Advanced Engineering Analysis

by Dukkipati, Rao V.

• ISBN13: 9781842652411
• ISBN10: 1842652419
• Format: Hardcover
• Copyright: 2006-01-30
• Publisher: Alpha Science International

Summary

Discusses in a concise but thorough manner the various areas of Advanced Engineering, including the fundamental statement of the theory, principles and methods on vectors and vector spaces, matrix analysis and many other complex areas. In addition, the book delves into the topics of vector differential amd vector integral calculus, variational calculus, canonical transformations, and Hamilton-Jacobi theory.

Table of Contents

Preface	vii
1. Vectors and Vector Spaces	1	(22)
1.1 Vector Algebra	1	(1)
1.2 Vector Operations	2	(1)
1.2.1 Product of a Scalar and Vector	2	(1)
1.2.2 Sum of Vectors	2	(1)
1.2.3 Vector Algebra	2	(1)
1.3 Properties of the Norm	3	(1)
1.4 Triangle Inequality	3	(2)
1.5 The Dot Product	5	(4)
1.5.1 Properties of the Dot Product	5	(1)
1.5.2 Angle between Two Vectors	6	(1)
1.5.3 Cauchy-Schwarz Inequality	7	(2)
1.6 The Cross Product	9	(4)
1.6.1 Properties of the Cross Product	10	(3)
1.7 The Vector Space Rn	13	(3)
1.7.1 Norm of an n-vector	13	(1)
1.7.2 Vector Operations	13	(1)
1.7.3 Dot Product of n-vectors	14	(1)
1.7.4 Cauchy-Schwarz Inequality	14	(1)
1.7.5 Subspace of n-space	15	(1)
1.8 Linear Independence and Dimension	16	(3)
1.8.1 Linear Combination of n-vectors	16	(1)
1.8.2 Linear Dependence of n-vectors	16	(1)
1.8.3 Linear Independence	17	(1)
1.8.4 Tests for Linear Independence	17	(1)
1.8.5 Basis	17	(1)
1.8.6 Dimension	18	(1)
1.9 Summary	19	(1)
Problems	20	(3)
2. Linear Algebra	23	(47)
2.1 Introduction	23	(1)
2.2 Definitions of Matrices	23	(6)
2.3 Elementary Matrix Operations	29	(2)
2.4 Determinants	31	(7)
2.5 Matrix Inversion	38	(3)
2.5.1 Transpose, Inverse and Determinant of a Product of Matrices	39	(2)
2.6 System of Algebraic Equations	41	(4)
2.6.1 Cramer's Rule	41	(1)
2.6.2 Inversion of a Matrix	42	(3)
2.7 Eigenvalues and Eigenvectors	45	(4)
2.7.1 Determination of Eigenvectors	45	(3)
2.7.2 Properties of Eigenvalues and Eigenvectors	48	(1)
2.8 Diagonalization	49	(6)
2.8.1 Properties of Diagonal Matrices	51	(1)
2.8.2 Condition for Diagonalizability	52	(3)
2.9 Orthogonal and Symmetric Matrices	55	(1)
2.9.1 Orthogonal Matrices	55	(1)
2.9.2 Symmetric Matrices	55	(1)
2.10 Unitary, Hermitian, and Skew Hermitian Matrices	56	(2)
2.10.1 Unitary Matrices	56	(1)
2.10.2 Hermitian Matrix	57	(1)
2.10.3 Skew Hermitian Matrix	57	(1)
2.11 Quadratic Forms	58	(1)
2.12 Positive Definite Matrix	58	(1)
2.13 Negative Definite Matrix	59	(1)
2.14 Indefinite Matrix	60	(1)
2.15 Norm of a Vector	60	(1)
2.16 Partitioning of Matrices	61	(1)
2.17 Augmented Matrix	62	(1)
2.18 Matrix Calculus	62	(1)
2.19 Summary	63	(1)
Problems	64	(6)
3. Laplace Transforms	70	(53)
3.1 Laplace Transformation	70	(1)
3.1.1 Linearity of the Laplace Transform	71	(1)
3.2 Existence of Laplace Transform	71	(2)
