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Mastering Technical Mathematics, Third Edition, 3rd Edition

Book Description

A thorough revision of the classic tutorial of scientific and engineering mathematics

For more than fifteen years, Mastering Technical Mathematics has been the definitive self-teaching guide for those wishing to boost their career by learning the principles of mathematics as they apply to science and engineering. Featuring the same user-friendly pedagogy, practical examples, and detailed illustrations that have made this resource a favorite of the scientific and technical communities, the new third edition delivers four entirely new chapters and expanded treatment of cutting-edge topics.

Table of Contents

  1. Cover 
  2. Half Title
  3. Title
  4. Copyright
  5. Dedication
  6. Contents 
  7. Introduction
  8. Part 1: Working with Numbers
    1. 1. From Counting to Addition
      1. Numbers and Numerals
      2. Counting in Tens and Dozens
      3. Writing Numerals Greater Than 10
      4. Why Zero Is Used in Counting
      5. By Tens and Hundreds to Thousands
      6. Don’t Forget the Zeros
      7. Millions and More
      8. What About Infinity?
      9. Addition Is Counting on
      10. Adding Three or More Numbers
      11. Adding Larger Numbers
      12. Carrying
      13. Checking Answers
      14. Weights
      15. Using the Grocer’s Balance
      16. Questions and Problems
    2. 2. Subtraction
      1. Taking Away
      2. Checking Subtraction by Addition
      3. Borrowing
      4. Subtracting with Larger Numbers
      5. Working with Money
      6. Making Change
      7. Subtracting Weights
      8. Subtracting Liquid Measures
      9. Questions and Problems
    3. 3. Multiplication
      1. It’s a Shortcut
      2. Making a Table
      3. Patterns in Multiplication Tables
      4. How Calculators Multiply
      5. Carrying in Multiplication
      6. Using Your Calculator to Verify This Process
      7. Put Those Zeros to Work!
      8. Using Subtraction to Multiply
      9. Multiplying by Factors
      10. Multiplying in Nondecimal Systems
      11. Questions and Problems
    4. 4. Division
      1. Let’s Think Like a Calculator
      2. Division “Undoes” Multiplication
      3. Alternative Notation
      4. Dividing into Longer Numbers
      5. Multiplication Checks Division
      6. More About How a Calculator Does It
      7. Dividing by Larger Numbers (The People Way)
      8. Division by Factors
      9. When a Remainder Is Left
      10. How a Calculator Handles Fractions
      11. Decimal Equivalents of Fractions
      12. More Difficult Fractions
      13. Where Groups of Digits Repeat
      14. Converting Recurring Decimals to Fractions
      15. Questions and Problems
    5. 5. Fractions
      1. Slicing up a Pie
      2. Factors Help Simplify Fractions
      3. Spotting the Factors
      4. Prime Numbers
      5. Factoring into Primes
      6. Prime Factors of 139
      7. Prime Factors of 493
      8. Prime Factors of 546
      9. Adding and Subtracting Fractions
      10. Finding the Common Denominator
      11. Significant Figures
      12. Approximate Division and Multiplication
      13. Approximate Addition and Subtraction
      14. Questions and Problems
    6. 6. Plane Polygons
      1. Length Times Length Is Area
      2. What Is a Square?
      3. Different Shapes with the Same Area
      4. Square Measure Vs. Linear Measure
      5. Right Triangles
      6. Parallelograms
      7. Area of a Parallelogram
      8. Area of an Acute Triangle
      9. Area of an Obtuse Triangle
      10. Metric Measure
      11. Area Problems
      12. Questions and Problems
    7. 7. Time, Percentages, and Graphs
      1. Time as a Dimension
      2. Time and Speed
      3. Average Speed
      4. Making up for Lost Time
      5. Fractional Growth
      6. Percentages
      7. Percentages with Money
      8. Percentages up and down
      9. The Bar Graph
      10. Other Graphs
      11. Interpolation
      12. Graphs Can Help You Evaluate Data and Find Errors
      13. Questions and Problems
