Dealing with Mathematica:
Quick Tips

\(Off[General::spell]; 
Off[General::spell1]; 
Off[Plot::plnr]; 

    Off[ParametricPlot::ppcom]; 

    If[MemberQ[{"Global`", \ "Miscellaneous`SIUnits`", \ 
          "Graphics`Colors`"}, 
\ \ \ \ Context[Gray]], \ 
      Remove["*`Gray"]]; 
<< "Miscellaneous`Units`"; 

    Remove["*`Gray"]; 
<< "Graphics`Colors`"; 
Clear[Derivative]; 
    \)

Axes3D[u,v]

Axes3D::usage = "Axes3D[a, b] creates a Graphics3D object of cartesian axes with x, y, and z running from -a/3 to a, and with axes labels b units beyond the tips of the axes.  Axes3D[a] is Axes3D[a, a/8].";

Axes3D[u_,v_] := 
      Graphics3D[{
            {Blue,Line[{{-u/3,0,0},{u,0,0}}]},
                    Text["x",{u + v,0,0}],
            {Blue,Line[{{0,-u/3,0},{0,u,0}}]},
                    Text["y",{0, u + v,0}],
            {Blue,Line[{{0,0,-u/3},{0,0,u}}]},
                    Text["z",{0,0,u + v}]}]; 
Axes3D[u_] := Axes3D[u,u/8];

CMView  =  {2.7,  1.6,  1.2};

Mathematica's notebook building front end is amazing.  You'll see all sorts of terrific formatting and cell grouping and the like as you go through the Math Everywhere Courseware.  We only scratch the surface.  You might want to know how to do some of the things you see here.  You can look everything up in the Mathematica Help Browser, but sometimes it's a bit difficult to find just what you want.  
This notebook is meant to get you started with stuff like entering mathematical expressions that look like standard math rather than computer commands, for example.  Play with it if you like.  Ignore it if you would rather do that.  We offer it for your enjoyment.

Basic Navigation through closed groups of cells.

Look below at those titles.  Look to the right.  Those blue sort of brackety looking things are, indeed, called cell brackets.  The double ones, the ones with the little harpoon hooks at the bottom, the farthest right, are the container brackets, the collectors, whatever.  That harpoon at the bottom means that there is stuff hidden from view inside.  To open them, move the cursor down to one of them.  Watch for when it looks like |<- (an approximation).  When it does, double click.  The cell opens up and more cells appear.  To close up a group, go to its outer bracket, look for the |<-, and double click.  Bingo.

Entering Mathematical Expressions

One dimensional, terminal entry style.

You probably know how to enter mathematical expressions that involve only arithmetic and algebra into mathematical software.  There's a very short list of things to know in Mathematica.  First, get yourself an input cell by moving the mouse down until the cursor turns into a horizontal I-beam.  While it's a horizontal I-beam, click the mouse.  That line across your window is automatically an empty Input Cell.  Type the stuff you want, and it will appear as bold-face Courier type with a dark vertical line on the left.

4 + 5  x + 6  x y - 3  x^2 - Pi x^2  y^3 + 
      (13 - 2  y)/(11 + 7  x)

So all you need to know for expressions like that is that a space suffices for multiplication, a slash, /, does division, and a carat is ^, shift-6. You also have to be careful about using parentheses to group things correctly.  Notice the output from that last command.  It's displayed in a form closer to what you'd see in print.  
Here are a couple of other standard symbols to check out.  The thing to notice is that native Mathematica always names things starting with capital letters.

Sqrt[x]

Standard constants like e and i appear as E and I.  Look

I^2

N[E, 25]

N[Pi, 30]

N[Sqrt[2], 35]

Function names are the same for the most part, but capitalized, and always bracketed with square brackets, not parentheses.

Sin[x]

Cos[x]

Tan[x]

One to be a bit careful about is the log function.  In Mathematica, Log[x] is the natural logarithm, the one which is frequently written as ln(x).  Your favorite base 10 logarithm, that appears as log or Log on calculators is

Log[10, x]

a handy formula in itself.  That natural log?

E^Log[x]

You'll encounter many others as you go.  Get used to the capital letters, and the square brackets, though.

Making input look more like standard math

In the stuff above, the input was easy to type, and the output was in more traditional format.  You'll see rather nicely formatted input expressions a lot.  How did they do that?  For example,

6 x y - 3 (x^2) - π (x^2) (y^3) + ((13 - 2 y)/(11 + 7 x)) + 
      (Sqrt[x])

How can you get your input cells to look like that?  The easiest way is to start with what you type in that one-dimensional format like this.

6  x y - 3  x^2 - Pi x^2  y^3 + (13 - 2  y)/(11 + 7  x) + 
      Sqrt[x]

Then select all of the typing in that line, go to the Cell menu, to ConvertTo/StandardForm.  Do that and see.
That little trick will work with even more complicated, more mathematical looking expressions like the unformatted

Integrate[f[x], {x, 0, T}]

Do the "select, and convert to standard form" stunt to that one.

(∫_0^T) f[x] \[DifferentialD]x

It works.

You can do the same thing with mathematical expressions inside any cell.  You can do it inside Text cells like this, for example.  Suppose you have the very same expression as above:  
	6x y-3x^2-Pi x^2  y^3+(13-2y)/(11+7x)+Sqrt[x].
Do the same thing.  Select the expression and go to Cell/ConvertTo/StandardForm.  The expression doesn't have to be dispalyed on a line of its own.  Here's that integral: (∫_0^T) f[x] dx.

Special symbols, Greek letters and such.

