Efnisyfirlit

• Inside Front Cover
• Title page
• Copyright page
• Dedication
• Contents
• Preface
• To the Student
• To the Instructor
• Acknowledgments
• What Is Calculus?
• P Preliminaries
  • P.1 Real Numbers and the Real Line
    • Intervals
    • The Absolute Value
    • Equations and Inequalities Involving Absolute Values
  • P.2 Cartesian Coordinates in the Plane
    • Axis Scales
    • Increments and Distances
    • Graphs
    • Straight Lines
    • Equations of Lines
  • P.3 Graphs of Quadratic Equations
    • Circles and Disks
    • Equations of Parabolas
    • Reflective Properties of Parabolas
    • Scaling a Graph
    • Shifting a Graph
    • Ellipses and Hyperbolas
  • P.4 Functions and Their Graphs
    • The Domain Convention
    • Graphs of Functions
    • Even and Odd Functions; Symmetry and Reflections
    • Reflections in Straight Lines
    • Defining and Graphing Functions with Maple
  • P.5 Combining Functions to Make New Functions
    • Sums, Differences, Products, Quotients, and Multiples
    • Composite Functions
    • Piecewise Defined Functions
  • P.6 Polynomials and Rational Functions
    • Roots, Zeros, and Factors
    • Roots and Factors of Quadratic Polynomials
    • Miscellaneous Factorings
  • P.7 The Trigonometric Functions
    • Some Useful Identities
    • Some Special Angles
    • The Addition Formulas
    • Other Trigonometric Functions
    • Maple Calculations
    • Trigonometry Review
• 1 Limits and Continuity
  • 1.1 Examples of Velocity, Growth Rate, and Area
    • Average Velocity and Instantaneous Velocity
    • The Growth of an Algal Culture
    • The Area of a Circle
  • 1.2 Limits of Functions
    • One-Sided Limits
    • Rules for Calculating Limits
    • The Squeeze Theorem
  • 1.3 Limits at Infinity and Infinite Limits
    • Limits at Infinity
    • Limits at Infinity for Rational Functions
    • Infinite Limits
    • Using Maple to Calculate Limits
  • 1.4 Continuity
    • Continuity at a Point
    • Continuity on an Interval
    • There Are Lots of Continuous Functions
    • Continuous Extensions and Removable Discontinuities
    • Continuous Functions on Closed, Finite Intervals
    • Finding Roots of Equations
  • 1.5 The Formal Definition of Limit
    • Using the Definition of Limit to Prove Theorems
    • Other Kinds of Limits
    • Chapter Review
• 2 Differentiation
  • 2.1 Tangent Lines and Their Slopes
    • Normals
  • 2.2 The Derivative
    • Some Important Derivatives
    • Leibniz Notation
    • Differentials
    • Derivatives Have the Intermediate-Value Property
  • 2.3 Differentiation Rules
    • Sums and Constant Multiples
    • The Product Rule
    • The Reciprocal Rule
    • The Quotient Rule
  • 2.4 The Chain Rule
    • Finding Derivatives with Maple
    • Building the Chain Rule into Differentiation Formulas
    • Proof of the Chain Rule (Theorem 6)
  • 2.5 Derivatives of Trigonometric Functions
    • Some Special Limits
    • The Derivatives of Sine and Cosine
    • The Derivatives of the Other Trigonometric Functions
  • 2.6 Higher-Order Derivatives
  • 2.7 Using Differentials and Derivatives
    • Approximating Small Changes
    • Average and Instantaneous Rates of Change
    • Sensitivity to Change
    • Derivatives in Economics
  • 2.8 The Mean-Value Theorem
    • Increasing and Decreasing Functions
    • Proof of the Mean-Value Theorem
  • 2.9 Implicit Differentiation
    • Higher-Order Derivatives
    • The General Power Rule
  • 2.10 Antiderivatives and Initial-Value Problems
    • Antiderivatives
    • The Indefinite Integral
    • Differential Equations and Initial-Value Problems
  • 2.11 Velocity and Acceleration
    • Velocity and Speed
    • Acceleration
    • Falling under Gravity
    • Chapter Review
• 3 Transcendental Functions
  • 3.1 Inverse Functions
    • Inverting Non–One-to-One Functions