  9. Part 2: Algebra, Geometry, and Trigonometry
    1. 8. First Notions in Algebra
      1. Shortcuts for Long Problems
      2. Graphs Help You Find Errors and Solutions
      3. Manipulating Equations
      4. Writing It as Algebra
      5. Italics—or Not?
      6. Grouping Symbols
      7. Solving Problems with Grouping Symbols
      8. Putting a Problem into Algebra
      9. Magic by Algebra
      10. A Boat-in-the-River Problem
      11. Number Problems
      12. Questions and Problems
    2. 9. Some “School” Algebra
      1. Orderly Writing in Algebra
      2. Showing “Place” in Algebra
      3. Dimension in Algebra
      4. Equations and Inequalities
      5. A Consecutive-Number Problem
      6. Simultaneous Equations
      7. A Fraction Problem
      8. A Little More Practice
      9. Solving by Substitution
      10. Solving for Reciprocals
      11. Long Division and Factoring
      12. Questions and Problems
    3. 10. Quadratic Equations
      1. A Quadratic Example
      2. Linear Vs. Quadratic Forms
      3. A Quadratic Graph Is Symmetric
      4. Solving a Quadratic Equation
      5. Factors of a Quadratic
      6. Finding Factors
      7. Completing the Square
      8. What the Answers Mean
      9. The Quadratic Formula
      10. A Quadratic Problem
      11. Questions and Problems
    4. 11. Some Useful Shortcuts
      1. Sum and Difference in Geometry
      2. Difference of Squares Finds Factors
      3. Finding Square Roots
      4. Signs in Successive Roots
      5. Real Vs. Imaginary Numbers
      6. Complex Numbers
      7. Complex Numbers Find New Roots
      8. A Problem Using Simultaneous Equations
      9. Questions and Problems
    5. 12. Mechanical Mathematics
      1. What Is Force?
      2. Units of Force
      3. Acceleration, Speed, and Distance
      4. Force and Work
      5. Work and Energy
      6. Energy and Power
      7. Gravity as a Source of Energy
      8. Weight as Force
      9. Gravitational Measure of Work
      10. Energy for Constant Acceleration
      11. Potential and Kinetic Energy
      12. Kinetic Energy and Speed
      13. Acceleration at Constant Power
      14. How a Spring Stores Energy
      15. How a Spring Transfers Energy
      16. Resonance Cycle
      17. Travel and Speed in Resonant System
      18. Questions and Problems
    6. 13. Ratio and Proportion
      1. Fractions, Projections, Extremes, and Means
      2. Proportions in Practice
      3. Shape and Size
      4. About Angles in Triangles
      5. Use of Right Triangles
      6. Angles Identified by Ratios
      7. Theorem of Pythagoras
      8. Names for Angle Ratios
      9. Spotting the Triangle
      10. Degree Measure of Angles
      11. Three Applications of Trigonometry
      12. Questions and Problems
    7. 14. Trigonometric and Geometric Calculations
      1. Ratios for Sum Angles
      2. Finding Tan (A + B)
      3. An Example: Ratios for 75°
      4. Ratios for Angles Greater Than 90°
      5. Ratios for Difference Angles
      6. Ratios Through the Four Quadrants
      7. Pythagoras in Trigonometry
      8. Double Angles
      9. Higher Multiple Angles
      10. Properties of the Isosceles Triangle
      11. Angles in a Circle
      12. Angles in a Semicircle
      13. Definitions
      14. Questions and Problems
  10. Part 3: Analysis and Calculus
    1. 15. Systems of Counting
      1. Mechanical Counters
      2. Duodecimal Numbers
      3. Conversion from Decimal to Duodecimal
      4. Conversion from Duodecimal to Decimal
      5. Octal Numbers
      6. Hexadecimal numbers
      7. Binary Numbers
      8. Comparing the Values
      9. Converting Decimal to Binary
      10. Binary Multiplication
      11. Binary Division
      12. Indices
      13. Negative and Fractional Exponents
      14. Surds
      15. Questions and Problems
    2. 16. Progressions, Permutations, and Combinations
      1. Arithmetic Progression
      2. Geometric Progression
      3. Harmonic Progression
      4. Arithmetic Series
      5. Geometric Series
      6. Convergence
      7. Rate of Convergence
