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Doing Mathematics with Scientific WorkPlace® & Scientific Notebook® Version 5.5
Darel W. Hardy and Carol L. Walker
(c)2005
536 pages
ISBN: 0-9741652-6-3
Doing Mathematics with Scientific WorkPlace® & Scientific Notebook® Version 5.5 describes how to use the built-in computer algebra system to do a wide range of mathematics.
This book is organized around the undergraduate mathematics curriculum for ease of use by beginners through professionals. It contains
- Basic procedures for using the software, illustrated with material from arithmetic through the standard precalculus courses
- Examples and instructions for creating graphs of curves and surfaces, including animated plots
- Procedures for using the software for problems in analytic geometry and calculus, linear algebra, vector analysis, differential equations, statistics, and applied modern algebra
- Exercises and sample solutions to help you practice the ideas presented and to suggest possibilities for further exploration.
Preface xxi
1 Basic Techniques for Doing Mathematics
- Inserting Text and Mathematics
- Basic Guidelines
- Displaying Mathematics
- Centering Plots, Graphics, and Text
- Basic Guidelines for Computing
- Evaluating Expressions
- Interpreting Expressions
- The Compute Menu and Toolbar
- Selecting Mathematical Expressions
- Computing in Place
- Stopping a Computation
- Computational Engine
- Error Handling
- Frequently Asked Questions
2 Numbers, Functions, and Units
- Integers and Fractions
- Addition and Subtraction
- Multiplication and Division
- Mixed Numbers and Long Division
- Elementary Number Theory
- Prime Factorization
- Greatest Common Divisor and Least Common Multiple
- Factorials
- Binomial Coefficients
- Real Numbers
- Basic Operations
- Powers and Radicals
- Rationalizing a Denominator
- Numerical Approximations
- Scientific Notation
- Computation and Display of Numerical Results
- Functions and Relations
- Absolute Value
- Maximum and Minimum
- Greatest and Smallest Integer Functions
- Checking Equality and Inequality
- Union, Intersection, and Difference
- Complex Numbers
- Basic Operations
- Real Powers and Roots of Complex Numbers
- Real and Imaginary Parts of a Complex Number
- Absolute Value
- Complex Conjugate
- Numerical Approximations of Complex Numbers
- Units and Measurements
- Units
- Physical Quantities, Symbols and Keyboard Shortcuts
- Compound Units
- Arithmetic Operations With Units
- Converting Units
- Exercises
3 Algebra
- Polynomials and Rational Expressions
- Sums, Differences, Products, and Quotients of Polynomials
- Summation Notation
- Sums and Differences of Rational Expressions
- Partial Fractions
- Products and Powers of Polynomials
- Division by Polynomials
- Collecting and Ordering Terms
- Factoring Polynomials
- Greatest Common Divisor of Two Polynomials
- Roots of Polynomials
- Defining Variables and Functions
- Assigning Values to Variables
- Defining Functions of One Variable
- Defining Functions of Several Variables
- Showing and Removing Definitions
- Solving Polynomial Equations
- Equations with One Variable
- Equations with Several Variables
- Systems of Equations
- Numerical Solutions
- Inequalities
- Substitution
- Substituting for a Variable
- Evaluating at Endpoints
- Exponents and Logarithms
- Exponents and Exponential Functions
- Logarithms and Logarithmic Functions
- Solving Exponential and Logarithmic Equations
- Exercises
4 Trigonometry
- Trigonometric Functions
- Radians and Degrees
- Solving Trigonometric Equations
- Trigonometric Identities
- Combining and Simplifying Trigonometric Expressions
- Inverse Trigonometric Functions and Trigonometric Equations
- Combining and Rewriting Inverse Trigonometric Functions
- Trigonometric Equations and Inverse Trigonometric Functions
- Hyperbolic Functions
- Inverse Hyperbolic Functions
- Complex Numbers and Complex Functions
- Arguments of a Complex Number
- Forms of a Complex Number
- Complex Powers and Roots of Complex Numbers
- DeMoivre's Theorem
