Advanced Engineering Mathematics, 5/e
暫譯: 高等工程數學，第五版

Peter V. O'Neil

Advanced Engineering Mathematics, 5/e

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Summary

Through four editions, Peter O'Neil has made rigorous engineering mathematics topics accessible to thousands of students by emphasizing visuals, numerous examples, and interesting mathematical models. ADVANCED ENGINEERING MATHEMATICS featuries a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts. And problem sets incorporate the use of such leading software packages as MAPLE. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight-parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Special Functions, Partial Differential Equations, Complex Analysis, and Historical Notes.

Table of Contents

Part I: ORDINARY DIFFERENTIAL EQUATIONS.

1. First Order Differential Equations.

Preliminary Concepts. General and Particular Solutions. Implicitly Defined Solut ions. Integral Curves. The Initial Value Problem. Direction Fields. Separable Eq uations. Some Applications of Separable Differential Equations. Linear Different ial Equations. Exact Differential Equations. Integrating Factors. Separable Equa tions and Integrating Factors. Linear Equations and Integrating Factors. Homogen eous, Bernoulli and Riccati Equations. Homogeneous Differential Equations. The B ernoulli Equation. The Riccati Equation. Applications to Mechanics, Electrical C ircuits and Orthogonal Trajectories. Mechanics. Electrical Circuits. Orthogonal Trajectories. Existence and Uniqueness for Solutions of Initial Value Problems.

2. Second Order Differential Equations.

Preliminary Concepts. Theory of Solutions of y" + p(x)y' + q(x)y = f(x). The Hom ogeneous Equation y" + p(x)y' + q(x) = 0. The Nonhomogeneous Equation y" + p(x)y ' + q(x)y = f(x). Reduction of Order. The Constant Coefficient Homogeneous Linea r Equation. Case 1 A² – 4B > 0. Case 2 A² – 4B = 0. Case 3 A² – 4B < 0. An Alter native General Solution In the Complex Root Case. Euler's Equation. The Nonhomog eneous Equation y" + p(x)y' + q(x)y = f(x). The Method of Variation of Parameter s. The Method of Undetermined Coefficients. The Principle of Superposition. High er Order Differential Equations. Application of Second Order Differential Equati ons to a Mechanical System. Unforced Motion. Forced Motion. Resonance. Beats. An alogy With An Electrical Circuit.

3. The Laplace Transform.

Definition and Basic Properties. Solution of Initial Value Problems Using the La place Transform. Shifting Theorems and the Heaviside Function. The First Shiftin g Theorem. The Heaviside Function and Pulses. The Second Shifting Theorem. Analy sis of Electrical Circuits. Convolution. Unit Impulses and the Dirac Delta Funct ion. Laplace Transform Solution of Systems. Differential Equations With Polynomi al Coefficients.

4. Series Solutions.

Power Series Solutions of Initial Value Problems. Power Series Solutions Using R ecurrence Relations. Singular Points and the Method of Frobenius. Second Solutio ns and Logarithm Factors. Appendix on Power Series. Convergence of Power Series. Algebra and Calculus of Power Series. Taylor and Maclaurin Expansions. Shifting Indices.
Part II: VECTORS AND LINEAR ALGEBRA.

5. Vectors and Vector Spaces.

The Algebra and Geometry of Vectors. The Dot Product. The Cross Product. The Vec tor Space Rn. Linear Independence, Spanning Sets and Dimension in Rn. Abstract V ector Spaces.

6. Matrices and Systems of Linear Equations.

Matrices. Matrix Algebra. Matrix Notation for Systems of Linear Equations. Some Special Matrices. Another Rationale for the Definition of Matrix Multiplication. Random Walks in Crystals. Elementary Row Operations and Elementary Matrices. Th e Row Echelon Form of a Matrix. The Row and Column Spaces of a Matrix and Rank o f a Matrix. Solution of Homogeneous Systems of Linear Equations. The Solution Sp ace of AX = O. Nonhomogeneous Systems of Linear Equations. The Structure of Solu tions of AX = B. Existence and Uniqueness of Solutions of AX = B. Summary for Li near Systems. Matrix Inverses. A Method for Finding A-1.