    • Derivatives of Inverse Functions
  • 3.2 Exponential and Logarithmic Functions
    • Exponentials
    • Logarithms
  • 3.3 The Natural Logarithm and Exponential Functions
    • The Natural Logarithm
    • The Exponential Function
    • General Exponentials and Logarithms
    • Logarithmic Differentiation
  • 3.4 Growth and Decay
    • The Growth of Exponentials and Logarithms
    • Exponential Growth and Decay Models
    • Interest on Investments
    • Logistic Growth
  • 3.5 The Inverse Trigonometric Functions
    • The Inverse Sine (or Arcsine) Function
    • The Inverse Tangent (or Arctangent) Function
    • Other Inverse Trigonometric Functions
  • 3.6 Hyperbolic Functions
    • Inverse Hyperbolic Functions
  • 3.7 Second-Order Linear DEs with Constant Coefficients
    • Recipe for Solving ay" + by' + cy = 0
    • Simple Harmonic Motion
    • Damped Harmonic Motion
    • Chapter Review
• 4 More Applications of Differentiation
  • 4.1 Related Rates
    • Procedures for Related-Rates Problems
  • 4.2 Finding Roots of Equations
    • Discrete Maps and Fixed-Point Iteration
    • Newton's Method
    • "Solve" Routines
  • 4.3 Indeterminate Forms
    • l'Ĥopital's Rules
  • 4.4 Extreme Values
    • Maximum and Minimum Values
    • Critical Points, Singular Points, and Endpoints
    • Finding Absolute Extreme Values
    • The First Derivative Test
    • Functions Not Defined on Closed, Finite Intervals
  • 4.5 Concavity and Inflections
    • The Second Derivative Test
  • 4.6 Sketching the Graph of a Function
    • Asymptotes
    • Examples of Formal Curve Sketching
  • 4.7 Graphing with Computers
    • Numerical Monsters and Computer Graphing
    • Floating-Point Representation of Numbers in Computers
    • Machine Epsilon and Its Effect on Figure 4.45
    • Determining Machine Epsilon
  • 4.8 Extreme-Value Problems
    • Procedure for Solving Extreme-Value Problems
  • 4.9 Linear Approximations
    • Approximating Values of Functions
    • Error Analysis
  • 4.10 Taylor Polynomials
    • Taylor's Formula
    • Big-O Notation
    • Evaluating Limits of Indeterminate Forms
  • 4.11 Roundoff Error, Truncation Error, and Computers
    • Taylor Polynomials in Maple
    • Persistent Roundoff Error
    • Truncation, Roundoff, and Computer Algebra
    • Chapter Review
• 5 Integration
  • 5.1 Sums and Sigma Notation
    • Evaluating Sums
  • 5.2 Areas as Limits of Sums
    • The Basic Area Problem
    • Some Area Calculations
  • 5.3 The Definite Integral
    • Partitions and Riemann Sums
    • The Definite Integral
    • General Riemann Sums
  • 5.4 Properties of the Definite Integral
    • A Mean-Value Theorem for Integrals
    • Definite Integrals of Piecewise Continuous Functions
  • 5.5 The Fundamental Theorem of Calculus
  • 5.6 The Method of Substitution
    • Trigonometric Integrals
  • 5.7 Areas of Plane Regions
    • Areas Between Two Curves
    • Chapter Review
• 6 Techniques of Integration
  • 6.1 Integration by Parts
    • Reduction Formulas
  • 6.2 Integrals of Rational Functions
    • Linear and Quadratic Denominators
    • Partial Fractions
    • Completing the Square
    • Denominators with Repeated Factors
  • 6.3 Inverse Substitutions
    • The Inverse Trigonometric Substitutions
    • Inverse Hyperbolic Substitutions
    • Other Inverse Substitutions
    • The tan( θ/2) Substitution
  • 6.4 Other Methods for Evaluating Integrals
    • The Method of Undetermined Coefficients
    • Using Maple for Integration
    • Using Integral Tables
    • Special Functions Arising from Integrals
  • 6.5 Improper Integrals
    • Improper Integrals of Type I
    • Improper Integrals of Type II
    • Estimating Convergence and Divergence
  • 6.6 The Trapezoid and Midpoint Rules
    • The Trapezoid Rule
    • The Midpoint Rule
    • Error Estimates
  • 6.7 Simpson's Rule
