Preface
This is a reference to the basic Mathematica commands that you are likely to encounter in your lab
work at Virginia Tech. You are encouraged to print it out and retain it for future reference. Please
send corrections, suggestions and other feedback to Heath Hart ([email protected]).
This is not a guide to installing Mathematica on your computer; if you have installation questions,
contact your instructor.
This document is version 1.0 (September 4, 2007).
Contents
Preface
1
2
3
1
The Mathematica Environment
1.1 Notebooks and cells . . . . . . . . . . .
1.2 Your first cell . . . . . . . . . . . . . . .
1.3 How to create a new cell . . . . . . . .
1.4 How to delete a cell . . . . . . . . . . .
1.5 How to change the cell type . . . . . .
1.6 How to change the appearance of text
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2
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2
4
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5
5
Calculations with Mathematica
2.1 Basic operations . . . . . . . . . . . . . . . . .
2.2 Calculator-like functions . . . . . . . . . . . .
2.2.1 Roots . . . . . . . . . . . . . . . . . . .
2.2.2 Trigonometric Functions . . . . . . . .
2.2.3 Logarithms . . . . . . . . . . . . . . .
2.3 Significant Digits and Special Commands . .
2.3.1 N . . . . . . . . . . . . . . . . . . . . .
2.3.2 % . . . . . . . . . . . . . . . . . . . . .
2.3.3 ; . . . . . . . . . . . . . . . . . . . . .
2.3.4 How to enter other special characters
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5
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8
Variables in Mathematica
3.1 One-time assignment using = . .
3.2 Ongoing assignment using := . .
3.3 Evaluation and solving using ==
3.4 Using Solve and NSolve . . . . .
3.5 The Clear command . . . . . . .
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4
Functions in Mathematica
4.1 How to create a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Some nifty things you can do with functions . . . . . . . . . . . . . . . . . . . . . . .
11
11
11
5
Tables and matrices in Mathematica
5.1 Creating a table manually . . . . .
5.2 Matrix operations . . . . . . . . . .
5.3 Solving linear systems of equations
5.4 The Table command . . . . . . . .
5.5 TableForm and MatrixForm . . . .
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12
12
14
14
14
15
Graphs in Mathematica
6.1 The Plot command . . . . . .
6.2 The PlotStyle option . . . .
6.3 Other Plot options . . . . . .
6.4 Two graphs on the same axes
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15
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18
6
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Common mistakes in Mathematica
18
The Mathematica Environment
1
1.1
Notebooks and cells
A Mathematica session is called a “notebook”, in the same way that a Word session is called a “document” or an Excel session is called a “spreadsheet”. When you first start Mathematica, a blank
notebook will be created for you.
Notebooks are made up of cells. A cell may contain commands, data, graphs, comments and other
types of information. Cells may even contain other cells. Every cell has a cell bracket positioned
at the right of the cell to indicate not only where the cell begins and ends, but also what kind of
information is contained in the cell:
1.2
•
A diagonal line in the upper corner of the cell bracket indicates that the cell is an input cell.
Content in these cells will be interpreted by Mathematica as commands.
•
A horizontal line in the upper corner of the cell bracket indicates that the cell is not evaluatable. These cells can be used for comments to the reader of the notebook, and are often called
text cells. However, these cells do not strictly have to contain text; they may contain graphics
as well.
•
A diagonal line and a horizontal line indicates an output cell. These contain the results of
successful commands (or error messages for unsuccessful commands.)
Your first cell
Start Mathematica and wait for the blank notebook to appear. If you do not have a blank notebook
available, click on “File” and drag down to “New”. (From now on, we’ll abbreviate this sort of
2
Figure 1: An example notebook.
3
instruction as “File → New”.)
Click anywhere within the blank notebook window and type this easy command (don’t press EN TER just yet, though!):
4 + 3
Hold down the SHIFT key and press ENTER. The combination of SHIFT-ENTER tells Mathematica to
process the commands in the current cell. (The ENTER key alone just takes you to a new line within
the current cell.)
A few things just happened:
• The obvious result is that Mathematica responded to your command to add 4 and 3 together
by writing 7 to the screen.
• Mathematica also created a cell for the result, and a grouping cell to group the command and
the result together.