7. Determinants.

Permutations. Definition of the Determinant. Properties of Determinants. Evaluat ion of Determinants by Elementary Row and Column Operations. Cofactor Expansions . Determinants of Triangular Matrices. A Determinant Formula for a Matrix Invers e. Cramer's Rule. The Matrix Tree Theorem.

8. Eigenvalues, Diagonalization, and Special Matrices.

Eigenvalues and Eigenvectors. Gerschgorin's Theorem. Diagonalization of Matrices . Orthogonal and Symmetric Matrices. Quadratic Forms. Unitary, Hermitian and Ske w Hermitian Matrices.
Part III: SYSTEMS OF DIFFERENTIAL EQUATIONS AND QUALITATIVE METHODS.

9. Systems of Linear Differential Equations.

Theory of the Homogeneous System X' = AX. General Solution of the Nonhomogeneous System X' = AX + G. Solution of X' = AX When A Is Constant. Solution of X' = AX When A Has Complex Eigenvalues. Solution of X' = AX When A Does Not Have n Line arly Independent Eigenvectors. Solution of X' = AX By Diagonalizing A. Exponenti al Matrix Solutions of X' = AX. Solution of X' = AX + G. Variation of Parameters . Solution of X' = AX + G By Diagonalizing A.

10. Qualitative Methods and Systems of Nonlinear Differential Equations.

Nonlinear Systems and Existence of Solutions. The Phase Plane, Phase Portraits a nd Direction Fields. Phase Portraits of Linear Systems. Critical Points and Stab ility. Almost Linear Systems. Predator/Prey Population Models. A Simple Predator /Prey Model. An Extended Predator/Prey Model. Competing Species Models. A Simple Competing Species Model. An Extended Competing Species Model. Lyapunov's Stabil ity Criteria. Limit Cycles and Periodic Solutions.
Part IV: VECTOR ANALYSIS.

11. Vector Differential Calculus.

Vector Functions of One Variable. Velocity, Acceleration, Curvature and Torsion. Tangential and Normal Components of Acceleration. Curvature As a Function of t. The Frenet Formulas. Vector Fields and Streamlines. The Gradient Field and Dire ctional Derivatives. Level Surfaces, Tangent Planes and Normal Lines. Divergence and Curl. A Physical Interpretation of Divergence. A Physical Interpretation of Curl.

12. Vector Integral Calculus.

Line Integrals. Line Integral With Respect to Arc Length. Green's Theorem. An Ex tension of Green's Theorem. Independence of Path and Potential Theory In the Pla ne. A More Critical Look at Theorem 12.5. Surfaces in 3- Space and Surface Integ rals. Normal Vector to a Surface. The Tangent Plane to a Surface. Smooth and Pie cewise Smooth Surfaces. Surface Integrals. Applications of Surface Integrals. Su rface Area. Mass and Center of Mass of a Shell. Flux of a Vector Field Across a Surface. Preparation for the Integral Theorems of Gauss and Stokes. The Divergen ce Theorem of Gauss. Archimedes's Principle. The Heat Equation. The Divergence T heorem As A Conservation of Mass Principle. Green's Identities. The Integral The orem of Stokes. An Interpretation of Curl. Potential Theory in 3- Space.
Part V: FOURIER ANALYSIS, ORTHOGONAL EXPANSIONS AND WAVELETS.

13. Fourier Series.

Why Fourier Series? The Fourier Series of a Function. Even and Odd Functions. Co nvergence of Fourier Series. Convergence at the End Points. A Second Convergence Theorem. Partial Sums of Fourier Series. The Gibbs Phenomenon. Fourier Cosine a nd Sine Series. The Fourier Cosine Series of a Function. The Fourier Sine Series of a Function. Integration and Differentiation of Fourier Series. The Phase Ang le Form of a Fourier Series. Complex Fourier Series and the Frequency Spectrum. Review of Complex Numbers. Complex Fourier Series.