      8. Permutations
      9. Combinations
      10. Powers of a Binomial
      11. Alternative Notation for Multiplying Numbers
      12. Questions and Problems
    3. 17. Introduction to Derivatives
      1. Rates of Change
      2. Infinitesimal Changes
      3. Successive Differentiation
      4. Differentiating a Polynomial
      5. Successive Differentiation of Movement
      6. Circular Measure of Angles
      7. Derivatives of Sine and Cosine
      8. Successive Derivatives of Sine
      9. Questions and Problems
    4. 18. More About Differentiation
      1. Frequency and Period of a Sine Wave
      2. Sinusoidal Motion
      3. Harmonic Motion
      4. Linear and Nonlinear Relationships
      5. Multiples and Powers
      6. Functions
      7. Derivative of Constant
      8. Derivative of Sum of Two Functions
      9. Derivative of Difference of Two Functions
      10. Derivative of Function Multiplied by a Constant
      11. Derivative of Product of Two Functions
      12. Derivative of Product of Three Functions
      13. Derivative of Quotient of Two Functions
      14. Reciprocal Derivatives
      15. Derivative of Function Raised to a Power
      16. Chain Rule
      17. Questions and Problems
    5. 19. Introduction to Integrals
      1. Adding up the Pieces
      2. The Antiderivative
      3. Patterns in Calculations
      4. The Constant of Integration
      5. Definite Integrals
      6. Finding Area by Integration
      7. Area of a Circle
      8. Curved Areas of Cylinders and Cones
      9. Surface Area of a Sphere
      10. Volumes of Wedges and Pyramids
      11. Volumes of Cones and Spheres
      12. Questions and Problems
    6. 20. Combining Calculus with Other Tools
      1. Local Maximum and Minimum
      2. Point of Inflection
      3. Maximum and Minimum Slope
      4. Maximum Area with Constant Perimeter
      5. Box with Minimum Surface Area
      6. Cylinder with Minimum Surface Area
      7. Cone with Minimum Surface Area
      8. Equations for Circles, Ellipses, and Parabolas
      9. Directrix, Focus, and Eccentricity
      10. The Ellipse and the Circle
      11. Relationships Between Focus, Directrix, and Eccentricity
      12. Focus Property of Parabola
      13. Focus Property of Ellipse
      14. Hyperbola: Eccentricity Greater Than 1
      15. Geometry of Conic Sections
      16. Questions and Problems
    7. 21. Alternative Coordinate Systems
      1. Mathematician’s Polar Coordinates
      2. Some Examples
      3. Compressed Polar Coordinates
      4. Mathematician’s Polar vs. Cartesian Coordinates
      5. Navigator’s Polar Coordinates
      6. Mathematician’s Vs. Navigator’s Polar Coordinates
      7. Navigator’s Polar Vs. Cartesian Coordinates
      8. Latitude and Longitude
      9. Celestial Coordinates
      10. Cartesian Three-Space
      11. Cylindrical Coordinates
      12. Spherical Coordinates
      13. Questions and Problems
    8. 22. Complex Numbers
      1. Imaginary Quantities
      2. The Complex Number Plane
      3. Multiplying Complex Quantities
      4. Three Cube Root Examples
      5. Reciprocal of a Complex Quantity
      6. Division of Complex Quantities
      7. Rationalization
      8. Checking Results
      9. Operation Summary
      10. Use of a Complex Plane
      11. Roots by Complex Quantities
      12. Questions and Problems
  11. Part 4: Tools of Applied Mathematics
    1. 23. Trigonometry in the Real World
      1. Parallax
      2. The Base-line
      3. Accuracy
      4. Distance to an Object
      5. Stadimetry
      6. Interstellar Distance Measurement
      7. Distance Vs. Parallax
      8. A Point of Confusion
      9. Radar
      10. Laws of Sines and Cosines
      11. Use of Radar
      12. How to Hunt a “Fox”
      13. The Fox Finds Itself
      14. Position of a Ship
      15. Questions and Problems
    2. 24. Logarithms and Exponentials
      1. The Logarithm Base
      2. Common Logarithms
      3. Natural Logarithms
      4. Basic Properties of Logarithms
      5. Conversion of Log Functions
      6. Logarithm Comparisons