  • 6.8 Other Aspects of Approximate Integration
    • Approximating Improper Integrals
    • Using Taylor's Formula
    • Romberg Integration
    • The Importance of Higher-Order Methods
    • Other Methods
    • Chapter Review
• 7 Applications of Integration
  • 7.1 Volumes by SlicingÑSolids of Revolution
    • Volumes by Slicing
    • Solids of Revolution
    • Cylindrical Shells
  • 7.2 More Volumes by Slicing
  • 7.3 Arc Length and Surface Area
    • Arc Length
    • The Arc Length of the Graph of a Function
    • Areas of Surfaces of Revolution
  • 7.4 Mass, Moments, and Centre of Mass
    • Mass and Density
    • Moments and Centres of Mass
    • Two- and Three-Dimensional Examples
  • 7.5 Centroids
    • Pappus's Theorem
  • 7.6 Other Physical Applications
    • Hydrostatic Pressure
    • Work
    • Potential Energy and Kinetic Energy
  • 7.7 Applications in Business, Finance, and Ecology
    • The Present Value of a Stream of Payments
    • The Economics of Exploiting Renewable Resources
  • 7.8 Probability
    • Discrete Random Variables
    • Expectation, Mean, Variance, and Standard Deviation
    • Continuous Random Variables
    • The Normal Distribution
    • Heavy Tails
  • 7.9 First-Order Differential Equations
    • Separable Equations
    • First-Order Linear Equations
    • Chapter Review
• 8 Conics, Parametric Curves, and Polar Curves
  • 8.1 Conics
    • Parabolas
    • The Focal Property of a Parabola
    • Ellipses
    • The Focal Property of an Ellipse
    • The Directrices of an Ellipse
    • Hyperbolas
    • The Focal Property of a Hyperbola
    • Classifying General Conics
  • 8.2 Parametric Curves
    • General Plane Curves and Parametrizations
    • Some Interesting Plane Curves
  • 8.3 Smooth Parametric Curves and Their Slopes
    • The Slope of a Parametric Curve
    • Sketching Parametric Curves
  • 8.4 Arc Lengths and Areas for Parametric Curves
    • Arc Lengths and Surface Areas
    • Areas Bounded by Parametric Curves
  • 8.5 Polar Coordinates and Polar Curves
    • Some Polar Curves
    • Intersections of Polar Curves
    • Polar Conics
  • 8.6 Slopes, Areas, and Arc Lengths for Polar Curves
    • Areas Bounded by Polar Curves
    • Arc Lengths for Polar Curves
    • Chapter Review
• 9 Sequences, Series, and Power Series
  • 9.1 Sequences and Convergence
    • Convergence of Sequences
  • 9.2 Infinite Series
    • Geometric Series
    • Telescoping Series and Harmonic Series
    • Some Theorems About Series
  • 9.3 Convergence Tests for Positive Series
    • The Integral Test
    • Using Integral Bounds to Estimate the Sum of a Series
    • Comparison Tests
    • The Ratio and Root Tests
    • Using Geometric Bounds to Estimate the Sum of a Series
  • 9.4 Absolute and Conditional Convergence
    • The Alternating Series Test
    • Rearranging the Terms in a Series
  • 9.5 Power Series
    • Algebraic Operations on Power Series
    • Differentiation and Integration of Power Series
    • Maple Calculations
  • 9.6 Taylor and Maclaurin Series
    • Maclaurin Series for Some Elementary Functions
    • Other Maclaurin and Taylor Series
    • Taylor's Formula Revisited
  • 9.7 Applications of Taylor and Maclaurin Series
    • Approximating the Values of Functions
    • Functions Defined by Integrals
    • Indeterminate Forms
  • 9.8 The Binomial Theorem and Binomial Series
    • The Binomial Series
    • The Multinomial Theorem
  • 9.9 Fourier Series
    • Periodic Functions
    • Fourier Series
    • Convergence of Fourier Series
    • Fourier Cosine and Sine Series
    • Chapter Review
• 10 Vectors and Coordinate Geometry in 3-Space
  • 10.1 Analytic Geometry in Three Dimensions
    • Euclidean n-Space
    • Describing Sets in the Plane, 3-Space, and n-Space
  • 10.2 Vectors
    • Vectors in 3-Space
    • Hanging Cables and Chains
