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" is with entering expressions and evaluating them. Here are some examples:"
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You hit Shift-Return to evaluate the expression. Return just moves you down \
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"."
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Numbers with a decimal point are treated as machine floating point numbers \
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which is very often NOT what you want in this class.\
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You can convert anything to decimal form with \[OpenCurlyDoubleQuote]N\
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\[CloseCurlyDoubleQuote], you can see all the digits.\
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" "
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brackets on the right hand side of the notebook denote \
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This is an (item) cell. You can select the cell type from the Format->Style \
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You can change fonts (or the look of the entire document) with \
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This makes it a lot easier to check that you\[CloseCurlyQuote]re coding \
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You really can\[CloseCurlyQuote]t do this in any other programming \
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You can do surprisingly complicated integrals with Integrate.\
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Since this answer doesn\[CloseCurlyQuote]t really mean anything to us, let\
\[CloseCurlyQuote]s evaluate it numerically.\
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"."
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Notice the syntax: Series[function,{variable,point to expand around,highest \
order term}]. You can convert this to a polynomial with Normal\
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Now poly isn\[CloseCurlyQuote]t a function yet. Why not? Because x isn\
\[CloseCurlyQuote]t a variable. To convert poly into a function \
automatically, we\[CloseCurlyQuote]ll need some trickery. In the meantime, we \
can always put it into InputForm and copy and paste to use it to define a \
function.\
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The other thing we probably want to do is think about the coefficients as \
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",
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The \[OpenCurlyDoubleQuote]=\[CloseCurlyDoubleQuote] means \
\[OpenCurlyDoubleQuote]the left-hand side is forever after whatever the right \
hand side evaluates to right now\[CloseCurlyDoubleQuote]. This is a \
\[OpenCurlyDoubleQuote]dead assignment\[CloseCurlyDoubleQuote], because the \
result is always the same.\
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On the other hand, the \[OpenCurlyDoubleQuote]:=\[CloseCurlyDoubleQuote] \
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evaluate the left hand side, replace it with the right hand side and evaluate \
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\[CloseCurlyQuote]s so important to use the \[OpenCurlyDoubleQuote]:=\
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Search and replace gets a lot more interesting if you\[CloseCurlyQuote]re \
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The most basic pattern (you\[CloseCurlyQuote]ve already seen this) is the \
function definition:\
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Here, the x_ means \[OpenCurlyDoubleQuote]a pattern (blank) matching any \
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list is a \[OpenCurlyDoubleQuote]single object\[CloseCurlyDoubleQuote], you \
can pass a list of numbers\
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But you can\[CloseCurlyQuote]t pass a pair of objects, since that doesn\
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You can define a new function with the SAME NAME, but a different pattern.\
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" will give you no warning when you redefine a variable name you\
\[CloseCurlyQuote]ve already used, reassign something you needed elsewhere, \
or otherwise mess yourself up. So I recommend:"
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tried a lot of different things (and erased them), some previous definition \
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Don\[CloseCurlyQuote]t do too much in one notebook. It\[CloseCurlyQuote]s not \
unreasonable to answer different homework questions in different notebooks, \
for example.\
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Don\[CloseCurlyQuote]t do a lot of editing and erasing, or evaluating your \
notebook out of sequence. The results can be very weird.\
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Sometimes, Simplify won\[CloseCurlyQuote]t do what you want it to and you \
want to transform an expression in a more controlled way. You do this with a \
transformation rule, written x->y and \[OpenCurlyDoubleQuote]applied\
\[CloseCurlyDoubleQuote] with /.\
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I took calculus, so I know that I really want to multiply by the conjugate of \
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Sometimes, one application of a transformation rule doesn\[CloseCurlyQuote]t \
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In this case, you can use the //. operation to apply the transformation rules \
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The syntax n_Head restricts the pattern to matching only objects with the \
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We can also apply a function (which gives True/False as the outputs) to check \
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Every object has a wrapper called a \[OpenCurlyDoubleQuote]Head\
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As you\[CloseCurlyQuote]ve probably guessed, you can address parts of an \
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You can change the head directly with the Apply function, which is \
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The default expression type has the head \[OpenCurlyDoubleQuote]List\
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Lists have \[OpenCurlyDoubleQuote]levels\[CloseCurlyDoubleQuote], which \
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Most functions will \[OpenCurlyDoubleQuote]thread\[CloseCurlyDoubleQuote] \
over a list, meaning that they will replace a list with a new list contained \
