Engineering mathematics. Vol.1
Sastry, S. S.
Engineering mathematics. Vol.1 - Ed. 4 Ed.4 - New Delhi PHI Learning Pvt Ltd 2009 - xii,676p.
CONTENTS Preface ix Preface to the First Edition xi 1. HIGHER ALGEBRA 1-174 1.1 Partial Fractions7 1.1.1 Linear Factors 3 1.1.2 Repeated Linear Factors 4 1.1.3 Nonlinear Factors6 1.1.4 Repeated Nonlinear Factors7 Exercise 1.18 1.2 Infinite Series9 1.2.1 Comparison Tests 12 1.2.2 D'Alembert's Ratio Test16 1.2.3 Cauchy's Root Test19 1.2.4 Absolute and Conditional Convergence 20 1.2.5 Alternating Series 22 Exercise 1.2 24 1.3 Binomial, Exponential and Logarithmic Series 26 1.3.1 The Binomial Theorem 26 1.3.2 Applications of Binomial Theorem to Approximation and Interpolation 37 1.3.3 The Exponential Series 37 1.3.4 The Logarithmic Series 44 Exercise 1.3 49 1.4 Continued Fractions52 1.4.1 Convergents of a Continued Fraction 53 1.4.2 Estimation of Error 57 1.4.3 Application to Interpolation 59 Exercise 1.462 1.5 Theory of Equations63 1.5.1 The Fundamental Theorem of Algebra 64 1.5.2 Relations between the Roots and Coefficients 65 1.5.3 Imaginary and Irrational Roots 67 1.5.4 Symmetric Functions of the Roots 69 1.5.5 Reciprocal Equations73 1.5.6 Transformation of Equations76 1.5.7 Descartes' Rule of Signs79 1.5.8 Homer's Method for Finding a Real Root 80 Exercise 1.5 83 1.6 Matrices 85 1.6.1 Basic Definitions 85 1.6.2 Matrix Operations 87 1.6.3 Transpose, Adjoint and Inverse of a Matrix 91 1.6.4 Some Special Matrices101 1.6.5 Rank of a Matrix and Linear Dependence of Vectors109 1.6.6 Linear Algebraic Systems118 1.6.7 Eigenvalues and Eigenvectors 130 1.6.8 Diagonalization by Orthogonal Transformation150 1.6.9 Quadratic Forms153 Exercise 1.6164 Answers to Selected Exercises170 2. GEOMETRY, VECTORS AND COMPLEX NUMBERS 175-370 2.1 Coordinate Systems 175 2.1.1 Rectangular Coordinates in a Plane 175 2.1.2 Distance between Two Points 176 2.1.3 Polar Coordinates178 2.1.4 Equation for a Line in Polar Coordinates179 2.1.5 Rectangular Coordinates in Space180 2.1.6 Cylindrical Polar Coordinates 182 2.1.7 Spherical Polar Coordinates183 Exercise 2.1184 2.2 Elementary Coordinate Geometry 185 2.2.1 The Straight Line 185 2.2.2 The Circle197 2.2.3 The Conic Sections 205 2.2.4 Translation of Axes 235 2.2.5 Equation of a Conic in Polar Coordinates 239 Exercise 2.2 241 2.3 Analytical Geometry of Three Dimensions 247 2.3.1 Direction Cosines and Angle between Two Lines 247 2.3.2 Equation of a Plane 255 2.3.3 The Straight Line 265 2.3.4 Shortest Distance between Two Skew Lines 277 2.3.5 The Sphere 282 2.3.6 The Cylinder 288 2.3.7 The Cone 293 2.3.8 Quadric Surfaces 299 Exercise 2.3304 2.4 Vector Algebra 306 2.4.1 The Concept of a Vector 307 2.4.2 Addition and Subtraction of Vectors 307 2.4.3 Resolution of Vectors 309 2.4.4 Scalar or Dot Product of Two Vectors 376 2.4.5 Vector or Cross Product of Two Vectors 320 2.4.6 Equations of Lines and Planes 324 2.4.7 Products of Three or More Vectors 337 Exercise 2.4 335 2.5 Complex Numbers 338 2.5.1 The Argand Diagram 342 2.5.2 Multiplication and Division in Polar Forms 344 2.5.3 De Moivre's Theorem 344 2.5.4 Roots of a Complex Number 346 2.5.5 Expansions for Powers of sine and cosine 348 2.5.6 