Contents. Part I Vector Analysis

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1 Contents Part I Vector Analysis 1 Vectors BoundandFreeVectors Vector Operations Multiplication by a Scalar Unit Vector Addition and Subtraction Dot Product Vector Components Cross Product Triple Products LinesandPlanes Straight Lines Planes in Space Exercises Vector Calculus The Time Derivative Velocity and Acceleration Angular Velocity Vector Differentiation in Noninertial Reference Systems TheoryofSpaceCurve The Gradient Operator The Gradient of a Scalar Function Geometrical Interpretation of Gradient Line Integral of a Gradient Vector The Divergence of a Vector The Flux of a Vector Field Divergence Theorem Continuity Equation

2 VIII Contents 2.6 TheCurlofaVector Stokes Theorem Further Vector Differential Operations Product Rules Second Derivatives FurtherIntegralTheorems Green s Theorem Other Related Integrals Classification of Vector Fields Irrotational Field and Scalar Potential Solenoidal Field and Vector Potential Theory of Vector Fields Functions of Relative Coordinates Divergence of R/ R 2 as a Delta Function Helmholtz s Theorem Poisson s and Laplace s Equations Uniqueness Theorem Exercises Curved Coordinates CylindricalCoordinates Differential Operations Infinitesimal Elements Spherical Coordinates Differential Operations Infinitesimal Elements General Curvilinear Coordinate System Coordinate Surfaces and Coordinate Curves Differential Operations in Curvilinear Coordinate Systems Elliptical Coordinates Coordinate Surfaces Relations with Rectangular Coordinates Prolate Spheroidal Coordinates Multiple Integrals Jacobian for Double Integral Jacobians for Multiple Integrals Exercises Vector Transformation and Cartesian Tensors Transformation Properties of Vectors Transformation of Position Vector Vector Equations Euler Angles Properties of Rotation Matrices

3 Contents IX Definition of a Scalar and a Vector in Terms of Transformation Properties CartesianTensors Definition Kronecker and Levi-Civita Tensors Outer Product Contraction Summation Convention Tensor Fields Quotient Rule Symmetry Properties of Tensors Pseudotensors SomePhysicalExamples Moment of Inertia Tensor Stress Tensor Strain Tensor and Hooke s Law Exercises Part II Differential Equations and Laplace Transforms 5 Ordinary Differential Equations First-Order Differential Equations Equations with Separable Variables Equations Reducible to Separable Type Exact Differential Equations Integrating Factors First-Order Linear Differential Equations Bernoulli Equation Linear Differential Equations of Higher Order Homogeneous Linear Differential Equations with Constant Coefficients Characteristic Equation with Distinct Roots Characteristic Equation with Equal Roots Characteristic Equation with Complex Roots Nonhomogeneous Linear Differential Equations with Constant Coefficients Method of Undetermined Coefficients Use of Complex Exponentials Euler Cauchy Differential Equations Variation of Parameters Mechanical Vibrations Free Vibration Free Vibration with Viscous Damping Free Vibration with Coulomb Damping

4 X Contents Forced Vibration without Damping Forced Vibration with Viscous Damping ElectricCircuits Analog Computation Complex Solution and Impedance Systems of Simultaneous Linear Differential Equations The Reduction of a System to a Single Equation Cramer s Rule for Simultaneous Differential Equations Simultaneous Equations as an Eigenvalue Problem Transformation of an nth Order Equation into a System of n First-Order Equations Coupled Oscillators and Normal Modes Other Methods and Resources for Differential Equations Exercises Laplace Transforms Definition and Properties of Laplace Transforms Laplace Transform A Linear Operator Laplace Transforms of Derivatives Substitution: s-shifting Derivative of a Transform A Short Table of Laplace Transforms Solving Differential Equation with Laplace Transform Inverse Laplace Transform Solving Differential Equations Laplace Transform of Impulse and Step Functions The Dirac Delta Function The Heaviside Unit Step Function Differential Equations with Discontinuous Forcing Functions Convolution The Duhamel Integral The Convolution Theorem FurtherPropertiesofLaplaceTransforms Transforms of Integrals Integration of Transforms Scaling Laplace Transforms of Periodic Functions Inverse Laplace Transforms Involving Periodic Functions Laplace Transforms and Gamma Functions Summary of Operations of Laplace Transforms Additional Applications of Laplace Transforms Evaluating Integrals

5 Contents XI Differential Equation with Variable Coefficients Integral and Integrodifferential Equations Inversion by Contour Integration Computer Algebraic Systems for Laplace Transforms Exercises References Index...335

ADVANCED ENGINEERING MATHEMATICS

ADVANCED ENGINEERING MATHEMATICS DENNIS G. ZILL Loyola Marymount University MICHAEL R. CULLEN Loyola Marymount University PWS-KENT O I^7 3 PUBLISHING COMPANY E 9 U Boston CONTENTS Preface xiii Parti ORDINARY