The Student's Introduction to MATHEMATICA ®: A Handbook for Precalculus, Calculus, and Linear Algebra

by Bruce F. Torrence , Eve A. Torrence

• ISBN13: 9780521594615
• ISBN10: 0521594618
• Format: Paperback
• Copyright: 1999-05-13
• Publisher: Cambridge University Press

Summary

The unique feature of this compact student's introduction is that it presents concepts in an order that closely follows a standard mathematics curriculum, rather than structure the book along features of the software. As a result, the book provides a brief introduction to those aspects of the Mathematica software program most useful to students. Furthermore, Mathematica commands are introduced as a means of solving problems and illuminating the underlying mathematical principles. No prerequisites other than high school level mathematics are assumed. This book can be used in a variety of courses, from precalculus to linear algebra. Used as a supplementary text it will aid in bridging the gap between the mathematics in the course and Mathematica. In addition to its course use, this book will serve as an excellent tutorial for those wishing to learn Mathematica and brush up on their mathematics at the same time.

Author Biography

Bruce F. Torrence received his Ph.D. in mathematics from the University of Virginia. He is currently serving as assistant chair in the Department of Mathematics and as codirector of the Honors Program at Randolph-Macon College. Eve A. Torrence received her Ph.D. in mathematics from the University of Virginia. She is currently professor of mathematics at Randolph-Macon College.