• Mathematica also gave the input and the output matching numbers. Your command was labeled In[1]:= and the answer Mathematica gave you was labeled Out[1]:=. Commands and
their output will always have matching numbers, and the numbers will be in the order the
commands were entered (which may not be the same order in which they appear in the notebook.)
1.3
How to create a new cell
To add a new cell to the worksheet, use the mouse to place the cursor anywhere in the notebook
other than within an existing cell, then click the mouse and begin typing. You may add cells before,
after, or between existing cells. The cursor changes to a horizontal edit bar to indicate a position
where a new cell can be added.
If you have not moved the mouse since entering your last command, you can just continue typing
and Mathematica will automatically create a new cell for you just below the cell containing your last
command.
Keyboard shortcut: You can also use the up and down keys on the keyboard to move through the worksheet.
When you’re at a position where you can begin a new cell, a horizontal line will appear all the way across the
notebook. When you see this line in the position where you want your new cell to appear, start typing.
1.4
How to delete a cell
To delete a cell from the worksheet, click on the cell bracket (the cell bracket will highlight) and
press the DELETE key.
Keyboard shortcut: Use the arrow keys on the keyboard to put the cursor somewhere in the cell you wish
to delete. While holding down the CTRL key, press the period (.) key once or twice until the cell bracket is
highlighted. Then press the DELETE key.
4
1.5
How to change the cell type
By default, Mathematica creates all new cells as input cells. To change the format to a text cell, click
the cell bracket to highlight it. Then from the menu bar select “Format → Style → Text”. (There are
other formats for titles and headers that you may wish to experiment with as well.)
Keyboard shortcut: To change the cell type, use CTRL-. (period) to select the cell bracket, and then press
-7 (Mac) to change the cell to a text cell.
ALT -7 (Windows) or
1.6
How to change the appearance of text
To change the appearance of any text, highlight the text, and select “Format” from the menu bar.
In the Format menu, you’ll see options to change the font, face, size, color, and justification of the
text.
Note: You may change the appearance of any portion of an input cell or text cell independently. (For example,
you can make one word of a line of text bold without affecting the rest of the line.) Mathematica allows
formatting to be applied only to the entire content of an output cell.
2
2.1
Calculations with Mathematica
Basic operations
Many basic operations—like addition and subtraction, for example—are entered in exactly the way
one would expect to enter them. Mathematica uses the symbols ?, /, and ^ to indicate multiplication, division, and exponentiation. The algebraic order of operations is carefully followed, and
parentheses may be used to specify that a particular operation be done first.
Entering this input. . .
3 + 8 - 2
13 ? 4
6 - 2 ? 5
(6 - 2) ? 5
2 ^ 6
100 / 4
would produce this output
9
52
-4
20
64
25
If an operation is omitted, Mathematica will assume that multiplication is intended. Therefore, you
can enter expressions like 3(2 + 5), and Mathematica will understand this to mean 3 times (2 + 5)
and will output 21.
Warning: Because Mathematica assumes a multiplication sign when an operation is omitted, you cannot enter
mixed numbers like 3 12 in Mathematica. The expression 3 1/2 would be interpreted by Mathematica as an
instruction to multiply 3 by 1/2, resulting in a value of 1.5, and not 3.5.
5
2.2
Calculator-like functions
Mathematica also knows the functions you might see on an scientific calculator. Each of them starts
with a capital letter, and their arguments are enclosed in square brackets, not parentheses!
Entering this input. . .
Sqrt[9]
Sin[Pi / 2]
Cos[45 Degree]
ArcSec[2]
E^2.5
2.2.1
would produce this output
3
1
Mathematica works in “radians mode”
1
√
Degree is always singular1
2
π
ArcSec[x] = sec−1 ( x )
3
12.1825
Roots
Mathematica has a built-in function Sqrt[x] to perform square roots. In order to calculate higher
roots, you may use fractional exponents. For example, 64^(1/3) will produce the output 4.
2.2.2
Trigonometric Functions
Mathematica knows all six trigonometric functions and their inverses: Sin[x], Cos[x], Tan[x],
Sec[x], Csc[x], Cot[x], ArcSin[x], ArcCos[x], ArcTan[x], ArcSec[x], ArcCsc[x], ArcCot[x].
Mathematica does not use the notation Sin^-1[x] to denote an inverse trig function.
Mathematica always assumes angles are given in radians unless the word Degree is explicitly used.
