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Notebook[{
Cell[TextData[{
  "Calculus",
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  "VectorAnalysis",
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  CellTags->{"S5.16.1", "1.1"}],

Cell[TextData[{
  "A three\[Hyphen]dimensional coordinate system assigns three numbers to \
each point in space. In defining a coordinate system, you have to make a \
choice about what to measure and how to measure it. Frequently, physical \
systems exhibit special symmetries or structures that make a particular \
coordinate system especially useful. In a mathematically elegant solution to \
problems related to these systems, often the main step is choosing the \
correct coordinates. \nA variety of tools for doing calculus in various three\
\[Hyphen]dimensional coordinate systems are provided in this package. Because \
a given symbolic or numeric expression can mean different things in different \
coordinate systems, ",
  StyleBox["Mathematica", "TI"],
  " must know what coordinate system you are using and what the coordinate \
variables are. The initial default coordinate system is ",
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  ", with coordinate variables ",
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  ", ",
  StyleBox["Yy", "MR"],
  ", and ",
  StyleBox["Zz", "MR"],
  ". If you frequently work in another system, it may be useful to change the \
default to that system using ",
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  ". "
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  CellTags->{"S5.16.1", "1.2"}],

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Cell["Coordinate systems and coordinate variables. ", "Caption",
  CellTags->"S5.16.1"],

Cell["This loads the package. ", "MathCaption",
  CellTags->{"S5.16.1", "1.4"}],

Cell["<<Calculus`VectorAnalysis`", "Input",
  CellLabel->"In[1]:= ",
  CellTags->"S5.16.1"],

Cell["This resets the default coordinate system and variables. ", \
"MathCaption",
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Cell["Here are the new defaults. ", "MathCaption",
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  "All coordinate systems can be viewed as arising from three \
one\[Hyphen]parameter families of surfaces. The three numbers assigned to \
each point (",
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  ", its coordinates) simply tell you to which member of each family the \
point belongs. Members of these families are called ",
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  ". In the simplest example, the ",
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  " system, the coordinate surfaces are members of families of planes \
parallel to the coordinate planes. In more complicated systems, the \
coordinate surfaces often belong to families of quadric surfaces such as \
ellipsoids, hyperboloids or elliptic paraboloids. In certain cases the \
families of coordinate surfaces depend on parameters such as focal distances \
or radii. In these cases the coordinate system as a whole depends on the \
parameters. These parameters are initially given default values and can be \
reset with ",
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  ". "
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Cell["Coordinate systems.", "Caption",
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Cell[TextData[{
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in the literature. When you use a coordinate system with this package, you \
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  " ",
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",
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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parameter ",
  Cell[BoxData[
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  " is ",
  Cell[BoxData[
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  " ",
  StyleBox["Bispherical[", "MR"],
  StyleBox["u", "TI"],
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  StyleBox["lambda", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["mu", "TI"],
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  StyleBox["nu", "TI"],
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  StyleBox["a", "TI"],
  StyleBox[",", "MR"],
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  StyleBox["b", "TI"],
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  ", parameterized by ",
  Cell[BoxData[
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  " and ",
  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  Cell[BoxData[
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  StyleBox["mu", "TI"],
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  StyleBox["nu", "TI"],
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  ", ",
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  Cell[BoxData[
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  Cell[BoxData[
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\\\"MR\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"vars\\\\\\\"\\\", \
\\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \\!\
\\(\\*StyleBox[\\\"\\\\\\\"param\\\\\\\"\\\", \
\\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]]\\\\\\\"\\\", \\\"MR\\\"]\\) \"\
\>", "\<\"set the default coordinate system to be \
\\!\\(\\*StyleBox[\\\"\\\\\\\"coordsys\\\\\\\"\\\", \\\"TI\\\"]\\) with \
variables \\!\\(\\*StyleBox[\\\"\\\\\\\"vars\\\\\\\"\\\", \\\"TI\\\"]\\) and \
parameter values \\!\\(\\*StyleBox[\\\"\\\\\\\"param\\\\\\\"\\\", \\\"TI\\\"]\
\\) \"\>"}
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Cell["The range of coordinate variables and parameters. ", "Caption",
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Cell[TextData[{
  "This gives the default coordinate variables and their ranges for the ",
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  " system. "
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Cell[CellGroupData[{

Cell["\<\
{Coordinates[Conical],
 CoordinateRanges[Conical]}\
\>", "Input",
  CellLabel->"In[4]:= ",
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Cell[BoxData[
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      RowBox[{\({Llambda, Mmu, Nnu}\), ",", 
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Cell[TextData[{
  "This means that the ",
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  " system has one parameter that must be a positive number. The default \
value of this parameter is ",
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Cell[CellGroupData[{