- Complex Trigonometric and Hyperbolic Functions
- Exercises
5 Function Definitions
- Function and Expression Names
- Valid Names for Functions and Expressions
- Custom Names
- Automatic Substitution
- Defining Variables and Functions
- Assigning Values to Variables, or Naming Expressions
- Functions of One Variable
- Subscripts as Function Arguments
- Piecewise-Defined Functions
- Defining Generic Functions
- Defining Generic Constants
- Functions of Several Variables
- Handling Definitions
- Showing and Removing Definitions
- Saving and Restoring Definitions
- Assumptions About Variables
- Formula
- External Functions
- Accessing Functions in MuPAD Libraries
- User-Defined MuPAD Functions
- Tables of Equivalents
- Constants
- Compute Menu Items
- Equivalents for Functions and Expressions
- Trigtype Functions
- Determining the Argument of a Trigtype Function
- Exercises
6 Plotting Curves and Surfaces
- Getting Started With Plots
- The Frame, the View, and the Plot Properties Dialog
- Layout
- Resizing the Frame
- Frame Placement
- Screen Display and Print Attributes
- Plot Intervals and View Intervals for 2D Plots
- Rectangular Coordinates
- Polar Coordinates
- Implicit Plots
- Parametric Plots
- Plotting Tools for 2D Plots
- Zooming In and Out
- Translating the View
- Plot Coordinates Dialog Bar
- Items Plotted
- Expressions and Relations
- Intervals and Sample Size
- Plot Color and Style
- Adjust Plot for Discontinuities
- Axes and Axis Scaling
- Plot Captions, Keys, and Names
- Plot Labels
- 2D Plots of Functions and Expressions
- Expressions
- Defined Functions
- Continuous and Discontinuous Plots
- Plotting Piecewise-Defined Functions
- Special Functions
- Polygons and Point Plots
- Log and Log-Log Plots
- Parametric Plots
- Envelopes
- Implicit Plots
- Polar Coordinates
- Parametric Polar Plots
- Animated 2D Plots and the VCAM Window
- Animated Plots in Rectangular Coordinates
- Animated Plots in Polar Coordinates
- Animated Implicit Plots
- The View for 3D Plots
- Plotting Tools and Dialogs for 3D Plots
- The Plot Orientation Tool
- The 3D Plot Properties Dialog
- 3D Plots of Functions and Expressions
- Defined Functions
- Parametric Plots
- Implicit Plots
- Curves in Space
- Polygonal Paths
- Cylindrical Coordinates
- Spherical Coordinates
- The VCAM Window and 3D Plots
- Animated 3D Plots
- Animated Plots in Rectangular Coordinates
- Animated Plots in Cylindrical Coordinates
- Animated Plots in Spherical Coordinates
- Animated Implicit Plot
- Animated Tube Plot
- Plot Snapshots
- Snapshot Generation and Removal
- Snapshot Resolution
- Snapshots as Pictures
- Setting Plot Default Options
- Universal Default Options For Plots
- Default Plot Options for a Document
- Exercises
7 Calculus
- Evaluating Calculus Expressions
- Limits
- Notation for Limits
- Special Limits
- Tables of Values and Plots
- Differentiation
- Notation for Derivative
- Plotting Derivatives
- Generic Functions
- Implicit Differentiation
- Numerical Solutions to Equations
- Optimization
- Curve Sketching
- Indefinite Integration
- Interpreting an Expression
- Sequences of Operations
- Methods of Integration
- Integration by Parts
- Change of Variables
- Partial Fractions
- Definite Integrals
- Entering and Evaluating Definite Integrals
- Methods of Integration with Definite Integrals
- Improper Integrals
- Assumptions About Variables
- Definite Integrals from the Definition
- Pictures of Riemann Sums
- Approximation Methods
- Numerical Integration
- Visualizing Solids of Revolution
- Sequences and Series
- Sequences
- Series
- Multivariable Calculus
- Optimization
- Taylor Polynomials in Two Variables
- Total Differential
- Iterated Integrals
- Exercises
8 Matrix Algebra
- Introduction
- Changing the Appearance of Matrices
- Creating Matrices
- Revising Matrices
- Concatenating and Stacking Matrices
- Reshaping Lists and Matrices
- Standard Operations
- Matrix Addition and Scalar Multiplication
- Inner Products and Matrix Multiplication
- Rows and Columns
- Identity and Inverse Matrices