2.2.3
Logarithms
In Mathematica, the expression Log[x] represents the natural logarithm of x, which is often written
ln x. For example, Log[3] would have the value of ln 3, and Log[E] would equal 1.
In order to calculate logarithms with other bases, Mathematica offers the alternate form Log[b, x],
where b represents the new base. To calculate log2 256, enter Log[2,256]; Mathematica will respond
with 8.
Remember: If no base is specified, the base defaults to e.
2.3
Significant Digits and Special Commands
Mathematica will attempt to provide exact mathematical values in algebraic, and not decimal approximations, unless a decimal is used as the input. This leads to some results that may seem
1 You
may also enter Cos[45◦ ], but entering the degree symbol requires a trick we’ll see later in the section on Palettes.
6
Figure 2: Using % to refer to previous results
strange at first until you get used to how Mathematica operates. For example,√
entering Sqrt[17.1]
produces the output 4.13521, but entering Sqrt[17] produces the output 17. This allows for
Mathematica to perform very elaborate calculations without accumulating round-off errors in its
intermediate results.
2.3.1 N
The command N[x, n] will provide a decimal approximation to x accurate to n significant digits..
For example, N[Sqrt[200],5] produces the output 14.142. Note that the five significant digits
includes digits to the left of the decimal point as well. If you want this result to five decimal places,
enter N[Sqrt[200],7] instead.
If the number of decimal places is omitted, Mathematica provides six digits by default. For example,
entering N[Pi] gives 3.14159 as a result.
2.3.2 %
The percent sign % can be used to indicate the result of a previous command. When Mathematica
encounters a percent sign immediately followed by a number n (no spaces), it replaces that term
with the result of the nth calculation (indicated by the Out[n]:= in front of that result).
Figure 2 shows the use of the percent sign.
A percent sign without a number always refers to the last computed result. % is often used in conjunction with N, because N[%] will give a decimal value for the result you just got in algebraic form.
2.3.3 ;
The semicolon is used at the end of a command to instruct Mathematica to perform the command
but not display the output. (See Figure 1 for an example.) This allows you to save space by not
7
showing intermediate results along the way to a final answer.
2.3.4
How to enter other special characters
To enter special characters, like Greek letters or the degree symbol, which are not easily accessible
from your keyboard, Mathematica provides you with a choice of editing palettes. These palettes are
turned off by default, but can be enabled by choosing “File → Palettes” from the menu bar. The
“BasicInput” palette contains most of the symbols you’re likely to encounter early on.
Keyboard shortcut: Most, but not all, of the special characters have a keyboard shortcut which is displayed
on the bottom of the palette window if you hover the mouse pointer over the character for a second. These
keyboard shortcuts begin and end with the ESC key. For example, typing ESC th ESC produces the Greek letter
theta (θ).
3
Variables in Mathematica
Mathematica allows you to use variables to store results of calculations and use them later. Variable
names must begin with a letter, may contain numbers (but not in the first position) and may not
duplicate any built-in words in Mathematica. Since the built-in words in Mathematica begin with
capital letters (and there are over 1000 built-in words!), we encourage you to use lowercase letters
in your variable names to avoid errors. Each of the following is a legal variable name:
x
y
z3
3.1
volume
pressure
height
r2d2
c3p0
One-time assignment using =
The = symbol is used to assign values to variables in the way you are probably most accustomed
to: the value of the expression is computed one time and immediately assigned to that variable. We
can then use that variable in other calculations. (This is called immediate assignment.)
When a variable is assigned a value, that value lasts until one of the following events occur:
• the variable is assigned a new value,
• the variable is unassigned using the Clear command, described below.
• you quit Mathematica.
It’s important to remember that the = symbol performs the calculation on the right-hand side only
once.
8
Figure 3: The difference between immediate and delayed assignment
3.2
Ongoing assignment using :=
The := symbol assigns values to variables in a different way, by perfoming the computation every
time the variable is actually used. What this means, in practical terms, is that the values of other
variables will have a lasting effect on the variable being assigned. (This is called delayed assignment.)
Using the := symbol means that the expression on the right-hand side will be calculated every time
the variable is used.
Figure 3 illustrates the difference between immediate assignment and delayed assignment. Notice
that when the assignment of y is delayed, Mathematica continues to maintain the relationship that y =
x + 3, so that even as x changes, y changes along with it. When the assignment of y is immediate,
that “closes” the relationship so that future changes in x no longer affect y.