14. The Fourier Integral and Fourier Transforms.

The Fourier Integral. Fourier Cosine and Sine Integrals. The Complex Fourier Int egral and the Fourier Transform. Additional Properties and Applications of the F ourier Transform. The Fourier Transform of a Derivative. Frequency Differentiati on. The Fourier Transform of an Integral. Convolution. Filtering and the Dirac D elta Function. The Windowed Fourier Transform. The Shannon Sampling Theorem. Low pass and Bandpass Filters. The Fourier Cosine and Sine Transforms. The Discrete Fourier Transform. Linearity and Periodicity. The Inverse N – Point DFT. DFT App roximation of Fourier Coefficients. Sampled Fourier Series. Approximation of a F ourier Transform by an N – Point DFT. Filtering. The Fast Fourier Transform. Com putational Efficiency of the FFT. Use of the FFT in Analyzing Power Spectral Den sities of Signals. Filtering Noise From a Signal. Analysis of the Tides in Morro Bay.

15. Special Functions, Orthogonal Expansions and Wavelets.

Legendre Polynomials. A Generating Function for the Legendre Polynomials. A Recu rrence Relation for the Legendre Polynomials. Orthogonality of the Legendre Poly nomials. Fourier-Legendre Series. Computation of Fourier-Legendre Coefficients. Zeros of the Legendre Polynomials. Derivative and Integral Formulas for Pn(x). B essel Functions. The Gamma Function. Bessel Functions of the First Kind and Solu tions of Bessel's Equation. Bessel Functions of the Second Kind. Modified Bessel Functions. Some Applications of Bessel Functions. A Generating Function for Jn( x). An Integral Formula for Jn(x). A Recurrence Relation for Jv(x). Zeros of Jv( x). Fourier-Bessel Expansions. Fourier-Bessel Coefficients. Sturm-Liouville Theo ry and Eigenfunction Expansions. The Sturm-Liouville Problem. The Sturm-Liouvill e Theorem. Eigenfunction Expansions. Approximation In the Mean and Bessel's Ineq uality. Convergence in the Mean and Parseval's Theorem. Completeness of the Eige nfunctions. Orthogonal Polynomials. Chebyshev Polynomials. Laguerre Polynomials. Hermite Polynomials. Wavelets. The Idea Behind Wavelets. The Haar Wavelets. A W avelet Expansion. Multiresolution Analysis With Haar Wavelets. General Construct ion of Wavelets and Multiresolution Analysis. Shannon Wavelets.
Part VI: PARTIAL DIFFERENTIAL EQUATIONS.

16. The Wave Equation.

The Wave Equation and Initial and Boundary Condition. Fourier Series Solutions o f the Wave Equation. Vibrating String with Given Initial Velocity and Zero Initi al Displacement. Vibrating String With Initial Displacement and Velocity. Verifi cation of Solutions. Transformation of Boundary Value Problems Involving the Wav e Equation. Effects of Initial Condition and Constants on the Motion. Wave Motio n Along Infinite and Semi-Infinite String. Fourier Transform Solution of Problem s on Unbounded Domains. Characteristics and d'Alembert's Solution. A Nonhomogene ous Wave Equation. Forward and Backward Waves. Normal Modes of Vibration of a Ci rcular Elastic Membrane. Vibrations of a Circular Elastic Membrane, Revisited. V ibrations of a Rectangular Membrane.

17. The Heat Equation.

The Heat Equation and Initial and Boundary Conditions. Fourier Series Solutions of the Heat Equation. Ends of the Bar Kept at Temperature Zero. Temperature in a Bar With Insulated Ends. Temperature Distribution in a Bar With Radiating End. Transformations of Boundary Value Problems Involving the Heat Equation. A Nonhom ogeneous Heat Equation. Effects of Boundary Conditions and Constants on Heat Con duction. Heat Conduction in Infinite Media. Heat Conduction in an Infinite Bar. Heat Conduction in a Semi-Infinite Bar. Integral Transform Methods for the Heat Equation in an Infinite Medium. Heat Conduction in an Infinite Cylinder. Heat Co nduction in a Rectangular Plate.

18. The Potential Equation.

Harmonic Functions and the Dirichlet Problem. Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. Poisson's Integral Formula for the Disk. Dirichle t Problems in Unbounded Regions. Dirichlet Problem for the Upper Half Plane. Dir ichlet Problem for the Right Quarter Plane. An Electrostatic Potential Problem. A Dirichlet Problem for a Cube. The Steady-State Heat Equation for a Solid Spher e. The Neumann Problem. A Neumann Problem for a Rectangle. A Neumann Problem for a Disk. A Neumann Problem for the Upper Half Plane.

