Calculus

The following sections are included:

• Preface

• Contents

The following sections are included:

• Matrix Arithmetic
  • Addition And Scalar Multiplication Of Matrices
  • Multiplication Of Matrices
  • The ij^th Entry Of A Product
  • Properties Of Matrix Multiplication
  • The Transpose
  • The Identity And Inverses

• Linear Transformations

• Constructing The Matrix Of A Linear Transformation

• Exercises

The following sections are included:

• Basic Techniques And Properties
  • Cofactors And 2 × 2 Determinants
  • The Determinant Of A Triangular Matrix
  • Properties Of Determinants
  • Finding Determinants Using Row Operations

• Applications
  • A Formula For The Inverse
  • Cramer's Rule

• Exercises

The following sections are included:

• Eigenvalues And Eigenvectors Of A Matrix
  • Definition Of Eigenvectors And Eigenvalues
  • Finding Eigenvectors And Eigenvalues
  • Volumes
  • Complex Eigenvalues

• Exercises

The following sections are included:

• Vector Valued Functions

• Vector Fields

• Exercises

• Continuous Functions
  • Sufficient Conditions For Continuity

• Limits Of A Function

• Properties Of Continuous Functions

• Exercises

• Open And Closed Sets

• Exercises

• Some Fundamentals
  • Combinations Of Continuous Functions
  • The Nested Interval Lemma
  • Convergent Sequences, Sequential Compactness
  • Continuity And The Limit Of A Sequence
  • The Extreme Value Theorem And Uniform Continuity

• Exercises

The following sections are included:

• Limits Of A Vector Valued Function Of One Real Variable

• The Derivative And Integral
  • Geometric And Physical Significance Of The Derivative
  • Differentiation Rules
  • Leibniz's Notation

• Exercises

• Line Integrals
  • Arc Length And Orientations
  • Line Integrals And Work
  • Another Notation For Line Integrals

• Exercises

• Independence Of Parametrization
  • Hard Calculus
  • Independence Of Parametrization

The following sections are included:

• Space Curves
  • Some Simple Techniques

• Geometry Of Space Curves

• Exercises

The following sections are included:

• The Graph Of A Function Of Two Variables

• Review Of Limits

• Exercises

• The Directional Derivative And Partial Derivatives
  • The Directional Derivative
  • Partial Derivatives

• Exercises

• Mixed Partial Derivatives

• Partial Differential Equations

• Exercises

The following sections are included:

• The Derivative Of Functions Of One Variable, o(v)

• Exercises

• The Derivative Of Functions Of Many Variables

• Exercises

• C¹ Functions

• The Chain Rule
  • The Chain Rule For Functions Of One Variable
  • The Chain Rule For Functions Of Many Variables

• Exercises
  • Related Rates Problems
  • The Derivative Of The Inverse Function
  • Proof Of The Chain Rule

• Exercises

• The Gradient

• The Gradient And Tangent Planes

• Exercises

The following sections are included:

• Local Extrema

• Exercises

• The Second Derivative Test

• Exercises

• Lagrange Multipliers

• Exercises

• Proof Of The Second Derivative Test

The following sections are included:

• Methods For Double Integrals
  • Density And Mass

• Exercises

• Methods For Triple Integrals
  • Definition Of The Integral
  • Iterated Integrals

• Exercises
  • Mass And Density

• Exercises

The following sections are included:

• Polar Coordinates

• Exercises

• Cylindrical And Spherical Coordinates

• Exercises

• The General Procedure

• Exercises

• The Moment Of Inertia And Center Of Mass

• Exercises

The following sections are included:

• The Two Dimensional Area In ℝ³
  • Surfaces Of The Form z = f (x, y)

• Flux Integrals

• Exercises

The following sections are included:

• Some Algebraic Preliminaries

• Divergence And Curl Of A Vector Field
  • Vector Identities
  • Vector Potentials
  • The Weak Maximum Principle

• Exercises

• The Divergence Theorem
  • Coordinate Free Concept Of Divergence

• Some Applications Of The Divergence Theorem
  • Hydrostatic Pressure
  • Archimedes Law Of Buoyancy
  • Equations Of Heat And Diffusion
  • Balance Of Mass
  • Balance Of Momentum
  • Frame Indifference
  • Bernoulli's Principle
  • The Wave Equation
  • A Negative Observation
  • Volumes Of Balls In ℝⁿ
  • Electrostatics

• Exercises

The following sections are included:

• Green's Theorem

• Exercises

• Stoke's Theorem From Green's Theorem
  • The Normal And The Orientation
  • The Mobeus Band
  • Conservative Vector Fields
  • Some Terminology

• Maxwell's Equations And The Wave Equation

• Exercises

The following sections are included:

• Appendix A The Mathematical Theory Of Determinants

• Appendix B Implicit Function Theorem

• Appendix C The Theory Of The Riemann Integral

• Appendix D Volumes In Higher Dimensions

• Appendix E Some Physical Applications

• Appendix F Curvilinear Coordinates

• Bibliography

• Appendix G Answers To Selected Exercises

• Index