    • The Dot Product and Projections
    • Vectors in n-Space
  • 10.3 The Cross Product in 3-Space
    • Determinants
    • The Cross Product as a Determinant
    • Applications of Cross Products
  • 10.4 Planes and Lines
    • Planes in 3-Space
    • Lines in 3-Space
    • Distances
  • 10.5 Quadric Surfaces
  • 10.6 Cylindrical and Spherical Coordinates
    • Cylindrical Coordinates
    • Spherical Coordinates
  • 10.7 A Little Linear Algebra
    • Matrices
    • Determinants and Matrix Inverses
    • Linear Transformations
    • Linear Equations
    • Quadratic Forms, Eigenvalues, and Eigenvectors
  • 10.8 Using Maple for Vector and Matrix Calculations
    • Vectors
    • Matrices
    • Linear Equations
    • Eigenvalues and Eigenvectors
    • Chapter Review
• 11 Arc length, Metric Spaces, and Applications
  • 11.1 Spherical Geometry and Trigonometry
    • Geodesics on a Sphere Are Great Circles
    • Spherical Triangles
    • The Area of a Spherical Triangle
    • Spherical Trigonometry
    • Polar Triangles
    • Right-Angled and Quadrantal Triangles
    • Solving a Class 6 Triangle
    • An Application to Aircraft and Ship Routings and Distances
  • 11.2 Relativity and Minkowskian Geometry
    • Extension of Metrics for a 4-Dimensional Space
    • The Minkowskian Metric and Physics
    • Space-time and Dynamics Through Geometry
    • Reference Frames and Observers
    • Minkowskian 2-space and Time Dilation
    • Lorentz Transformations and Length Contraction
    • Proper Time, Mass, Momentum, and Energy.
• 12 Vector Functions and Curves
  • 12.1 Vector Functions of One Variable
    • Differentiating Combinations of Vectors
  • 12.2 Some Applications of Vector Differentiation
    • Motion Involving Varying Mass
    • Circular Motion
    • Rotating Frames and the Coriolis Effect
  • 12.3 Curves and Parametrizations
    • Parametrizing the Curve of Intersection of Two Surfaces
    • Arc Length
    • Piecewise Smooth Curves
    • The Arc-Length Parametrization
  • 12.4 Curvature, Torsion, and the Frenet Frame
    • The Unit Tangent Vector
    • Curvature and the Unit Normal
    • Torsion and Binormal, the Frenet-Serret Formulas
  • 12.5 Curvature and Torsion for General Parametrizations
    • Tangential and Normal Acceleration
    • Evolutes
    • An Application to Track (or Road) Design
    • Maple Calculations
  • 12.6 Kepler's Laws of Planetary Motion
    • Ellipses in Polar Coordinates
    • Polar Components of Velocity and Acceleration
    • Central Forces and Kepler's Second Law
    • Derivation of Kepler's First and Third Laws
    • Conservation of Energy
    • Chapter Review
• 13 Partial Differentiation
  • 13.1 Functions of Several Variables
    • Graphs
    • Level Curves
    • Using Maple Graphics
  • 13.2 Limits and Continuity
  • 13.3 Partial Derivatives
    • Tangent Planes and Normal Lines
    • Distance from a Point to a Surface: A Geometric Example
  • 13.4 Higher-Order Derivatives
    • The Laplace and Wave Equations
  • 13.5 The Chain Rule
    • Homogeneous Functions
    • Higher-Order Derivatives
  • 13.6 Linear Approximations, Differentiability, and Differentials
    • Proof of the Chain Rule
    • Differentials
    • Functions from n-Space to m-Space
    • Differentials in Applications
    • Differentials and Legendre Transformations
  • 13.7 Gradients and Directional Derivatives
    • Directional Derivatives
    • Rates Perceived by a Moving Observer
    • The Gradient in Three and More Dimensions
  • 13.8 Implicit Functions
    • Systems of Equations
    • Choosing Dependent and Independent Variables
    • Jacobian Determinants
    • The Implicit Function Theorem
  • 13.9 Taylor's Formula, Taylor Series, and Approximations
    • Approximating Implicit Functions
    • Chapter Review
• 14 Applications of Partial Derivatives
  • 14.1 Extreme Values
    • Classifying Critical Points