19. Canonical Forms, Existence and Uniqueness of Solutions, and Well-Posed Probl ems.

Canonical Forms. Existence and Uniqueness of Solutions. Well-Posed Problems.
Part VII: COMPLEX ANALYSIS.

20. Geometry and Arithmetic of Complex Numbers.

Complex Numbers. The Complex Plane. Magnitude and Conjugate. Complex Division. I nequalities. Argument and Polar Form of a Complex Number. Ordering. Binomial Exp ansion of (z = w) n. Loci and Sets of Points in the Complex Plane. Distance. Cir cles and Disks. The Equation |z-a| = |z-b|. Other Loci. Interior Points, Boundar y Points, and Open and Closed Sets. Limit Points. Complex Sequences. Subsequence s. Compactness and the Bolzano-Weierstrass Theorem.

21. Complex Functions.

Limits, Continuity and Derivatives. Limits. Continuity. The Derivative of a Comp lex Function. The Cauchy-Riemann Equations. Power Series. Series of Complex Numb ers. Power Series. The Exponential and Trigonometric Functions. The Complex Loga rithm. Powers. Integer Powers. z1/n for Positive Integer n. Rational Powers. Pow ers zw.

22. Complex Integration.

Curves in the Plane. The Integral of a Complex Function. The Complex Integral in Terms of Real Integrals. Properties of Complex Integrals. Integrals of Series o f Functions. Cauchy's Theorem. Proof of Cauchy's Theorem for a Special Case. Pro of of Cauchy's Theorem for a Rectangle. Consequences of Cauchy's Theorem. Indepe ndence of Path. The Deformation Theorem. Cauchy's Integral Formula. Cauchy's Int egral Formula for Higher Derivatives. Bounds on Derivatives and Liouville's Theo rem. An Extended Deformation Theorem.

23. Series Representations of Functions.

Power Series Representations. Isolated Zeros and the Identity Theorem. The Maxim um Modulus Theorem. The Laurent Expansion.

24. Singularities and the Residue Theorem.

Singularities. The Residue Theorem. Some Applications of the Residue Theorem. Th e Argument Principle. Rouchés Theorem. Summation of Real Series. An Inversion Fo rmula for the Laplace Transform. Evaluation of Real Integrals.

25. Conformal Mappings.

Functions as Mappings. Conformal Mappings. Linear Fractional Transformations. Co nstruction of Conformal Mappings Between Domains. Schwarz-Christoffel Transforma tion. Harmonic Functions and the Dirichlet Problem. Solution of Dirichlet Proble ms by Conformal Mapping. Complex Function Models of Plane Fluid Flow.
Part VIII: HISTORICAL NOTES.

26. Development of Areas of Mathematics.

Ordinary Differential Equations. Matrices and Determinants. Vector Analysis. Fou rier Analysis. Partial Differential Equations. Complex Function Theory.

27. Biographical Sketches.

Galileo Galilei (1564 – 1642). Isaac Newton (1642 – 1727). Gottfried Wilhelm Lei bniz (1646 – 1716). The Bernoulli Family. Leonhard Euler (1707 – 1783). Carl Fri edrich Gauss (1777 – 1855). Joseph-Louis Lagrange (1736 – 1813). Pierre-Simon de Laplace (1749 – 1827). Augustin-Louis Cauchy (1789 – 1857). Joseph Fourier (176 8 – 1830). Henri Poincaré (1854 – 1912).