- Polynomials with Matrix Values
- Operations on Matrix Entries
- Row Operations and Echelon Forms
- Gaussian Elimination and Row Echelon Form
- Elementary Row Operations
- Equations
- Systems of Linear Equations
- Matrix Equations
- Matrix Operators
- Trace
- Transpose and Hermitian Transpose
- Determinant
- Adjugate
- Permanent
- Maximum and Minimum Matrix Entries
- Matrix Norms
- Spectral Radius
- Condition Number
- Exponential Functions
- Polynomials and Vectors Associated With a Matrix
- Characteristic Polynomial and Minimum Polynomial
- Eigenvalues and Eigenvectors
- Positive Definite Matrices
- Vector Spaces Associated With a Matrix
- T he Row Space
- The Column Space
- The Left and Right Nullspaces
- Orthogonal Matrices
- The QR Factorization and Orthonormal Bases
- Rank and Dimension
- Normal Forms of Matrices
- Smith Normal Form
- Hermite Normal Form
- Companion Matrix and Rational Canonical Form
- Jordan Form
- Matrix Decompositions
- Singular Value Decomposition (SVD)
- PLU Decomposition
- QR Decomposition
- Cholesky Decomposition
- Exercises
9 Vector Calculus
- Vectors
- Notation for Vectors
- Vector Sums and Scalar Multiplication
- Dot Product
- Cross Product
- Vector Norms
- Planes and Lines in R3
- Gradient, Divergence, and Curl
- Gradient
- Divergence
- Curl
- Laplacian
- Directional Derivatives
- Plots of Vector Fields and Gradients
- Plots and Animated Plots of 2D Vector Fields
- Plots and Animated Plots of 3D Vector Fields
- Plots and Animated Plots of 2D Gradient Fields
- Plots and Animated Plots of 3D Gradient Fields
- Scalar and Vector Potentials
- Scalar Potentials
- Vector Potential
- Matrix-Valued Operators
- Hessian
- Jacobian
- Wronskian
- Plots of Complex Functions
- Conformal Plots
- Animated Conformal Plots
- Exercises
10 Differential Equations
- Ordinary Differential Equations
- Exact Solutions
- Series Solutions
- Heaviside and Dirac Funtions
- Laplace Transforms
- Fourier Transforms
- Initial-Value Problems and Systems of Ordinary Differential Equations
- Exact Solutions
- Series Solutions
- Numerical Methods For Ordinary Differential Equations
- Numerical Solutions for Initial-Value Problems
- Graphical Solutions to Initial-Value Problems
- Numerical Solutions to Systems of Differential Equations
- Graphical Solutions to Systems of ODEs
- Bessel Functions
- Exercises
11 Statistics
- Introduction to Statistics
- Lists and Matrices
- Importing Data from an ASCII File
- Measures of Central Tendency
- Arithmetic Mean
- Median
- Quantile
- Mode
- Geometric Mean
- Harmonic Mean
- Measures of Dispersion
- Mean Deviation
- Variance and Standard Deviation
- Covariance
- Moment
- Correlation
- Distributions and Densities
- Cumulative Distribution Functions
- Inverse Distribution Functions
- Distribution Tables
- Families of Continuous Distributions
- Gamma Function
- Normal Distribution
- Student's t Distribution
- Chi-Square Distribution
- F Distribution
- Exponential Distribution
- Weibull Distribution
- Gamma Distribution
- Beta Distribution
- Cauchy Distribution
- Uniform Distribution
- Families of Discrete Distributions
- Binomial Distribution
- Poisson Distribution
- Hypergeometric Distribution
- Random Numbers
- Curve Fitting
- Linear Regression
- Polynomial Fit
- Overdetermined Systems of Equations
- Exercises
12 Applied Modern Algebra
- Solving Equations
- Integer Solutions
- Continued Fractions
- Recursive Solutions
- Integers Modulo m
- Multiplication Tables Modulo m
- Inverses Modulo m
- Solving Congruences Modulo m
- Pairs of Linear Congruences
- Systems of Linear Congruences
- Extended Precision Arithmetic
- Powers Modulo m
- Generating Large Primes
- Other Systems Modulo m
- Matrices Modulo m
- Polynomials Modulo m
- Poynomials Modulo Polynomials
- Greatest Common Divisor of Polynomials
- Multiplicity of Roots of Polynomials
- The Galois Field GFpn
- Linear Programming
- The Simplex Algorithm
- Feasible Systems
- Standard Form
- The Dual of a Linear Program
- Exercises
Index
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