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Notebook[{
Cell["Mathematica Primer (the first one.)", "Title",
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Cell[TextData[{
  "This notebook has been written in ",
  StyleBox["Mathematica ",
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  "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University of Notre Dame\nNotre Dame IN 46556\nUSA\n\n",
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  "\n",
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    ButtonData:>{
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    ButtonStyle->"Hyperlink"],
  "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \
the World Wide Web and the Internet.\n\nI would hope that as a professional \
courtesy that this notice remain visible to other users. \nThere is no charge \
for copying and dissemination   \n\nVersion:  8/20/00\nMore recent versions \
of this notebook should be available at the web site:\n",
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  "Basic ",
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  " commands"
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Cell["Arithmetic", "Subsubsection"],

Cell["\<\
A system of such colossal power probably can do simple \
mathematics\
\>", "Text"],

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Cell["\<\
Note that for multiplication, there is no need for a sign.  Just \
use a space.\
\>", "Text"],

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  " does not switch to floating point, which is inherently less precise, \
unless you want it to.\n\nWe could do N[%], which says make the previous \
result floating point"
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Cell[TextData[StyleBox["Vectors and Matrices", "Subsubsection"]], \
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Cell["\<\
This looks good for the scalar multiplication of vectors.  We can \
also do:\
\>", "Text"],

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Cell["\<\
The normal cross product can also be obtained from the determinant \
of a matrix as you will see below.\
\>", "Text"],

Cell["You can also do", "Text"],

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Cell["\<\
But this multiplication does not arise in any physical problems \
that we will be considering this semester.  It could be useful for numerical \
solution techniques.    \
\>", "Text"]
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  " is a 3 by 3 square matrix. \n\nA matrix could correspond to many \
different physical entities.  For example a 3 by 3 could be the elements in a \
second order tensor which corresponds to the stresses in a fluid at a point \
and some time.  Of course, if this were the case, the matrix would be \
symmetric for all simple fluids. (Yes, this course will be complicated at \
times!)\n\nHow can we get a symmetric matrix from ",
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  " and what is \"symmetric\". \n\nFirst define transpose, which is a \
switching of a[i,j] --> a[j,i]"
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Cell["\<\
You see that we left the diagonal elements unchanged and \
\"reflected\" the off diagonal elements across the diagonal.  For example \
elements a[i,j] , where i is row and j is column were transformed to a[ \
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\>", "Text",
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Cell["\<\
We see that the antisymmetric part does not have any diagonal \
elements because a[i,i] cannot be equal to -a[i,i].  Thus our definition of \
an anti symmetric matrix is that a[i,j]=-a[j,i].  \
\>", "Text",
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Symmetric and antisymmetric matrixes are useful, for example, in \
defining the basic elements of fluid motion in a study of \"kinematics\".  It \
is convenient that\
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Cell["\<\
As mentioned above, the cross product can be defined with the \
determinant\
\>", "Text"],

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Where we had to work a little too hard with the {ex,ey,ez} being a \
unit vector and the Coefficient being used to recover the components of each \
direction.  \
\>", "Text"],

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From these operations we see that addition and subtraction of \
matrices is done by simple addition and subtraction of corresponding \
elements. \
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Cell["Making plots, special functions and Series expansion", "Subsubsection"],

Cell["\<\
It knows all of the functions, including yours and my favorite \
orthogonal polynomials.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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HkmS_`02001EHkl00`00HkmS_`02Hkl200000f>o001S_`02001JHkl20002Hkl00`00HkmS_`0SHkl0
0`00HkmS_`06Hkl000MS_`03001S_f>o02AS_`<005IS_`03001S_f>o00=S_`D005aS_`<002IS_`03
001S_f>o00IS_`00Qf>o00<006>oHkl01F>o00<006>oHkl0SV>o0026Hkl20007Hkl00`00HkmS_`2>
Hkl008MS_`03001S_f>o00ES_`03001S_f>o08iS_`00Sf>o00<006>oHkl0SV>o0000\
\>"],
  ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.05622, -1.05362, \
0.00736043, 0.0119094}}]
}, Open  ]],

Cell["\<\
You have already figured out that if the first number in the \
argument, is odd, the polynomial is an odd function.  If it is even, the \
polynomial is an even function.  \
\>", "Text"],

Cell["From math you may already recall:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Integral]\_\(-1\)\%1\( ChebyshevT[2, y]\ ChebyshevT[1, 
            y]\ 1\/\@\(1 - y\^2\)\) \[DifferentialD]y\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`0\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(2/\[Pi]\ Integrate[
        ChebyshevT[8, y]\ ChebyshevT[3, y]\ 1\/\@\(1 - y\^2\), {y, \(-1\), 
          1}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`0\)], "Output"]
}, Open  ]],

Cell["Was this magic?", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(2/\[Pi]\ Integrate[
        ChebyshevT[4, y]\ ChebyshevT[7, y]\ 1\/\@\(1 - y\^2\), {y, \(-1\), 
          1}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`0\)], "Output"]
}, Open  ]],

Cell["This is also magic.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(2/\[Pi]\ Integrate[
        ChebyshevT[4, y]\ ChebyshevT[4, y]\ 1\/\@\(1 - y\^2\), {y, \(-1\), 
          1}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell["One of the most fun things to do is expand things.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Series[BesselJ[0, x], {x, 0, 5}]\)], "Input"],

Cell[BoxData[
    FormBox[
      InterpretationBox[
        RowBox[{"1", "-", \(x\^2\/4\), "+", \(x\^4\/64\), "+", 
          InterpretationBox[\(O(x\^6)\),
            SeriesData[ x, 0, {}, 0, 6, 1]]}],
        SeriesData[ x, 0, {1, 0, 
          Rational[ -1, 4], 0, 
          Rational[ 1, 64]}, 0, 6, 1]], TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
A secret command that you always need with \"Series\" is \
\"Normal\".  Normal turns the expression with the error term into just a \
regular expression.\
\>", "Text"],

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Cell["Try some other lengths", "Text"],

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Cell["Do you see the pattern?", "Text"],

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Cell[TextData[{
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  " we do not do this with just an equal sign.  An equal sign in ",
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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[CellGroupData[{

Cell[BoxData[
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