Table of Contents

Preface	xv
Why Mathematica?	xvi
How to Use this Book	xvi
Acknowledgments	xvii
Getting Started	1	(21)
Launching Mathematica	1	(1)
The Basic Technique for Using Mathematica	1	(1)
The First Computation Is Slow	2	(1)
Commands for Basic Arithmetic	2	(1)
Input and Output	3	(1)
The BasicInput Palette and Two-Dimensional Input	4	(2)
Decimal In, Decimal Out	6	(1)
Use Parentheses to Group Terms	7	(2)
Three Well-Known Constants	9	(1)
Typing Commands in Mathematica	9	(7)
Numerical Approximation and Scientific Notation	10	(1)
Trigonometric Functions	11	(1)
Logarithms	12	(1)
Factoring Integers	13	(1)
Factoring and Expanding Polynomials	13	(1)
Plotting Functions	14	(1)
Square Root Function	15	(1)
Real and Imaginary Parts of Complex Numbers	16	(1)
Naming Things	16	(2)
Saving Your Work and Quitting Mathematica	18	(1)
Some Frequently Asked Questions Regarding Mathematica's Syntax	19	(3)
Why Do All Built-In Mathematica Objects Begin with Capital Letters?	19	(1)
Why Are Some Command Names so Long?	19	(1)
Why Are the Arguments of Commands Enclosed in Square Brackets?	20	(1)
What Happens If I Use Incorrect Syntax?	20	(2)
Working with Mathematica	22	(23)
Opening Saved Notebooks	22	(1)
Adding Text to Notebooks - An Introduction to Mathematica's Cell Structure	22	(2)
Printing	24	(1)
Creating Web Pages	25	(1)
Mathematica's Kernel	25	(2)
Numbering Input and Output	25	(1)
Reevaluating Previously Saved Notebooks	26	(1)
Tips for Working Effectively	27	(6)
Referring to Previous Output	27	(1)
Referring to Previous Input	28	(1)
Postfix Command Structure	28	(1)
Undoing Mistakes	29	(1)
Keyboard Shortcuts	30	(1)
Typesetting Input-More Shortcuts	30	(2)
Suppressing Output and Entering Sequences of Commands	32	(1)
Getting Help from Mathematica	33	(3)
Information on Known Commands	33	(1)
Command Completion	34	(1)
Command Templates	34	(1)
The Help Browser	34	(2)
Loading Packages	36	(3)
Problems: Dealing with Them and Preventing Them	39	(6)
Recognizing a Crash	41	(1)
Aborting Calculations and/or Recovering from a Crash	41	(2)
Running Efficiently: Preventing Crashes	43	(1)
Memory Gauges for the Mac OS	44	(1)
Functions and Their Graphs	45	(49)
Defining a Function	45	(2)
Clearing a Function	47	(1)
Producing a Table of Values	48	(3)
Plotting a Function	51	(4)
Using Mathematica's Plot Options	55	(11)
How to Get the Same Scaling on Both Axes	56	(1)
How to Get the Axes to Intersect at the Origin	57	(1)
How to Add Color, and Other Style Changes	57	(2)
How to Remove the Axes or Add a Frame	59	(1)
How to Add Grid Lines and Adjust Tick Marks	60	(1)
How to Label the Axes	61	(1)
How to Give Your Plot a Title	62	(1)
How to Change Fonts	63	(2)
How to Find Out About Other Options	65	(1)
Working with Piecewise Defined Functions	66	(2)
Plotting Implicitly Defined Functions	68	(3)
Superimposing Plots	71	(6)
Animating Plots	77	(1)
Working with Data	77	(5)
Manipulating Data - An Introduction to Lists	82	(6)
Importing Data into Mathematica	88	(2)
On a Machine Running the Mac OS	89	(1)
On a PC Running Windows	90	(1)
Extracting Columns from the Data (All Platforms)	90	(1)
Working with Difference Equations	90	(4)
Algebra	94	(36)
Factoring and Expanding Polynomials	94	(3)
Finding Roots of Polynomials with Solve and NSolve	97	(7)
Suppressing Complex Output	104	(3)
Simplifying and Manipulating Rational Functions	107	(4)
Solving Equations	107	(1)
Simplifying Rational Expressions	107	(2)
Vertical Asymptotes	109	(1)
Long Division	109	(1)
Partial Fractions	110	(1)
Simplifying and Manipulating Other Expressions	111	(6)
Simplifying Things	111	(3)
Manipulating Trigonometric Expressions	114	(3)
Solving General Equations	117	(7)
Solving Difference Equations	124	(2)
Solving Systems of Equations	126	(4)
Calculus	130	(41)
Computing Limits	130	(3)
Working with Difference Quotients	133	(2)
Producing and Simplifying Difference Quotients	133	(1)
Average Rate of Change	134	(1)
Instantaneous Rate of Change	135	(1)
The Derivative	135	(3)
Higher Derivatives	138	(1)
Maxima and Minima	139	(5)
Inflection Points	144	(2)
Implicit Differentiation	146	(1)
Differential Equations	147	(3)
Integration	150	(3)
Definite and Improper Integrals	153	(9)
Computing Definite Integrals	153	(1)
Riemann Sums	154	(2)
Producing Filled Plots	156	(3)
Computing Improper Integrals	159	(1)
Defining Functions with Integrals	159	(1)
Integrands with Discontinuities	160	(2)
Some Integrals Are Bad	162	(1)
Numerical Integration	162	(4)
Series	166	(5)
Multivariable Calculus	171	(61)
Vectors	171	(5)
Vectors and the Dot Product	171	(2)
Rendering Vectors in the Plane	173	(2)
The Cross Product	175	(1)
Real-Valued Functions of Two or More Variables	176	(25)
Defining a Real-Valued Function of Two or More Variables	176	(1)
Plotting Functions of Two Variables	177	(1)
Three-Dimensional Plotting Tips	178	(16)
Differentiation of Functions of Two or More Variables	194	(6)
Integrating Functions of Two or More Variables	200	(1)
Parametric Curves and Surfaces	201	(7)
Parametric Curves in the Plane	201	(3)
Parametric Curves in Space	204	(2)
Parametric Surfaces in Space	206	(2)
Other Coordinate Systems	208	(12)
Polar Coordinates	208	(5)
Cylindrical and Spherical Coordinates	213	(6)
Integrating in Other Coordinate Systems	219	(1)
Vector Fields	220	(7)
Defining a Vector Field	220	(1)
Plotting a Two-Dimensional Vector Field	221	(2)
Plotting a Three-Dimensional Vector Field	223	(2)
Divergence and Curl of a Three-Dimensional Vector Field	225	(2)
Line Integrals and Surface Integrals	227	(5)
Line Integrals	227	(2)
Surface Integrals	229	(3)
Linear Algebra	232	(43)
Matrices	232	(9)
Entering Matrices	232	(6)
Editing Matrices	238	(3)
Performing Gaussian Elimination	241	(3)
Referring to Parts of Matrices	241	(1)
Gaussian Elimination	242	(2)
Matrix Operations	244	(4)
Solving Systems of Linear Equations	248	(5)
Nonhomogeneous Systems of Linear Equations	248	(3)
Homogeneous Systems of Equations	251	(1)
Using LinearSolve and NullSpace to Solve Nonhomogeneous Systems	252	(1)
Vector Spaces	253	(6)
Span and Linear Independence	253	(1)
Bases	254	(2)
Rank and Nullity	256	(1)
Orthonormal Bases and the Gram-Schmidt Process	256	(3)
Eigenvalues and Eigenvectors	259	(5)
Finding Eigenvalues and Eigenvectors Automatically	259	(3)
Finding Eigenvalues and Eigenvectors Manually	262	(1)
Diagonalization	263	(1)
Visualizing Linear Transformations	264	(11)
Index	275