 

 

商品描述(中文翻譯)

摘要

透過四個版本，彼得·奧尼爾（Peter O'Neil）使嚴謹的工程數學主題對數千名學生變得易於理解，強調視覺效果、眾多範例和有趣的數學模型。《高級工程數學》（ADVANCED ENGINEERING MATHEMATICS）包含更多的範例和問題，並在整體上進行了微調，以改善思想的清晰流動。計算機在生成用於顯示概念的計算機圖形方面比以往任何時候都更為重要。問題集整合了如 MAPLE 等領先軟體包的使用。為了鼓勵學生利用這些計算工具，內容中包含了計算輔助、練習和專案。內容分為八個部分，涵蓋了廣泛的主題，包括常微分方程、向量與線性代數、微分方程系統、向量分析、傅立葉分析、正交展開、小波、特殊函數、偏微分方程、複變分析和歷史註釋。

目錄

第一部分：常微分方程。
1. 一階微分方程。
初步概念。一般解與特解。隱式定義的解。積分曲線。初值問題。方向場。可分離方程。一些可分離微分方程的應用。線性微分方程。精確微分方程。積分因子。可分離方程與積分因子。線性方程與積分因子。齊次方程、伯努利方程和里卡提方程。齊次微分方程。伯努利方程。里卡提方程。應用於力學、電路和正交軌跡。力學。電路。正交軌跡。初值問題解的存在性與唯一性。

2. 二階微分方程。
初步概念。y' + p(x)y' + q(x)y = f(x) 的解的理論。齊次方程 y' + p(x)y' + q(x) = 0。非齊次方程 y' + p(x)y' + q(x)y = f(x)。降階。常係數齊次線性方程。情況 1 A² – 4B > 0。情況 2 A² – 4B = 0。情況 3 A² – 4B < 0。複根情況下的替代通解。歐拉方程。非齊次方程 y' + p(x)y' + q(x)y = f(x)。變數變化法。未定係數法。疊加原理。高階微分方程。二階微分方程在機械系統中的應用。無強迫運動。強迫運動。共振。拍頻。與電路的類比。

3. 拉普拉斯變換。
定義及基本性質。使用拉普拉斯變換解初值問題。平移定理與海維賽德函數。第一次平移定理。海維賽德函數與脈衝。第二次平移定理。電路分析。卷積。單位脈衝與狄拉克δ函數。拉普拉斯變換解系統。具有多項式係數的微分方程。

4. 級數解。
初值問題的冪級數解。使用遞迴關係的冪級數解。奇異點與弗羅貝尼烏斯法。第二解與對數因子。冪級數附錄。冪級數的收斂性。冪級數的代數與微積分。泰勒展開與麥克勞林展開。指數變換。

第二部分：向量與線性代數。
5. 向量與向量空間。
向量的代數與幾何。點積。叉積。向量空間 Rn。線性獨立、生成集與 Rn 的維度。抽象向量空間。

6. 矩陣與線性方程組。
矩陣。矩陣代數。線性方程組的矩陣表示法。一些特殊矩陣。矩陣乘法定義的另一種理由。晶體中的隨機漫步。基本行運算與基本矩陣。矩陣的行階梯形。矩陣的行空間與列空間及矩陣的秩。齊次線性方程組的解。AX = O 的解空間。非齊次線性方程組。AX = B 的解結構。AX = B 的解的存在性與唯一性。線性系統的總結。矩陣的逆。尋找 A⁻¹ 的方法。

7. 行列式。
排列。行列式的定義。行列式的性質。通過基本行與列運算計算行列式。余子式展開。三角矩陣的行列式。矩陣逆的行列式公式。克拉默法則。矩陣樹定理。

8. 特徵值、對角化與特殊矩陣。
特徵值與特徵向量。格爾什戈林定理。矩陣的對角化。正交矩陣與對稱矩陣。二次型。單位矩陣、厄米矩陣與斜厄米矩陣。

第三部分：微分方程系統與定性方法。
9. 線性微分方程系統。
齊次系統 X' = AX 的理論。非齊次系統 X' = AX + G 的一般解。當 A 為常數時 X' = AX 的解。當 A 具有複特徵值時 X' = AX 的解。當 A 沒有 n 個線性獨立特徵向量時 X' = AX 的解。通過對角化 A 解 X' = AX。X' = AX 的指數矩陣解。X' = AX + G 的解。變數變化法。通過對角化 A 解 X' = AX + G。

10. 定性方法與非線性微分方程系統。
非線性系統與解的存在性。相平面、相圖與方向場。線性系統的相圖。臨界點與穩定性。幾乎線性系統。捕食者/獵物種群模型。一個簡單的捕食者/獵物模型。一個擴展的捕食者/獵物模型。競爭物種模型。一個簡單的競爭物種模型。一個擴展的競爭物種模型。李雅普諾夫穩定性準則。極限環與周期解。