If you have a deep need for special mathematical symbols, or greek letters, or pre-formatted cube root templates, the first thing to use is the BasicInput palette.  Go to the File menu, and down to Palettes->BasicInput.  There is also a complete character palette there.  Play with one of them for a while.  You can enter all of that stuff into your text cells just as well, you know.  That integral above done in this text cell using the basic input palette comes out like this:  (∫_0^T) f[x] dx.  You get from box to box in the template with the tab key, and you escape from the template with a control-0, or control-space.
Try out some of the basic typing here.
To get the Greek letter alpha, start the special character by hitting the escape key, esc, type a (for alpha), hit escape.  Here's α.  Here's β, done the same way, but with a "b".  
To get a superscript, say as in (E^x), type the E, then control-6 (control-^), then x, and escape from the formula mode with control-space followed by control-0.  To get a subscript, as in something like (Subscript[A, k + 1]), type the A, control-underscore, k+1, control-space, control-0.

This could go on and on and on, but it needn't.  If you want to do really fancy stuff, we refer you to the Help Browser...  ( Click here for help in inputting expressions )

The really good mathematical looking expressions, like you might see in a textbook, can be done in Mathematica, too.  It's called Traditional Form.  Unless you have a really good reason to typeset some expression in that form, avoid it.  Mathematica sometimes has trouble executing commands which contain TraditionalForm elements because they are frequently entered in a non-parsable sequence.  True, you can get stuff like 
	\(TraditionalForm
      \`∫\_0\%X\( e\^\(-x\^2\)\) 
          dx\), and \(TraditionalForm
      \`∏\+\(k = 0\)\%K\(∑\+\(i = 1\)\%20 a\_\(i, k\)\)\) , and fancier, but don't.

Cleaning house: Grouping and ungrouping, setting cell styles and doing away with old cells.

We at Math Everywhere have our default style sheet, MEImathstyle.nb, set so that there isn't any automatic cell grouping.  It was our choice.  Maybe we'll change, but for now, you are using our style sheet, so what we say applies to the setting you'll find under the Cell menu under CellGrouping -> ManualGrouping.

How to collect cells into a group

You have the following cells.

cell1

cell2

cell3

cell4

and cell5

Quite a mix.  You decide you want them bracketed together into a single group.  
Do this.  Select the inner cell bracket on that "cell1".  Hold the shift key down and select the cell bracket on "cell4".  (Yes, "cell4").  Now that top bunch of cells have all their brackets selected.  Good.  Go to Cell/CellGrouping/GroupCells.  What happened?  They got enclosed in a single outer cell.  
Now you decide you want to add "cell5" to the group.  No problem.  Select its cell bracket, hold down the shift key and click anywhere inside the already bracketed things and click.  Go to Cell/CellGrouping/GroupCells, and voila, it slid it right inside.

How to ungroup

This is your notebook.Open this group.

It has two cells grouped together.  You want them ungrouped for some reason.
Select the outer bracket for the pair.  Go to Cell/CellGrouping/UngroupCells.  Bingo.

Setting cell styles.

Cells have individual styles.  Go to Format/Style to see what this is.  The check is beside the Text.  Do the same by putting the cursor in the cell above this one and going to Format/Style.  What is it?
Now select the cell bracket for this particular cell.  Go to the Format/Style menu and release on Title.  Do the same for Subsection, and so on in that list.  See the differences.  Right now let it suffice to say that the styles mainly control the look of the cells.  That's far less than they really do, though.  In our setup, only input cells can evaluate expressions, for example.

\(Here' s\ a\ cell\ for\ you\ to\ play\ with . 
		Change\ it\ to\ 
      SmallText\)

Getting rid of unwanted cells

The cell directly below this one is full of trash.  You want to get rid of it.

empty sardine cans
broken light bulbs
Mathematically Correct newsletters
water-soaked magazines
yesterday's fish bones

Select the cell bracket.  Go to Edit/Clear.  Gone.

If you are going to send your notebook to someone, or archive it for later use, you might want to make it smaller.  The best way to do that is to delete all of the output in the notebook.  The best way to do that is to go to Kernel/DeleteAllOutput.  It will be gone.  All of the output, gone.  You can't get it back without re-evaluating everything.  That's OK.  Your machine and Mathematica make a fast team.

Brackets, Lists and Parts of things.

There are lots of things in Mathematica input and output as lists.  Some you see frequently.  In the Plot[ ] command here,

Plot[{Sin[u], Tan[u]}, {u, ((-π)/2), (π/2)}, 
      PlotStyle -> {{Red, Thickness[0.007]}, {Blue, Thickness[0.008]}}];

everything you see in curly brackets is a list.  There's the list of two functions to be plotted.  There's the list that says that u goes from ((-π)/2)to (π/2), and the list of the styles for the plots of Sin[u] and for Tan[u].

Lists from solve commands, and what to do.

Look at the output from this Solve[ ] command.

\(Clear[x, a, b, c, solutions]; 

    solutions = Solve[x\^2 + b\ x + c == 0, x]\)

The quadratic equation gave us two solutions.

solutions[[1]]

and

solutions[[2]]

Notice how easy it was to get the first solution rule out of the solutions, and the second.   
There's a puzzle here.  Look at those parts of solutions.  They both say things like x->number.  They don't say x=number.  Ask Mathematica what it thinks x is.

x

Nothing.  That's because that x->number only gives you a rule for assigning a value to x.  It doesn't do that assignment.  If you want the value assigned to x, tell Mathematica to do the substitution (replacement) this way.

x = x /. solutions[[1]]

Now check it.

x

That /. says replace according to the rule to the right.  It's flexible, too.

\(Clear[x]; 
x\^2 + b\ x + c /. solutions[\([1]\)] // Expand\)

Hmm.  It substituted the x in the rule for solutions[[1]], and got what it should have, namely zero.