3.3
Evaluation and solving using ==
The double-equal sign tells Mathematica not to assign a value to a variable at all. Instead, it tests to
see if two expressions are equal without disturbing either expression. Figure 4 illustrates the basic
use of the double-equal sign.
The double-equal sign is also used in the Solve and NSolve commands, described in the next paragraph:
9
Figure 4: Using == to test for equality
Figure 5: The Solve and NSolve commands
3.4
Using Solve and NSolve
The Solve command tells Mathematica to solve an equation algebraically. Mathematica uses its vast
library of algebraic rules, trig identities, log and exponent rules, and other tricks to identify solutions to equations, and presents those solutions as exact quantities. The NSolve command (both
the ‘N’ and the ‘S’ are capitalized!) also solves these equations, but presents solutions as decimal
values instead.
The form of these commands are the same:
Solve[equation,variable]
NSolve[equation,variable]
The equation uses the double-equal sign, because we are testing to see which values make the equation true, not assigning one of the values. The variable identifies which variable we are solving for
(after all, there may be more than one variable in the equation.) Figure 5 illustrates the use of these
commands.
Note that Solve and NSolve will sometimes give imaginary answers.
10
3.5
The Clear command
The Clear command resets a previously-defined variable so that it is undefined again; Mathematica
‘forgets’ that the variable was defined in the first place. Simply list the variables you want erased
separated by commas, for example:
Clear[x,y,z,c3p0,volume]
Tip: A special form of the Clear command exists, Clear["@"], which clears every symbol that does not have
a capital letter in it. (This prevents Mathematica from trying to delete its own built-in words.) If you’ve
followed our recommendation not to use capital letters in your variables, then this command can be used to
“wipe the slate clean” inside a notebook.
4
Functions in Mathematica
In addition to the hundreds of built-in functions in Mathematica, you may also define your own
functions. A function in Mathematica may be the sort of thing that you would ordinarily consider
to be a function, like f ( x ) = x2 − 3x + 1, but Mathematica allows for functions that are even more
powerful and elaborate than these.
4.1
How to create a function
The rules for naming functions are the same as the rules for naming a variable: function names
must start with a letter, but may contain numbers as well. You are encouraged to avoid capital
letters in the names of your functions.
Specify the name of your function, followed by a list of input variables in square brackets. Each
input variable requires an underscore ( ) character. Use the := symbol (we want to calculate the
function every time it is used), followed by the actual definition of the function. Here’s an example
of a command to create a function:
f[x_,y_] := 2x^3 - Sin[Pi ? y]
The underscores on the left-hand side tells Mathematica that these variables are blanks that may be
filled with any value later. If the underscore were omitted in the previous example, Mathematica
would define f ( x, y), but would not know that that definition should also be used for things like
f (0, 4) or f (q, s)—leaving out the underscore makes the definition apply only to the exact input
shown. (See Figure 6.)
4.2
Some nifty things you can do with functions
• Factor takes a polynomial and breaks it down into smaller factors, if possible. For example Factor[x^3 - 1] gives the result (-1 + x)(1 + x + x2 ). Expand does the opposite:
Expand[(x-3)(x+4)(x^2 - 2x + 11)] gives -132 + 35x - 3x2 - x3 + x4 .
11
Figure 6: How the underscore works in Mathematica.
• Solve and NSolve (Section 3.4) work with functions in the same way they work with variables.
• Plot is one way to create a graph of a function. We’ll look at the Plot command more closely
in section 6.
• The Derivative and Integrate commands allow you to compute derivatives and integrals of
functions. For single-variable functions, the apostrophe ’ can be used to indicate a derivative.
Figure 7 illustrates some of these commands.
5
5.1
Tables and matrices in Mathematica
Creating a table manually
To create a table manually, Mathematica uses curly braces, {}, to enclose the rows of the table, as
well as the table itself. Tables may be assigned to a variable in the same way that numbers can.
If we want to create the matrix:
M=
3 4
2 0
12
2
−3
−8
1
,
Figure 7: Some of the commands from Section 4.2
13
we would enter:
m = {{3,4,2,-8},{2,0,-3,1}}
Note that there are three sets of curly braces: one for each row of the matrix, and then one outer set
that encloses the whole matrix.
5.2
Matrix operations
If m and n are matrices which have the same dimensions, then m + n and m - n give the sum and
difference of the two matrices. If k is a scalar, then k m or k ? m gives the scalar product of k and m
(every entry of m is multiplied by k).