第四部分：向量分析。
11. 向量微分計算。
一變量的向量函數。速度、加速度、曲率與扭率。加速度的切向與法向分量。曲率作為 t 的函數。弗雷內公式。向量場與流線。梯度場與方向導數。等高面、切平面與法線。散度與旋度。散度的物理解釋。旋度的物理解釋。

12. 向量積分計算。
線積分。相對於弧長的線積分。格林定理。格林定理的擴展。路徑獨立性與平面中的勢能理論。對定理 12.5 的更深入分析。三維空間中的曲面與曲面積分。曲面的法向量。曲面的切平面。光滑與分段光滑曲面。曲面積分。曲面積分的應用。曲面面積。殼體的質量與質心。向量場穿過曲面的通量。為高斯與斯托克斯的積分定理做準備。高斯的散度定理。阿基米德原理。熱方程。散度定理作為質量守恆原則。格林的恆等式。斯托克斯的積分定理。旋度的解釋。三維空間中的勢能理論。

第五部分：傅立葉分析、正交展開與小波。
13. 傅立葉級數。
為什麼是傅立葉級數？函數的傅立葉級數。偶函數與奇函數。傅立葉級數的收斂性。在端點的收斂性。第二收斂定理。傅立葉級數的部分和。吉布斯現象。傅立葉餘弦級數與正弦級數。函數的傅立葉餘弦級數。函數的傅立葉正弦級數。傅立葉級數的積分與微分。傅立葉級數的相位角形式。複傅立葉級數與頻率譜。複數的回顧。複傅立葉級數。

14. 傅立葉積分與傅立葉變換。
傅立葉積分。傅立葉餘弦與正弦積分。複傅立葉積分與傅立葉變換。傅立葉變換的附加性質與應用。導數的傅立葉變換。頻率微分。積分的傅立葉變換。卷積。濾波與狄拉克δ函數。窗函數傅立葉變換。香農取樣定理。低通與帶通濾波器。傅立葉餘弦與正弦變換。離散傅立葉變換。線性與周期性。逆 N 點 DFT。DFT 對傅立葉係數的近似。取樣傅立葉級數。通過 N 點 DFT 近似傅立葉變換。濾波。快速傅立葉變換。FFT 的計算效率。使用 FFT 分析信號的功率譜密度。從信號中濾除噪聲。分析莫羅灣的潮汐。

15. 特殊函數、正交展開與小波。
勒讓德多項式。勒讓德多項式的生成函數。勒讓德多項式的遞迴關係。勒讓德多項式的正交性。傅立葉-勒讓德級數。計算傅立葉-勒讓德係數。勒讓德多項式的零點。Pn(x) 的導數與積分公式。貝塞爾函數。伽瑪函數。第一類貝塞爾函數與貝塞爾方程的解。第二類貝塞爾函數。修正貝塞爾函數。貝塞爾函數的一些應用。Jn(x) 的生成函數。Jn(x) 的積分公式。Jv(x) 的遞迴關係。Jv(x) 的零點。傅立葉-貝塞爾展開。傅立葉-貝塞爾係數。斯圖姆-柳維爾理論與特徵函數展開。斯圖姆-柳維爾問題。斯圖姆-柳維爾定理。特徵函數展開。均值近似與貝塞爾不等式。均值收斂與帕爾塞瓦爾定理。特徵函數的完備性。正交多項式。切比雪夫多項式。拉蓋爾多項式。赫爾米特多項式。小波。小波的概念。哈爾小波。小波展開。使用哈爾小波的多解析分析。小波與多解析分析的一般構造。香農小波。

第六部分：偏微分方程。
16. 波動方程。
波動方程及初始與邊界條件。波動方程的傅立葉級數解。具有給定初始速度和零初始位移的振動弦。具有初始位移和速度的振動弦。解的驗證。涉及波動方程的邊值問題的變換。初始條件與常數對運動的影響。沿著無限與半無限弦的波動運動。無界域問題的傅立葉變換解。特徵與達朗貝爾解。非齊次波動方程。前進與後退波。圓形彈性膜的正常振動模式。圓形彈性膜的振動。

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