Cell["\<\
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Cell[TextData[{
  "This resets the ",
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Cell[CellGroupData[{

Cell["SetCoordinates[Bipolar[u, v, z, 2]]", "Input",
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Cell[BoxData[
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Cell[TextData[{
  "In the ",
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Cell[CellGroupData[{

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\\!\\(\\*StyleBox[\\\"\\\\\\\"pt\\\\\\\"\\\", \\\"TI\\\"]\\) is given in \
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                {"\<\"\\!\\(\\*StyleBox[\\\"\\\\\\\"CoordinatesFromCartesian[\
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\\(\\*StyleBox[\\\"\\\\\\\"coordsys\\\\\\\"\\\", \
\\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \\\"MR\\\"]\\) \
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Cell["Converting to and from Cartesian coordinates. ", "Caption",
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Cell[TextData[{
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  " ",
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  " ",
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Cell[CellGroupData[{

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CoordinatesToCartesian[{1, Pi/2, Pi/4},
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Cell[BoxData[
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Cell["\<\
You get the general coordinate transformation formula when you give \
the variables in symbolic form. \
\>", "MathCaption",
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Cell[CellGroupData[{

Cell["\<\
CoordinatesToCartesian[{u, v, phi},
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Cell[BoxData[
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  CellTags->"S5.16.1"]
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Cell[TextData[{
  "Here are the formulas for changing from ",
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  CellTags->{"S5.16.1", "1.18"}],

Cell[CellGroupData[{

Cell["\<\
CoordinatesFromCartesian[{x, y, z},
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  CellTags->"S5.16.1"],

Cell[BoxData[
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Cell[TextData[{
  "The standard vector product operations, such as the dot and cross product, \
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have vectors given in a different coordinate system, you can compute vector \
products using ",
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  ", ",
  StyleBox["CrossProduct", "MR"],
  ", and ",
  StyleBox["ScalarTripleProduct", "MR"],
  ". These functions convert the given vectors into Cartesian coordinates and \
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\\!\\(\\*StyleBox[\\\"\\\\\\\"coordsys\\\\\\\"\\\", \
\\\"TI\\\"]\\)\\!\\(\\*StyleBox[\\\"\\\\\\\"]\\\\\\\"\\\", \\\"MR\\\"]\\),  \
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Cell["Computing vector products in various coordinate systems. ", "Caption",
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Cell[TextData[{
  "This sets the default coordinate system to ",
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Cell[CellGroupData[{

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Cell[TextData[{
  "This converts the vectors from the default system to ",
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Cell[BoxData[
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Cell["\<\
Viewed geometrically, the scalar triple product gives the volume of \
the parallelepiped spanned by the three vectors. \
\>", "MathCaption",
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Cell[CellGroupData[{

Cell["\<\
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Cell[BoxData[
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                      "InlineFormula"], \
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                      "InlineFormula"], \
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\\!\\(\\*StyleBox[\\\"\\\\\\\"t\\\\\\\"\\\", \\\"TI\\\"]\\)\\!\\(\\*StyleBox[\
\\\"\\\\\\\",\\\\\\\"\\\", \\\"MR\\\"]\\) \
\\!\\(\\*StyleBox[\\\"\\\\\\\"coordsys\\\\\\\"\\\", \
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Cell["Differential arc length element. ", "Caption",
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Cell[TextData[{
  "In Cartesian coordinates the differential arc length element is just ",
  Cell[BoxData[
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    "InlineFormula"],
  ". When you compute an integral along a parameterized curve in another \
coordinate system, you need to use the form appropriate to the coordinate \
system. To get the integral along the curve, you must plug the \
parameterization into the arc length element, compute the appropriate \
derivatives, and then integrate with respect to the parameter. The function \
",
  StyleBox["ArcLengthFactor", "MR"],
  " takes the parameterization as an argument and returns the formula for the \
derivative of the arc length along the curve. "
}], "Text",
  CellTags->{"S5.16.1", "1.25"}],

Cell["This gives the parameterization of an ascending helix. ", "MathCaption",
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Cell[CellGroupData[{

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Cell[TextData[{
  "This means that the derivative of the arc length, with respect to ",
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  " along the helix, is ",
  Cell[BoxData[
      \(TraditionalForm\`\@2\)], "InlineFormula"],
  ". If we think of the parameterization as describing the motion of a \
particle in ",
  StyleBox["Cartesian", "MR"],
  " coordinates, this just means that the speed of the particle is ",
  Cell[BoxData[
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Cell[CellGroupData[{

Cell["\<\
ArcLengthFactor[
 param, t, Cartesian] //Simplify\
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Cell[BoxData[
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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)