If m and n have dimensions which allow them to be multiplied, then m .
product of the two matrices.
n (m, period, n) gives the
The Transpose and Inverse commands provide the transpose of a matrix, and the inverse of a
square matrix, respectively.
5.3
Solving linear systems of equations
Consider the following system of equations:

 3x1 − 5x2
3x1 + 4x2

4x1 − 2x2
+ x3
− 2x3
+ 3x3
= −2
=
1
= 15
Here are two methods for solving this system of equations in Mathematica:
1. Use the Solve command to solve the equations directly. Using this method, the three equations have to be enclosed in one table, and the three variables have to be enclosed in a separate
table.
2. Create a matrix a containing the coefficients, a matrix x containing the variables, and a matrix
b containing the variables. Then use the Solve command to solve the matrix equation a.x =
b.
These two methods are shown in Figure 8.
5.4
The Table command
The Table command is also used to build a table, and is most often used when the entries in the
table follow a pattern. To build, for example, a table with the integers from −5 to +5 in the first
column, and the cubes of those numbers in the second column, we could type:
14
Figure 8: Two methods of solving linear systems of equations.
Table[{x, x^3}, {x, -5, 5}]
The first term, {x, x^3}, specifies what one row of the table looks like, and the second term, {x,
-5, 5}, specifies what variable to use and what range of numbers to substitute for that variable.
5.5
TableForm and MatrixForm
The TableForm and MatrixForm commands do no “math”, but specify how a table is to be displayed. The difference between the commands is that MatrixForm displays some extra borders
around the table.
The basic form of the commands are TableForm[table] and MatrixForm[table]. table can literally
be a table, or can be a variable which represents the table (like in Figure 9). The command takes
an optional second argument, TableHeadings which specifies the headings to be used on the rows
(first), and then the columns (second). The → arrow in the TableHeadings option can be created
by typing ->.
There is another optional argument, TableAlignments which specifies how the entries are to be
positioned. TableAlignments -> Center will cause the entries to be centered (other options are
Left and Right.)
6
6.1
Graphs in Mathematica
The Plot command
Mathematica has many ways of graphing information, but the most basic graphing command is the
Plot command. The basic syntax for the Plot command is,
15
Figure 9: TableForm and MatrixForm
Plot[function, interval]
For example, if we wanted to graph the function f ( x ) = 4x3 − 7 over the interval [−1, 5], we
would type: Plot[4x^3 - 7, {x, -1, 5}], or if we had already defined f[x], we could type
Plot[f[x],{x, -1, 5}] instead.
If you attempt
to graph a function which is undefined for part of the interval, such as graphing
√
f ( x ) = x over the interval [−2, 2], Mathematica will issue a warning, but will proceed with the
graph over the portion of the interval that is applicable.
6.2
The PlotStyle option
PlotStyle is an optional modifier to the Plot command which overrides the default graphing options. For example, Plot[4x^3 - 7, {x, -1, 5}, PlotStyle -> {Red}] would make the graph
red. Notice that the actual style is enclosed in curly braces.
Some of the most common plot styles you’ll use are:
• Red, Blue, Green, etc. The colors that are “built-in” to Mathematica are: Black, Blue, Brown,
Cyan, Gray, Green, Magenta, Orange, Pink, Purple, Red, White, Yellow.
You may also define custom colors using the commands RGBColor and CMYKColor. For example, the Virginia Tech colors can be created with the commands Maroon = RGBColor[.4,0,0],
16
Figure 10: An example of a graph created using PlotStyle
and HokieOrange = RGBColor[1, .4, 0] (this is slightly different from the built-in shade of
Orange.) Once you’ve defined these colors, you can use them in a PlotStyle.
• Thickness[n]. This specifies the thickness of the stroke which makes the graph. The thickness is given as a percentage of the whole graph, so Thickness[1] would obliterate everything. Good numbers for this style are tiny: Thickness[0.012] turns out to be fairly hefty.
• Dashing[{on,off }]. This specifies that the graph should be drawn using a dashed/dotted
stroke instead of a solid stroke. The first number specifies how long the dashes should be,
the second number specifies how long the spaces between the gap should be. Again, you
want to keep these numbers small: Dashing[{.05, .02}] works well. Dashing uses both
square brackets and curly braces around its values. (The redundancy is necessary to support
alternate forms of the Dashing option that create more elaborate dashing patterns.)