  • 14.2 Extreme Values of Functions Defined on Restricted Domains
    • Linear Programming
  • 14.3 Lagrange Multipliers
    • The Method of Lagrange Multipliers
    • Problems with More than One Constraint
  • 14.4 Lagrange Multipliers in n-Space
    • Using Maple to Solve Constrained Extremal Problems
    • Significance of Lagrange Multiplier Values
    • Nonlinear Programming
  • 14.5 The Method of Least Squares
    • Linear Regression
    • Applications of the Least Squares Method to Integrals
  • 14.6 Parametric Problems
    • Differentiating Integrals with Parameters
    • Envelopes
    • Equations with Perturbations
  • 14.7 Newton's Method
    • Implementing Newton's Method Using a Spreadsheet
  • 14.8 Calculations with Maple
    • Solving Systems of Equations
    • Finding and Classifying Critical Points
  • 14.9 Entropy in Statistical Mechanics and Information Theory
    • Boltzmann Entropy
    • Shannon Entropy
    • Information Theory
    • Chapter Review
• 15 Multiple Integration
  • 15.1 Double Integrals
    • Double Integrals over More General Domains
    • Properties of the Double Integral
    • Double Integrals by Inspection
  • 15.2 Iteration of Double Integrals in Cartesian Coordinates
  • 15.3 Improper Integrals and a Mean-Value Theorem
    • Improper Integrals of Positive Functions
    • A Mean-Value Theorem for Double Integrals
  • 15.4 Double Integrals in Polar Coordinates
    • Change of Variables in Double Integrals
  • 15.5 Triple Integrals
  • 15.6 Change of Variables in Triple Integrals
    • Cylindrical Coordinates
    • Spherical Coordinates
  • 15.7 Applications of Multiple Integrals
    • The Surface Area of a Graph
    • The Gravitational Attraction of a Disk
    • Moments and Centres of Mass
    • Moment of Inertia
    • Chapter Review
• 16 Vector Fields
  • 16.1 Vector and Scalar Fields
    • Field Lines (Integral Curves, Trajectories, Streamlines)
    • Vector Fields in Polar Coordinates
    • Nonlinear Systems and Liapunov Functions
  • 16.2 Conservative Fields
    • Equipotential Surfaces and Curves
    • Sources, Sinks, and Dipoles
  • 16.3 Line Integrals
    • Evaluating Line Integrals
  • 16.4 Line Integrals of Vector Fields
    • Connected and Simply Connected Domains
    • Independence of Path
  • 16.5 Surfaces and Surface Integrals
    • Parametric Surfaces
    • Composite Surfaces
    • Surface Integrals
    • Smooth Surfaces, Normals, and Area Elements
    • Evaluating Surface Integrals
    • The Attraction of a Spherical Shell
  • 16.6 Oriented Surfaces and Flux Integrals
    • Oriented Surfaces
    • The Flux of a Vector Field Across a Surface
    • Calculating Flux Integrals
    • Chapter Review
• 17 Vector Calculus
  • 17.1 Gradient, Divergence, and Curl
    • Interpretation of the Divergence
    • Distributions and Delta Functions
    • Interpretation of the Curl
  • 17.2 Some Identities Involving Grad, Div, and Curl
    • Scalar and Vector Potentials
    • Maple Calculations
  • 17.3 Green's Theorem in the Plane
    • The Two-Dimensional Divergence Theorem
  • 17.4 The Divergence Theorem in 3-Space
    • Variants of the Divergence Theorem
  • 17.5 Stokes's Theorem
  • 17.6 Some Physical Applications of Vector Calculus
    • Fluid Dynamics
    • Electromagnetism
    • Electrostatics
    • Magnetostatics
    • Maxwell's Equations
  • 17.7 Orthogonal Curvilinear Coordinates
    • Coordinate Surfaces and Coordinate Curves
    • Scale Factors and Differential Elements
    • Grad, Div, and Curl in Orthogonal Curvilinear Coordinates
    • Chapter Review
• 18 Differential Forms and Exterior Calculus
  • Differentials and Vectors
    • Derivatives versus Differentials
  • 18.1 k-Forms
    • Bilinear Forms and 2-Forms
    • k-Forms
    • Forms on a Vector Space
  • 18.2 Differential Forms and the Exterior Derivative