Expansion of cos no and sin no in Powers of cos o and sin o 352 2.5.7 Relation to Exponential Series 353 2.5.8 Inverse Hyperbolic Functions 357 2.5.9 The Logarithm of a Negative Quantity 367 2.5.10 Complex Numbers and Matrices 361 Exercise 2.5 362 Answers to Selected Exercises366 3. APPLICATIONS OF DIFFERENTIAL CALCULUS 371-459 3.1 Introduction377 3.1.1 Geometrical Applications 377 3.1.2 Related Rates and Indeterminate Forms 394 Exercise 3.1406 3.2 Maxima, Minima and Points of Inflexion411 Exercise 3.2 421 3.3 Expansions of Functions 422 Exercise 3.3431 3.4 Curvature, Envelopes and Evolutes 432 3.4.1 Curvature 432 3.4.2 Envelopes442 3.4.3 Evolutes and Involutes 448 Exercise 3.4455 Answers to Selected Exercises457 4. INTEGRATION 460-523 4.1 Introduction460 4.1.1 Theory of the Integral460 4.1.2 Properties of the Definite Integral465 4.1.3 Methods of Integration471 4.1.4 Reduction Formulae and Definite Integrals 480 Exercise 4.1 490 4.2 Applications of the Definite Integral492 4.2.1 Volumes and Surfaces of Solids of Revolution499 4.2.2 Length of an Arc of a Curve 505 4.2.3 Centroid of Area and Volume of Revolution 508 4.2.4 Theorems of Pappus 570 4.2.5 Areas and Volumes in Polar Coordinates 514 4.2.6 Mean Values and Root-mean-square Values 577 Exercise 4.2518 Answers to Selected Exercises521 5. ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 524-583 5.1 Introduction524 5.1.1 Analytical Methods of Solution 529 5.1.2 Method of Variation of Parameters 540 5.1.3 Equations of Higher Degree 543 Exercise 5.1549 5.2 Applications involving First-Order Equations 552 5.2.1 Trajectories 552 5.2.2 Flow of Electricity 556 5.2.3 Newton's Law of Cooling 560 5.2.4 Laws of Decay and Growth 562 5.2.5 Hooke's Law 564 5.2.6 Body Falling in a Resisting Medium 567 5.2.7 Satellite Launch Vehicles 575 Exercise 5.2 579 Answers to Selected Exercises581 6. NUMERICAL METHODS 584-669 6.1 Interpolation and Least Squares 584 6.1.1 Lagrange Interpolation 584 6.1.2 Newton's Divided Difference Formula 593 6.1.3 Finite Differences: Newton's Formulae 597 6.1.4 Central Differences: Stirling's, Bessel's and Everett's Formulae 603 6.1.5 Empirical Laws and Least Squares Curve Fitting 608 Exercise 6.1613 6.2 Solution of Nonlinear Equations 616 6.2.1 The Bisection Method617 6.2.2 The Regula-falsi Method 677 6.2.3 Newton-Raphson Method 619 6.2.4 Generalized Newton's Method622 6.2.5 Muller's Method623 6.2.6 Solution of Systems of Nonlinear Equations 625 Exercise 6.2 626 6.3 Numerical Differentiation627 6.3.1 Numerical Differentiation Formulae628 6.3.2 Maximum and Minimum Values of a Tabulated Function 633 Exercise 6.3 635 6.4 Numerical Integration 636 6.4.1Trapezoidal Rule637 6.4.2 Simpson's 1/3-rule638 6.4.3 Simpson's 3/8-rule 638 6.4.4 Romberg Integration 639 Exercise 6.4 644 6.5 Numerical Methods for First-Order Differential Equations 646 6.5.1 Solution by Taylor's Series647 6.5.2 Picard's Iterative Method 649 6.5.3 Euler's Method652 6.5.4 Runge-Kutta Methods 656 6.5.5 Milne's Method 662 Exercise 6.5 665 Answers to Selected Exercises 668 BIBLIOGRAPHY 671 INDEX 673-676
812033616X
510 / SAS
Engineering mathematics. Vol.1 - Ed. 4 Ed.4 - New Delhi PHI Learning Pvt Ltd 2009 - xii,676p.