3.3 Inverse Laplace Transform	73	(1)
3.4 Properties of the Laplace Transform	73	(1)
3.4.1 Multiplication by a Constant	73	(1)
3.4.2 Sum and Difference	73	(1)
3.5 Special Functions	74	(12)
3.5.1 Exponential Function	74	(7)
3.5.2 Step Function	81	(1)
3.5.3 Ramp Function	82	(1)
3.5.4 Pulse Function	83	(1)
3.5.5 Impulse Function	83	(1)
3.5.6 Dirac Delta Function	84	(1)
3.5.7 Sinusoidal Function	85	(1)
3.6 Multiplication of f(t) by e-at	86	(1)
3.7 Differentiation	86	(2)
3.8 Integration	88	(3)
3.9 Final-Value Theorem	91	(1)
3.10 Initial-Value Theorem	92	(1)
3.11 Shift in Time	93	(2)
3.12 Complex Shifting	95	(1)
3.13 Real Convolution (Complex Multiplication)	95	(2)
3.14 Inverse Laplace Transformation	97	(7)
3.14.1 Partial Fraction Expansions	97	(1)
3.14.2 Partial Fraction Expansion when Q(S) has Distinct Roots	98	(4)
3.14.3 Partial Fraction Expansion when Q(S) has Complex Conjugate Roots	102	(1)
3.14.4 Partial Fraction Expansion when Q(S) has Repeated Roots	102	(2)
3.15 Solution of Differential Equations	104	(3)
3.16 Example Problems and Solutions	107	(12)
3.17 Summary	119	(1)
Problems	120	(3)
4. Fourier Series and Integrals	123	(49)
4.1 Periodic Functions	123	(1)
4.2 Trigonometric Series	124	(1)
4.3 Fourier Series	125	(5)
4.3.1 Euler's Formulas	127	(3)
4.3.2 Orthogonality of the Trigonometric System	130	(1)
4.4 Functions of any Period p = 2	130	(2)
4.5 Even and Odd Functions	132	(3)
4.6 Convergence of Fourier Series	135	(6)
4.7 Term by Term Differentiation of Fourier Series	141	(1)
4.8 Term by Term Integration of Fourier Series	141	(4)
4.9 Fourier Half-Range Series	145	(6)
4.9.1 Sine Series on 0 less than or equal to x less than or equal to 146
4.9.2 Cosine Series on 0 less than or equal to x less than or equal to 146
4.10 Complex Form of Fourier Series	151	(1)
4.11 Fourier Integrals	152	(3)
4.12 Fourier Cosine and Sine Integrals	155	(5)
4.13 Complex Form of Fourier Integrals Representation	160	(3)
4.14 Frequency and Amplitude Spectra of a Function	163	(3)
4.15 Summary	166	(1)
Problems	167	(5)
5. Fourier Transforms	172	(31)
5.1 The Fourier Transform	172	(6)
5.2 Operational Properties of the Fourier Transform	178	(11)
5.2.1 Linearity of the Fourier Transform	178	(1)
5.2.2 Fourier Transform of a Derivative of f(x)	179	(1)
5.2.3 Fourier Transform of xnf(x)	180	(1)
5.2.4 Fourier Transform of xnf(n)(x)	181	(1)
5.2.5 Convolution Theorem for Fourier Transform	182	(3)
5.2.6 The Parseval Relation for the Fourier Transform	185	(1)
5.2.7 Fourier Transform Involving Scaling and Shifting	186	(1)
5.2.8 Amplitude Spectrum	187	(1)
5.2.9 Differentiation with respect to Frequency ω	187	(1)
5.2.10 Modulation	188	(1)
5.3 Fourier Transform of an Integral	189	(1)
5.4 Fourier Transform of the Dirac Delta Functioa	190	(1)
5.5 Fourier Cosine and Sine Transforms	191	(4)
5.6 Linearity of the Fourier Cosine and Sine Transforms	195	(1)
5.7 Fourier Cosine and Sine Transforms of Derivatives	196	(1)