      7. The Exponential Base
      8. Common Exponentials
      9. Natural Exponentials
      10. Alternative Expressions
      11. Logarithms Vs. Exponentials
      12. Basic Properties of Exponentials
      13. Exponential Comparisons
      14. Numerical Orders of Magnitude
      15. Questions and Problems
    3. 25. Scientific Notation Tutorial
      1. Standard Form
      2. Alternative Form
      3. The “Times” Sign
      4. Plain-Text Exponents
      5. Orders of Magnitude
      6. When to Use Scientific Notation
      7. Prefix Multipliers
      8. Multiplication
      9. Division
      10. Raising to a Power
      11. Taking a Root
      12. Addition
      13. Subtraction
      14. Significant Figures
      15. How Accurate Are You?
      16. What About 0?
      17. What About Exact Values?
      18. More About Addition and Subtraction
      19. Questions and Problems
    4. 26. Vectors
      1. Magnitude and Direction
      2. Sum of Two Vectors
      3. Multiplication of a Vector by a Scalar
      4. Dot Product of Two Vectors
      5. Magnitude and Direction in the Polar Plane
      6. Sum of Two Vectors in the Polar Plane
      7. Multiplication of a Vector by a Scalar in the Polar Plane
      8. Dot Product of Two Vectors in Polar Plane
      9. Magnitude in XYZ Space
      10. Direction in XYZ Space
      11. Sum of two Vectors in XYZ Space
      12. Multiplication of a Vector by a Scalar in XYZ Space
      13. Dot Product of Two Vectors in XYZ Space
      14. Cross Product of Two Vectors
      15. Negative Vector Magnitudes
      16. Questions and Problems
    5. 27. Logic and Truth Tables
      1. Sentences
      2. Negation (Not)
      3. Conjunction (And)
      4. Disjunction (Or)
      5. Implication (If/Then)
      6. Logical Equivalence (Iff)
      7. Truth Tables
      8. Tables for Basic Logic Functions
      9. The Equals Sign
      10. Q.E.D.
      11. Precedence
      12. Contradiction and Double Negation
      13. Commutative Laws
      14. Associative Laws
      15. Law of Implication Reversal
      16. De Morgan’s Laws
      17. Distributive Law
      18. Truth Table Proofs
      19. Proofs Without Tables
      20. Questions and Problems
    6. 28. Beyond Three Dimensions
      1. Imagine That!
      2. Time-Space
      3. Position and Motion
      4. Time as Displacement
      5. Universal Speed
      6. The Four-Cube
      7. The Rectangular Four-Prism
      8. Impossible Paths
      9. General Time-Space Hypervolume
      10. Cartesian N-Space
      11. Distance Formulas
      12. Vertices of a Tesseract
      13. Hypervolume of a Four-Prism
      14. Euclid’s Axioms
      15. The Parallel Postulate
      16. Geodesics
      17. Modified Parallel Postulate
      18. No Parallel Geodesics
      19. More Than One Parallel Geodesic
      20. Curved Space
      21. The “Hyperfunnel”
      22. Questions and Problems
    7. 29. A Statistics Primer
      1. Experiment
      2. Variable
      3. Population
      4. Sample, Event, and Census
      5. Random Variable
      6. Frequency
      7. Parameter
      8. Statistic
      9. Distribution
      10. Frequency Distribution
      11. Cumulative Frequency Data
      12. Mean
      13. Median
      14. Mode
      15. Questions and Problems
    8. 30. Probability Basics
      1. Sets and Set Notation
      2. Event Vs. Outcome
      3. Sample Space
      4. Mathematical Probability
      5. Empirical Probability
      6. Law of Large Numbers
      7. Independent Outcomes
      8. Mutually Exclusive Outcomes
      9. Complementary Outcomes
      10. Nondisjoint Outcomes
      11. Multiple Outcomes
      12. The Density Function
      13. Expressing the Pattern
      14. Area under the Curve
      15. Uniform Distribution
      16. Normal Distribution
      17. The Empirical Rule
      18. Questions and Problems
  12. Final Exam
  13. Appendices
    1. 1. Solutions to End-of-Chapter Questions and Problems
    2. 2. Answers to Final Exam Questions
    3. 3. Table of Derivatives
    4. 4. Table of Indefinite Integrals
  14. Suggested Additional Reading
  15. Index