Document Outline

• Front Matter
• Preface
• Table of Contents
• 1. Trigonometry and Cyclic Functions
  • 1.1 Units of Measurement
  • 1.2 Functions of Angles
  • 1.3 Definitions
  • 1.4 Quadrants
  • 1.5 Frequency of Cycling
  • 1.6 Sine of the Sum of Two Angles
  • 1.7 Cosine of a Sum
  • 1.8 Sine of a Difference Between Two Angles
  • 1.9 Cosine of a Difference
  • 1.10 Potentially Useful Relationships
    • 1.10.1 Example 1: Tan (X + Y)
    • 1.10.2 Example 2: sin 2X - sin 2Y
    • 1.10.3 Example 3: Radius of the Earth
    • 1.10.4 Example 4: The Third Side
    • 1.10.5 Example 5: Gain and Phase Lag
• 2. Differential Calculus
  • 2.1 Concept of Approaching a Limit
    • 2.1.1 Example 1: f(t) = t^2
  • 2.2 Procedure for Determining a Derivative
    • 2.2.1 Derivative of a Sum or Difference
    • 2.2.2 Derivative of a Product
    • 2.2.3 Derivative of a Quotient
    • 2.2.4 Dimensions and Units
    • 2.2.5 Derivative of a Sine Function
    • 2.2.6 Binomial Theorem
    • 2.2.7 Derivative of a Power
    • 2.2.8 Derivative of an Exponential
    • 2.2.9 A Function within a Function
    • 2.2.10 Example 2: Detecting a Maximum or Minimum
    • 2.2.11 Example 3: Watering the Lawn
• 3. Integral Calculus
  • 3.1 Problem Areas
  • 3.2 Practical Uses of Integration
    • 3.2.1 Example 1: Powers and Constants
    • 3.2.2 Example 2: sin^3 x
    • 3.2.3 Example 3: cos 2x
    • 3.2.4 Example 4: Substitution of Variables
    • 3.2.5 Example 5: Fractions
    • 3.2.6 Example 6: Using Partial Fractions
    • 3.2.7 Example 7: Numerator Higher Order than Denominator
    • 3.2.8 Example 8: Changing to an Angular Mode
    • 3.2.9 Example 9: sin^2 x
    • 3.2.10 Example 10: Square Root of (a^2 - x^2)
  • 3.3 Integration over a Specified Range
    • 3.3.1 Example 11: Tank Filling Case
    • 3.3.2 Example 12: Area of a Circle
    • 3.3.3 Example 13: Surface Area of a Sphere
    • 3.3.4 Example 14: Volume of a Sphere
    • 3.3.5 Example 15: The Escape Velocity
    • 3.3.6 Example 16: Area of a Segment of a Circle
  • 3.4 Table of Basic Integrals
• 4. Infinite Series
  • 4.1 Power Series
  • 4.2 The nth Term
  • 4.3 Test for Convergence
    • 4.3.1 Example 1: Convergence Test 1
    • 4.3.2 Example 2: Convergence Test 2
    • 4.3.3 Example 3: Convergence Test 3
    • 4.3.4 Example 4: The (n+1)th Term
  • 4.4 Maclaurin's Series
    • 4.4.1 Example 5: e^omegat
    • 4.4.2 Practical Disadvantage of Maclaurin's Series
  • 4.5 Taylor's Series
    • 4.5.1 Example 6: Value of a Sine Function Using Taylor's Series
• 5. Complex Quantities
  • 5.1 Background
  • 5.2 Graphical Representation
  • 5.3 The Complex Variable
  • 5.4 Trigonometric and Exponential Functions
    • 5.4.1 Sum of Two Complex Quantities
    • 5.4.2 Product of Two Complex Quantities
  • 5.5 Separating the Real and Imaginary Parts
    • 5.5.1 Example 1: Magnitude and Argument of a Complex Expression
• 6. Differential Equations
  • 6.1 Introduction
  • 6.2 Philosophy
  • 6.3 Definitions
  • 6.4 Application
  • 6.5 Differential Equations of the First Order and First Degree
    • 6.5.1 Category 1: Exact Differentials
    • 6.5.2 Category 2: Variables Separable
    • 6.5.3 Category 3: Homogeneous Equations
    • 6.5.4 Category 4: Linear Differential Equations
    • 6.5.5 Example 1: Time Constant
  • 6.6 Linear Differential Equations with Constant Coefficients
  • 6.7 Second Order Linear Differential Equation with Constant Coefficients
  • 6.8 The Oscillatory Case
  • 6.9 The Constant of Integration
    • 6.9.1 Commentary on the Result
    • 6.9.2 Example 2: The Spring/Mass System
  • 6.10 Units
  • 6.11 Partial Differential Equations
• 7. Laplace Transforms
  • 7.1 History
    • 7.1.1 Example 1: Step Change
  • 7.2 Transforms of Derivatives
    • 7.2.1 Example 2: Time Constant
    • 7.2.2 Example 3: Pendulum
    • 7.2.3 Example 4: Second Order Linear Differential Equations with Constant Coefficients
  • 7.3 The Oscillatory Case
    • 7.3.1 Consistency of Results
    • 7.3.2 Table of Laplace Transforms
• 8. Frequency Response Analysis
  • 8.1 Background
  • 8.2 The Bode Diagram
  • 8.3 Frequency Response of a Time Constant Element
  • 8.4 Frequency Response of a Dead Time Element
  • 8.5 Combinations of Components
  • 8.6 Period of Oscillation
  • 8.7 Summary
• 9. Transfer Functions and Block Diagrams
  • 9.1 Background
  • 9.2 Transfer Functions
  • 9.3 The Step Input Function
  • 9.4 Time Constants
  • 9.5 Dead Time
  • 9.6 The Value of the Transfer Function
    • 9.6.1 Example 1: Time Constant
    • 9.6.2 Example 2: Dead Time
  • 9.7 Block Diagrams
  • 9.8 Conditions for Continuous Oscillation
  • 9.9 The Transfer Function of a Closed Loop
  • 9.10 Evaluating the Closed Loop Transfer Function
• 10. The Z-N Approximation
  • 10.1 Historical
  • 10.2 The Z-N Approximation
  • 10.3 Estimating the Frequency of Oscillation
  • 10.4 Values for the Dead Time and Time Constant
  • 10.5 Just How Good Is the Approximation?
  • 10.6 Making a Process Reaction Curve
    • 10.6.1 Example 1: Reaction of a Real Process
• 11. Units, Best Values, Formulas, and Other Good Stuff
  • 11.1 True Value
  • 11.2 Errors
  • 11.3 Errors in Combinations of Quantities
  • 11.4 Correction Factor
  • 11.5 Significant Figures
  • 11.6 Conversion of Units
  • 11.7 Converting Formulas to New Units
    • 11.7.1 Example 1: Reynolds Number
    • 11.7.2 Example 2: Pressure of a Column of Liquid
  • 11.8 The Most Representative Value (MRV)
  • 11.9 Predicting Future Values
  • 11.10 How Much Confidence in the Most Representative Value?
  • 11.11 The Standard Deviation
  • 11.12 Curve Fitting
• Index
  • A
  • B
  • C
  • D
  • E
  • F
  • G
  • H
  • I
  • K
  • L
  • M
  • N
  • O
  • P
  • Q
  • R
  • S
  • T
  • V
  • W
  • Z