the function evaluated on each member of the list.\
\>", "Text",
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A standard way to generate lists is to use the Table function. \
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No sooner do you generate a table than you want to start messing with it.\
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Here the best I can do for you is to give you some examples and recommend to \
you the tutorial \[OpenCurlyDoubleQuote]Rearranging and Restructuring Lists\
\[CloseCurlyDoubleQuote]. However, this is really important stuff, and you\
\[CloseCurlyQuote]ll probably want to do a lot of it. Given sinTable, a \
common question would be \[OpenCurlyDoubleQuote]Can I get separate lists of \
the x and y coordinates?\[CloseCurlyDoubleQuote] Sure! Use Transpose:\
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Now I want to see the positive entries in this list. To do this, you use \
Select. This requires a function (to be called on each list member). You get \
the list entries for which this function returns \[OpenCurlyDoubleQuote]True\
\[CloseCurlyDoubleQuote].\
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There are weirder things to do with lists, which you often want to do. We \
start with the \[OpenCurlyDoubleQuote]range\[CloseCurlyDoubleQuote] operator \
x ;; y which is kind of like Matlab. You can use this as a general submatrix \
operation on a nested list if you need to.\
\>", "Text",
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It\[CloseCurlyQuote]s convenient to split things up into sublists sometimes, \
which you do with Partition.\
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And you can eliminate all that sometimes annoying nesting with Flatten:\
\>", "Text",
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A last move that comes in handy is the \[OpenCurlyDoubleQuote]riffle\
\[CloseCurlyDoubleQuote], which combines two lists:\
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The classic \[OpenCurlyDoubleQuote]riffle\[CloseCurlyDoubleQuote] is used for \
commas or spaces in text output, as in\
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A really important data structure in VHLs like Python is the \
\[OpenCurlyDoubleQuote]dict\[CloseCurlyDoubleQuote] or \
\[OpenCurlyDoubleQuote]associative array\[CloseCurlyDoubleQuote]; basically, \
an array which is indexed by something other than the numbers 1..n. First, a \
very common use case for this is the array indexed by numbers which are only \
a few of 1..n. This is a special case. If we have a 50x20 matrix of zeros \
with 1\[CloseCurlyQuote]s in positions (1,3) and (17,4), there\
\[CloseCurlyQuote]s no need to store the other 998 zeros explicitly.
We use SparseArray.\
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There is a more general \[OpenCurlyDoubleQuote]Association\
\[CloseCurlyDoubleQuote] type, which is happy to be indexed by anything you \
like:\
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Cell["Map! Map! ", "Section",
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Cell["\<\
This can be easily parallelized by replacing \[OpenCurlyDoubleQuote]Map\
\[CloseCurlyDoubleQuote] with \[OpenCurlyDoubleQuote]ParallelMap\
\[CloseCurlyDoubleQuote]. Nothing else changes, but there\[CloseCurlyQuote]s \
some overhead, particularly if your computer has only a few cores, or is \
doing something else at the time.\
\>", "Text",
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Cell[CellGroupData[{
Cell["Abbreviating Map. ", "Section",
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Cell["Map can be abbreviated /@ and usually is:", "Text",
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Cell["\<\
You say this as \[OpenCurlyDoubleQuote]Sin mapped onto (stuff)\
\[CloseCurlyDoubleQuote]. It\[CloseCurlyQuote]s equivalent to \
\>", "Text",
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Cell["\<\
There are additional options for Map which allow you to control exactly where \
in the expression the function is applied.\
\>", "Text",
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Cell["Pure functions.", "Section",
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Cell["\<\
A useful thing to combine with Map is the \[OpenCurlyDoubleQuote]pure \
function\[CloseCurlyDoubleQuote]. Suppose you want to square all the numbers \
in your list and multiply them by 3. Since \[OpenCurlyDoubleQuote]Square3\
\[CloseCurlyDoubleQuote] isn\[CloseCurlyQuote]t a built-in function, you \
could define it:\
\>", "Text",
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Cell["\<\
But it\[CloseCurlyQuote]s often easier to read if you use the /@ syntax and \
define the function \[OpenCurlyDoubleQuote]inline\[CloseCurlyDoubleQuote]:\
\>", "Text",
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Cell[TextData[{
"The alternative to a \[OpenCurlyDoubleQuote]Map\[CloseCurlyDoubleQuote] is \
the kind of iteration where each iteration applies a transformation to some \
output. This is really common in numerical algorithms, which tend to iterate \
on a solution in the hopes of improving it. Here\[CloseCurlyQuote]s a problem \
from Homework #3.\n",
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This shows the iteration converging as hoped. Note that unlike \
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Counting stuff in lists is a great place for the \
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You can construct more complicated patterns, like \[OpenCurlyDoubleQuote]e or \
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Similarly, you can do string transformations using StringReplace and ordinary \
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Manipulate can display anything you like, including numbers, formulae, \
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Note the extra \[OpenCurlyDoubleQuote]by 1\[CloseCurlyDoubleQuote] step in \
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If we didn\[CloseCurlyQuote]t have it, the iterator would try to iterate over \
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Actually, you can do a LOT with Manipulate, including having input which isn\
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What makes this work is a special kind of control called a \
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\[OpenCurlyDoubleQuote]MathematicaPlayer\[CloseCurlyDoubleQuote]. In general, \
these ONLY get Manipulate inputs because otherwise they would BE ",
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Demonstrations Project\[CloseCurlyDoubleQuote] is a curated collection of \
hundreds of ",
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