(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 2392316, 70853]*) (*NotebookOutlinePosition[ 2393370, 70889]*) (* CellTagsIndexPosition[ 2393326, 70885]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["CoordinateSystemsB"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Version 0.92\n9/12/91\nby John B. Schneider"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Polar and Cylindrical Coordinates"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Two dimensions (Polar)"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " Again, our goal is to move from the point (0,0) to the point (3,2) (as \ specified in Cartesian coordinates). In polar coordinates two numbers must \ be given to locate a point. These numbers are often written as an ordered \ pair, such as (rho,phi), where rho is the distance from the origin, and phi \ is the inverse tangent of y/x. Actually, the value of phi must take into \ account the quadrant \[Dash] so you cannot just plug arctan(y/x) into your \ calculator. You should be aware that the point (3,2)(Cartesian) could be \ specified directly in cylindrical coordinates as (3.61,.59)(polar, phi \ expressed in radians).\n\n If rho is given and phi is left unspecified, a \ circle with a radius of rho is obtained. If phi is given and rho is \ unspecified, a ray that starts at the origin and \ \[Trademark]points\[Integral] in the phi direction is obtained. In the cell \ below, the circle that corresponds to rho, the ray which corresponds to phi, \ and the point which is uniquely determined by the intersection of the circle \ and ray are shown as rho and phi vary.\n \n In the \ following cell the horizontal axis is the x axis and the vertical is the y \ axis. There are no rho and phi axes. In this animation, the \ \[Trademark]current\[Integral] point is moved away from the origin by \ increasing rho while leaving phi equal to zero. When phi is zero, the point \ moves along the positive x axis. Rho increases until it obtains a value \ equal to the square root of 13 (13 = 3^2 + 2 ^2). Next, phi increases from \ zero to a value of inverse tangent of 1.5. 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to the point (3,2,1) (as specified in \ Cartesian coordinates). The \[Trademark]third\[Integral] dimension in \ cylindrical coordinates is just like the third dimension in Cartesian \ coordinates. If a point (x,y,z) in Cartesian coordinates is equivalent to \ the point (rho,phi,z) in cylindrical coordinates, the z's are equal. If a z \ value is given and rho and phi are unspecified, we describe a plane. \ Specifying only a rho value describes a circular cylinder. This cylinder has \ radius rho and is concentric with the z axis. Specifying only a phi value \ describes a half plane (sometimes called a semi-infinite plane). This half \ plane has an edge along the z axis and is an angle phi away from the positive \ x axis (measured in the counter-clockwise direction and along the shortest \ arc from the positive x axis to the phi plane).\n \n The \ intersection of the rho cylinder, the phi plane and the z plane is a point. \ As before, the following animations show the point given by the current \ intersections of these surfaces. However, the surface is not shown if the \ value that specifies that surface is zero (e.g., the plane specifed by the \ value of z is not shown if z is zero). Now, on to the animations."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Increasing rho"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart % Scaling calculations 0.02381 0.95238 0.02381 0.95238 [ [(X)] 0.59131 0.1053 0.02018 1 Msboxa [(Y)] 0.22564 0.2194 1 0.177706 Msboxa [(Z)] 0.88933 0.42696 -1 -0.165205 Msboxa [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics3D [ ] 0 setdash 0 setgray gsave 0 setgray [(X)] 0.59131 0.1053 0.02018 1 Mshowa grestore gsave 0 setgray [(Y)] 0.22564 0.2194 1 0.177706 Mshowa grestore gsave 0 setgray [(Z)] 0.88933 0.42696 -1 -0.165205 Mshowa grestore 0 0 moveto 1 0 lineto 1 1 lineto 0 1 lineto closepath clip newpath gsave 0.002 setlinewidth 0.64182 0.4184 moveto 0.84746 0.17986 lineto stroke 0.64182 0.4184 moveto 0.65477 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