Figure 10 illustrates all three of these options in one graph.
6.3
Other Plot options
• PlotLabel creates a title for the overall graph, centered at the top of the graph. For example,
PlotLabel -> "This is my graph."
• AxesLabel labels the individual axes of the graph.
Example: AxesLabel -> {"Greatest x-axis ever!", "A y-axis Mom could be proud of!"}
• ImageSize specifies the size of the overall graph. For example, ImageSize -> 500 makes the
graph 500 pixels wide.
17
6.4
Two graphs on the same axes
You can use Plot to graph two or more functions on the same axes. When you do this, enclose both
functions in a set of curly braces (you’re graphing a list of functions, and Mathematica uses curly
braces to indicate a list or a table, as we saw in Section 5.1.) Also, if you use PlotStyle to alter the
color or shading of the graph, you need to specify a plot style for each function—these will also be
in curly braces.
This command would plot three graphs on the same axes using the default formatting options:
Plot[ {f[x], g[x], h[x]}, {x, -5, 5}]
The first set of curly braces is optional when graphing only one function.
This command would plot the same three graphs, giving each of them a different color and making
the graph of g( x ) dashed; we’ll use the AxesLabel option here as well. The command is listed on
two lines here for space reasons.
Plot[ {f[x], g[x], h[x]}, {x, -5, 5}, AxesLabel -> {"x", "y"},
PlotStyle -> {{Red}, {Blue, Dashing[{.05,.02}]},{Green}}]
7
Common mistakes in Mathematica
• Using the wrong kind of grouping symbol, for example, using parentheses where Mathematica
expects square brackets:
– Parentheses are used only for specifying the order of operations, never for anything else.
– Square brackets are the input to a named function, whether you named it, or the name
was built-in to Mathematica. We type Sqrt[5], for example, because “5” is an input to
the Sqrt function.
– Curly braces represent a list. We use curly braces to list the elements of the rows of a
matrix, for example, but we also use curly braces to form a list of PlotStyle options.
• Miscapitalization. Every word built-in to Mathematica starts with a capital letter, so typing
(for example) sin[x] will not give you the results you expect. (And sin(x) is even worse!)
Built-in constants, like E and Pi also start with capital letters, and some words, like PlotStyle
have capital letters inside the word; using Plotstyle instead will not work.
• Leaving out the underscore when creating a function. See Section 4.1 for an example of what
happens when you do this.
18
Index
’, 12
:=, 9
=, 8
==, 9
ArcCos, 6
ArcCot, 6
ArcCsc, 6
ArcSec, 6
ArcSin, 6
ArcTan, 6
assignment
delayed, 9
immediate, 8
AxesLabel, 17
BasicInput palette, 8
bracket, cell, 2
exponentiation (^), 5
Factor, 11
formatting a cell, 5
function
creating, 11
ImageSize, 17
immediate assignment, 8
In, 4
input cell, 2
Integrate, 12
Inverse, 14
Log, 6
logarithm
natural, 6
other bases, 6
cell, 2
bracket, 2
creating, 4
deleting, 4
format, 5
grouping, 4
input, 2
output, 2
text, 2
type, 5
Clear, 11
CMYKColor, 17
colors, 16
Cos, 6
Cot, 6
creating a cell, 4
creating a function, 11
Csc, 6
matrix multiplication, 14
MatrixForm, 15
multiplication (?), 5
Dashing, 17
Degree, 6
delayed assignment, 9
deleting a cell, 4
Derivative, 12
division (/), 5
Sec, 6
semicolon (;), 8
SHIFT - ENTER , 4
Sin, 6
Solve, 10
Sqrt, 6
systems of equations, 14
E, 6
editing palettes, 8
ESC , 8
Expand, 11
N, 7
notebook, 2
NSolve, 10
Out, 4
output cell, 2
percent sign (%), 7
Pi, 6
Plot, 16
PlotLabel, 17
PlotStyle, 16
RGBColor, 17
roots, 6
Table, 14
table
manual, 12
19
TableAlignments, 15
TableForm, 15
TableHeadings, 15
Tan, 6
testing for equality, 9
text cell, 2
Thickness, 17
Transpose, 14
trigonometric functions, 6
underscore ( ), 11
variables, 8
20