cos and cos are the direction cosines of with respect to the three coordinate directions. That is cos = ieˆ ˆ the cosine of the angle between the direction of eˆ and that of iˆ , the x direction. CS 4 Coordinate Systems Cylindrical Coordinates Orientation relative to the Cartesian standard system: The origins and z axes of the cylindrical system and of the Cartesian reference are coincident. The cylindrical radial coordinate is the perpendicular distance from the point to the z axis. The angle is the angle between the x axis and the projection of the position vector in the x-y plane. Coordinate ranges: 0 r < ∞, 0 < 2π, and -∞ < z < ∞. NOTATION ALERT: The radial coordinate r represents the distance from the axis in the cylindrical system. It is chosen to coincide with the standard notation used for 2D polar coordinates. This notation has a potential pitfall as r rrz22 , and it can be confused with the spherical radial coordinate r = r , the distance from the origin. Stay Alert! Why do we use this ambiguous notation? Unfortunately, there is no uniformly adopted notation that avoids the potential for confusion. Some authors use or perhaps s for the cylindrical radial coordinate, but just as many use r. The symbols and s are also multiply assigned as charge or mass densities or as distance. It is therefore the choice of this author to follow the common practice of using r.