Open Mathematica 8.0 by clicking on the icon.  It may take a while to open.

When the Mathematical window opens, click on "Create New Notebook."  When this opens,
you may start typing commands.

Try this:  Type 2+2, nothing else.

To obtain an answer (output), press "Shift + Enter".

Remember, Mathematica is case sensitive.  Be sure you use the right kind of
bracketts and upper case letters exactly as shown in any tutorials or examples..

Warning:  In mathematica, functions must have the first letter as upper case, that is,
Cos, not cos.  Angles must be in radians, where 1 rad = 180/ 3.14159 = 57.2958 degrees.

Also the argument of a function must be in squre brackets not parentheses.  E. G.,
Cos[.7854] = 0.707105

For example, to find the integral of y=(1+5x²)dx, from x=0 to x=10, type the following as
input:

   Integrate[(1+5*x^2) , {x, 0, 10}]   (If anything you type comes out red, your
syntax is wrong or incomplete, until you have balanced your brackets. )

Then press "Shift + Enter".

The answer is 5030/3.

Differentiation:   Dt[function, variable].  E. G.: Dt[5x^3+2x^4, x]   Then press "Shift + Enter".

Plotting: Plot[ function,  {x, xmin, xmax}]

To enlarge a plot to fit on a single page, click on the graph.  Then grab a corner marker and drag
with the mouse to expand the plot area.  After clicking on the graph you may also copy it to the
clipboard and then paste it into another notebook file.  Then expand to fit the page.  This gives a
 more professional looking graph for submitting in a report or paper.

Also, to specify ordinate range use:  Plot[ function,  {x, xmin, xmax}, PlotRange -> {ymin,ymax}]
 

MM1.  The following is an example from Statistical Thermal Physics:  Plot the following function, which
is the Maxwell-Boltzman velocity distriubtion function for oxygen moleclues in a gas at T=300K.:

Plot[(y = 3.6*10^-8)*(v^2)*Exp[(-v^2)/157800], {v,0,1000}].

Then press "Shift + Enter."

The number 157800 may also be written as (1.578*10^5), the parentheses being necessary.

In the above expression, y is the fraction of the total number of molecules that have speed v.
Hence, the integral of y over all velocities should be 1.00.

MM2. Integrate the above expression for y over the range v=0 to v= 3000.
You should get 0.9999.

MM3.  Plot the function y=e^-2xCos[20x] from x=0 to 90 degrees.

To learn more, click on "help" in the menu bar at the top of the page.  Then click on
"Virtual Book" and then "Introduction."  Go next tp "Getting Started" amd then click
on then "Your First Mathematica  Calcultions."  Then proceed to explore more from there.

The following is a summary of  some commands for Mathematica.

>= <=    grater than or equal to
<=    less than or equal to
&&   and
||      or

3.    Working with Ploynomials

        Simplify[expression]    This command tries to produce an expression that is shortest
                                        to write out.  Often, the result may be the same.

4.    Working with Equations

        Solve[{ equation1, equation2 }, {variables to folve for}]    Solves a set of equations simultaneously.

        FindRoot [ x==Cos [x], {x, 0.75} ]       Ans:  {x -> 0.739085}

To learn more about how to use Mathematica, click on "help" in the menu bar at the top of the Mathematica window.
When a menu opens, click on "Virtual Book."  Then you can click on any of the options to learn more.  I recommend
starting with "Introduction" and then "Getting Started."  Follow this with "your First Mathematica Calculations and
go from there.