Introduction to Mathematica

Mathematica Cells

Mathematica notebooks are subdivided into cells, each of which is marked by a blue bracket at the right of the window.  These cells come in several different types.  For instance, the text you are currently reading is in a "text" cell, while the expression below is in an "input" cell.

76 - 23

You can tell the different types of cells apart by the font they use: text cells use a normal weight proportional width font (Times), while input cells use a boldface typewriter-style font (Courier).  Text cells are just for show, containing instructions, descriptions, or explanations.  If you want Mathematica to evaluate a mathematical expression, you must put it in an input cell.  To create a new cell, position the cursor between two existing cells or after the last cell.  The cursor will then change from a vertical I-beam to a horizontal I-beam.  If you click there, Mathematica will draw a horizontal line across the window.  If you then start typing, Mathematica will create a new input cell for you.  Any new cell you create will automatically be an input cell.  (Creating a text cell takes an additional step, but you won't be needing to create any text cells in these labs.)

Try creating a few input cells and entering a few expressions in the space below.

Evaluating Expressions

All commands are entered into Mathematica as text.  To evalute an expression, click anywhere in the expression and press the <Enter> key on the keypad (not the one on the main keyboard) or hold down the <Shift> key and press <Enter>.  Mathematica will then think for a moment and write the answer below the expression.  Your input and Mathematica's output are labeled "In[#]:" and "Out[#]:" respectively, and Mathematica draws the input in boldface to distinguish it from the plain text output.  For example, to add the following two numbers, click the sum below and press <Enter>.

3 + 8

Throughout these Mathematica assignments, we will give you lots of expressions to evaluate, along with descriptive text.  Just go through it one cell at a time, reading the text and evaluating each of the expressions one by one.  In these notebooks, cells which require you to try out what you have learned will be boxed in blue.

Basic Operators

The basic operations are + (addition), - (subtraction), * (multiplication), / (division), and ^ (exponentiation).  Mathematica obeys the usual operator precedence, first performing exponentiation, then multiplication and division, and finally addition and subtraction.  If you want to group your operations differently, use parentheses ().  Try evaluating the following expressions:

12 - 3*5

(12 - 3)*5

6/2*3

6/(2*3)

-5^2

(-5)^2

If you put two expressions next to each other separated only by a space, Mathematica multiplies them, just like in normal mathematical notation.  This can be confusing if you just put two numbers next to each other, but it comes in handy with more complicated expressions.  Here are some examples:

5 2

3 (8 - 4)

Functions and Constants

Mathematica comes preprogrammed with lots of functions and constants.  Some of the functions we will be using are Sin[x], Cos[x], Exp[x] (the exponential (e^x)), Log[x] (the natural logarithm ln x), Sqrt[x] (square root (Sqrt[x])), and Abs[x] (the absolute value |x|).  Note that the first letter of all built-in functions are capitalized, and you use square brackets [] to apply them, as opposed to regular parentheses ().  Some useful constants that Mathematica knows are Pi (or π) and E (or e ≈ 2.71828, the base for natural logarithms).  Again, be sure to capitalize the first letter.  Alternatively, you can use the palette of templates and symbols on the right of your screen to type in some of these operators and constants.  (If you do not see the palette, select from the menu File - Palettes - BasicInput.)  Here are some examples.

Sin[0.123]

Just like in standard math notation, if you write two things next to each other with only a space between them, Mathematica multiplies them.

Cos[2 Pi]

When you evaluated this next expression, note that Mathematica is clever enough to give you the exact answer, rather than a decimal approximation.

Sin[π/4]

If you prefer a numeric approximation, add //N to the end of your expression.

Sin[π/4] // N

Defining Variables and Functions

Mathematica is more than just a fancy calculator.  It lets you do both symbolic and numerical calculations using variables and functions.  You can use any string of letters for a variable name.  In addition to the usual variable names x, y, t, you can also use more descriptive names like theta or radius.  You can assign variables a value and then use that value in further computations.  Here are some examples:

a = 5

a^2  -  3  a  +  4

b = -3

a b

Notice that if you write two variable names next to each other separated only by a space, Mathematica multiplies them, just like in normal mathematical notation.  However, you must take care to put in the space between your variable names.  Otherwise, Mathematica will think that you are talking about a single variable with a long name!  For instance, in the following cell, instead of multiplying a by b, Mathematica assumes you are using a new variable ab:

ab

To define a function for Mathematica, we write f[x_] := value, where f is the name of the function, x is the variable, and value is the expression for the function.  Note that when defining a function, you again use square brackets [] instead of parentheses. In Mathematica, this distinction is very important; parentheses () are used to group expressions, while square brackets [] are used when applying a function.  When defining a function, you use colon-equal := instead of a normal equal sign, and you put an underscore _ immediately after the variable name.  This tells Mathematica to evaluate the right side only after plugging in the value of the variable.  Note that Mathematica does not give you an output when you define a function, but once a function is defined, you can apply it to any number you want, or even to a variable or expression.  When you are finished typing out
your function definition in the input cell, you must type 'enter' on the keypad or hold down
'shift' and press 'enter' on the regular key board to tell Mathematica that your definition is ready.
(This tells Mathematica to read the input cell). Give the following functions a try:

f[x_] := x^2 - 3  x  +  4

f[5]

f[t + 1]

Here we define a function of two variables.  Note that your variable and function names can be as long or as descriptive as you like.

DistanceFromOrigin[x_, y_]  :=  Sqrt[x^2 + y^2]

DistanceFromOrigin[3, 5]

Exercise: Quadratic Formula

Do you remember the quadratic formula from high school?  It gives you the roots of a quadratic polynomial a (x^2) + b x + c.  In case you still don't remember, here it is: \(TraditionalForm\`\(\(-b\)\  ± \@\(b\^2 - 4  a\ c\)\)\/\(2  a\)\).  Sure that's one formula, but it still takes a while to plug in the values and compute the roots by hand.  However, using Mathematica, you can define functions to do all the hard work for you.  Try creating a few new cells below and define two functions PlusRoot[a,b,c] and MinusRoot[a,b,c] to compute these two roots (corresponding to the plus-or-minus sign in the numerator).  Then use your functions to find the roots of the polynomial 3  (x^2) + 2  x - 5.  (Hint: the answers are 1 and -(5/3))

Here are some things to remember:
•  To create a new input cell, position the cursor below this gray box so that Mathematica switches the cursor to a horizontal I-beam, click the mouse, and then start typing.  Make sure you are typing into a new input cell (with bold typewriter-style text in the Courier font) and not into one of the pre-existing text cells.  If you are typing into a text cell, you will not be able to evaluate your expression.
•  To take a square root, use the Sqrt[] function, and remember that the first letter is capitalized.
•  Remember to use parentheses () to group expressions and square brackets [] when applying functions.
•  To multiply two variables together, write them next to each other separated by a space.  If you leave out the space, Mathematica thinks it is one variable with a two letter name.
•  When you define a function, don't forget to put an underscore after your variable names (a_, b_, c_) on the left hand side, and use colon-equals := instead of just equals = in the function definition.  Otherwise, Mathematica will use the values of  a, b, c that you defined earlier in this notebook.
•  Remember that before you use your functions, you must first 'evaluate'your function definitions by pressing 'enter' on the key pad or holding down 'shift' and pressing 'enter' on the standard key board.
•  If you get the wrong answer, make sure that the expressions are grouped together correctly with parentheses.  Your functions will need to be of the form ( numerator ) / ( denominator ).

Graphing Functions

To graph a function in Mathematica, you need to tell Mathematica the function itself, the name of the dependent variable, and the domain of the function in the following format: 
	Plot[function to be plotted, {dependent variable, lower range, upper range}]  
Note that we use [] for the Plot function.  To describe the domain of the function, we enclose the dependent variable and the bounds in curly braces {}.  For instance, to plot the function f(x) = (e^(-((x^2)/2))) (called a Gaussian distribution function) in the domain –3 ≤ x ≤ 3 you would type:

f[x_]  :=  Exp[-x^2/2]

Plot[f[x], {x, -3,  3}]

In the Plot command, be sure to write f[x] including the variable name. If you just write f, Mathematica will think that f is a variable name, not a function. As a shorthand, you can skip the step where you define the function separately, and include that in the Plot command itself:

Plot[Exp[-x^2/2], {x, -3,  3}]

Try defining and plotting the following functions using the Plot command:
	y = ln x		0 < x ≤ 5
	g(t) = sin t		-2 π ≤ t ≤ 
        2  π
	F(
          x) = (∫_0^x) g(t) dt	-2 π ≤ x ≤ 
        2  π
	(Use the definite integral template (\(TraditionalForm\`∫\_□\%□ □ d□\)) from the toolbox on the right.)