5.8 The Parseval Relation for the Fourier Cosine and Sine Transforms	197	(2)
5.9 Shifting and Scaling in Fourier Cosine and Sine Transforms	199	(1)
5.10 Summary	200	(1)
Problems	201	(2)
6. Vector Differential Calculus	203	(30)
6.1 Basic Concepts	203	(2)
6.1.1 Scalar Field	203	(1)
6.1.2 Vector Field	204	(1)
6.1.3 Level Surfaces	205	(1)
6.1.4 Streamlines	205	(1)
6.2 Parametric Representations	205	(3)
6.2.1 Parametric Representation of Curves	206	(1)
6.2.2 Parametric Representation of Surfaces	207	(1)
6.3 Limit, Continuity and Differentiability of Vector Functions	208	(5)
6.4 Motion of a Body or Particle on a Curve	213	(2)
6.5 The Gradient of a Scalar Field and Directional Derivative	215	(5)
6.5.1 Geometrical Representation of the Gradient	217	(3)
6.6 Directional Derivative	220	(4)
6.6.1 Maximum Rate of Change of a Scalar Field	223	(1)
6.6.2 Conservative Vector Field	224	(1)
6.7 Divergence and Curl of a Vector	224	(7)
6.7.1 Divergence of a Vector	225	(1)
6.7.2 Curl of Vector	225	(2)
6.7.3 Physical Interpretation of Divergence	227	(3)
6.7.4 Physical Interpretation of Curl	230	(1)
6.8 Summary	231	(1)
Problems	232	(1)
7. Vector Integral Calculus	233	(46)
7.1 Line Integrals	235	(9)
7.1.1 Line Integral with Respect to Arc Length	236	(1)
7.1.2 Line Integral of Vector Fields	236	(1)
7.1.3 Line Integral of Scalar Fields	237	(1)
7.1.4 Application of Line Integrals	238	(1)
7.1.5 Line Integrals Independents of the Path	239	(5)
7.2 Green's Theorem	244	(7)
7.3 Surfaces	251	(2)
7.3.1 Tangent Plane and Surface Normal	252	(1)
7.4 Surface Integrals	253	(6)
7.4.1 Surface Area	255	(2)
7.4.2 Orientable Surfaces	257	(2)
7.4.3 Flux of a Vector Field Across a Surface	259	(1)
7.5 Divergence Theorem of Gauss	259	(6)
7.6 Green's Identities	265	(1)
7.7 Stoke's Theorem	266	(6)
7.8 Summary	272	(1)
Problems	273	(6)
8. Partial Differential Equations and Boundary Value Problems	279	(38)
8.1 Definitions	279	(1)
8.2 Classification of Linear Second Order Partial Differential Equations	280	(2)
8.3 Methods of Solving Boundary-Value Problems	282	(3)
8.3.1 General Solutions	282	(1)
8.3.2 Superposition Principle	283	(1)
8.3.3 Separation of Variables Method (Fourier Method)	283	(1)
8.3.4 Laplace Transform Method	284	(1)
8.4 Fourier Series Solution of the One-Dimensional Heat Equation	285	(15)
8.4.1 Ends of Bar kept at Zero Temperature	286	(3)
8.4.2 Temperature in a Bar with Insulated Ends	289	(2)
8.4.3 Thin Bar or Wire with a Radiating End	291	(4)
8.4.4 Thin Infinite Bar	295	(5)
8.5 Fourier Solution of the Wave Equation	300	(7)
8.5.1 One-Dimensional Wave Equation	300	(2)
8.5.2 Normal Modes of Vibration	302	(1)
8.5.3 D'Alembert Solution for An Infinitely Long Elastic String	303	(4)
8.6 Fourier Series Solution of Laplace's Equation	307	(4)
8.7 Summary	311	(1)
Problems	312	(5)
Answers to Selected Problems	317	(17)
Bibliography	334	(2)
Glossary of Terms	336	(3)
Index	339