    • The Exterior Derivative
    • 1-Forms and Legendre Transformations
    • Maxwell's Equations Revisited
    • Closed and Exact Forms
  • 18.3 Integration on Manifolds
    • Smooth Manifolds
    • Integration in n Dimensions
    • Sets of k-Volume Zero
    • Parametrizing and Integrating over a Smooth Manifold
  • 18.4 Orientations, Boundaries, and Integration of Forms
    • Oriented Manifolds
    • Pieces-with-Boundary of a Manifold
    • Integrating a Differential Form over a Manifold
  • 18.5 The Generalized Stokes's Theorem
    • Proof of Theorem 4 for a k-Cube
    • Completing the Proof
    • The Classical Theorems of Vector Calculus
• 19 Ordinary Differential Equations
  • 19.1 Classifying Differential Equations
  • 19.2 Solving First-Order Equations
    • Separable Equations
    • First-Order Linear Equations
    • First-Order Homogeneous Equations
    • Exact Equations
    • Integrating Factors
  • 19.3 Existence, Uniqueness, and Numerical Methods
    • Existence and Uniqueness of Solutions
    • Numerical Methods
  • 19.4 Differential Equations of Second Order
    • Equations Reducible to First Order
    • Second-Order Linear Equations
  • 19.5 Linear Differential Equations with Constant Coefficients
    • Constant-Coefficient Equations of Higher Order
    • Euler (Equidimensional) Equations
  • 19.6 Nonhomogeneous Linear Equations
    • Resonance
    • Variation of Parameters
    • Maple Calculations
    • Chapter Review
• 20 More Topics in Differential Equations
  • 20.1 The Laplace Transform
    • Some Basic Laplace Transforms
    • More Properties of Laplace Transforms
    • The Heaviside Function and the Dirac Delta Function
  • 20.2 Series Solutions of Differential Equations
  • 20.3 Dynamical Systems, Phase Space, and the Phase Plane
    • A Differential Equation as a First-Order System
    • Existence, Uniqueness, and Autonomous Systems
    • Second-Order Autonomous Equations and the Phase Plane
    • Fixed Points
    • Linear Systems, Eigenvalues, and Fixed Points
    • Implications for Nonlinear Systems
    • Predator–Prey Models
  • 20.4 Calculus of Variations
    • Variation of a Function
    • Variation of a Functional
    • Extreme Values of a Functional
    • Constrained Extrema
    • The Principle of Least Action
    • Maximum Entropy and the Flexibility of Variational Methods
• Appendices
  • Appendix I Complex Numbers
    • Definition of Complex Numbers
    • Graphical Representation of Complex Numbers
    • Complex Arithmetic
    • Roots of Complex Numbers
  • Appendix II Complex Functions
    • Limits and Continuity
    • The Complex Derivative
    • The Exponential Function
    • The Fundamental Theorem of Algebra
  • Appendix III Continuous Functions
    • Limits of Functions
    • Continuous Functions
    • Completeness and Sequential Limits
    • Continuous Functions on a Closed, Finite Interval
  • Appendix IV The Riemann Integral
    • Uniform Continuity
  • Appendix V Doing Calculus with Maple
    • List of Maple Examples and Discussion
  • Appendix VI Doing Calculus with Python
    • Sympy and Elementary Algebra
    • Vector Calculus with Sympy
    • Mathematical Graphics Using Matplotlib
• Answers to Odd-Numbered Exercises
  • Chapter P
  • Chapter 1
  • Chapter 2
  • Chapter 3
  • Chapter 4
  • Chapter 5
  • Chapter 6
  • Chapter 7
  • Chapter 8
  • Chapter 9
  • Chapter 10
  • Chapter 11
  • Chapter 12
  • Chapter 13
  • Chapter 14
  • Chapter 15
  • Chapter 16
  • Chapter 17
  • Chapter 18
  • Chapter 19
  • Chapter 20
  • Appendix I
  • Appendix II
  • Appendix III
  • Appendix IV
  • Appendix V
  • Appendix VI
• Index
  • A
  • B
  • C
  • D
  • E
  • F
  • G
  • H
  • I
  • J
  • K
  • L
  • M
  • N
  • O
  • P
  • Q
  • R
  • S
  • T
  • U
  • V
  • W
  • Z
• Inside Back Cover

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