CONTENTS Preface ix Preface to the First Edition xi 1. HIGHER ALGEBRA 1-174 1.1 Partial Fractions7 1.1.1 Linear Factors 3 1.1.2 Repeated Linear Factors 4 1.1.3 Nonlinear Factors6 1.1.4 Repeated Nonlinear Factors7 Exercise 1.18 1.2 Infinite Series9 1.2.1 Comparison Tests 12 1.2.2 D'Alembert's Ratio Test16 1.2.3 Cauchy's Root Test19 1.2.4 Absolute and Conditional Convergence 20 1.2.5 Alternating Series 22 Exercise 1.2 24 1.3 Binomial, Exponential and Logarithmic Series 26 1.3.1 The Binomial Theorem 26 1.3.2 Applications of Binomial Theorem to Approximation and Interpolation 37 1.3.3 The Exponential Series 37 1.3.4 The Logarithmic Series 44 Exercise 1.3 49 1.4 Continued Fractions52 1.4.1 Convergents of a Continued Fraction 53 1.4.2 Estimation of Error 57 1.4.3 Application to Interpolation 59 Exercise 1.462 1.5 Theory of Equations63 1.5.1 The Fundamental Theorem of Algebra 64 1.5.2 Relations between the Roots and Coefficients 65 1.5.3 Imaginary and Irrational Roots 67 1.5.4 Symmetric Functions of the Roots 69 1.5.5 Reciprocal Equations73 1.5.6 Transformation of Equations76 1.5.7 Descartes' Rule of Signs79 1.5.8 Homer's Method for Finding a Real Root 80 Exercise 1.5 83 1.6 Matrices 85 1.6.1 Basic Definitions 85 1.6.2 Matrix Operations 87 1.6.3 Transpose, Adjoint and Inverse of a Matrix 91 1.6.4 Some Special Matrices101 1.6.5 Rank of a Matrix and Linear Dependence of Vectors109 1.6.6 Linear Algebraic Systems118 1.6.7 Eigenvalues and Eigenvectors 130 1.6.8 Diagonalization by Orthogonal Transformation150 1.6.9 Quadratic Forms153 Exercise 1.6164 Answers to Selected Exercises170 2. GEOMETRY, VECTORS AND COMPLEX NUMBERS 175-370 2.1 Coordinate Systems 175 2.1.1 Rectangular Coordinates in a Plane 175 2.1.2 Distance between Two Points 176 2.1.3 Polar Coordinates178 2.1.4 Equation for a Line in Polar Coordinates179 2.1.5 Rectangular Coordinates in Space180 2.1.6 Cylindrical Polar Coordinates 182 2.1.7 Spherical Polar Coordinates183 Exercise 2.1184 2.2 Elementary Coordinate Geometry 185 2.2.1 The Straight Line 185 2.2.2 The Circle197 2.2.3 The Conic Sections 205 2.2.4 Translation of Axes 235 2.2.5 Equation of a Conic in Polar Coordinates 239 Exercise 2.2 241 2.3 Analytical Geometry of Three Dimensions 247 2.3.1 Direction Cosines and Angle between Two Lines 247 2.3.2 Equation of a Plane 255 2.3.3 The Straight Line 265 2.3.4 Shortest Distance between Two Skew Lines 277 2.3.5 The Sphere 282 2.3.6 The Cylinder 288 2.3.7 The Cone 293 2.3.8 Quadric Surfaces 299 Exercise 2.3304 2.4 Vector Algebra 306 2.4.1 The Concept of a Vector 307 2.4.2 Addition and Subtraction of Vectors 307 2.4.3 Resolution of Vectors 309 2.4.4 Scalar or Dot Product of Two Vectors 376 2.4.5 Vector or Cross Product of Two Vectors 320 2.4.6 Equations of Lines and Planes 324 2.4.7 Products of Three or More Vectors 337 Exercise 2.4 335 2.5 Complex Numbers 338 2.5.1 The Argand Diagram 342 2.5.2 Multiplication and Division in Polar Forms 344 2.5.3 De Moivre's Theorem 344 2.5.4 Roots of a Complex Number 346 2.5.5 Expansions for Powers of sine and cosine 348 2.5.6 Expansion of cos no and sin no in Powers of cos o and sin o 352 2.5.7 Relation to Exponential Series 353 2.5.8 Inverse Hyperbolic Functions 357 2.5.9 The Logarithm of a Negative Quantity 367 2.5.10 Complex Numbers and Matrices 361 Exercise 2.5 362 Answers to Selected Exercises366 3. APPLICATIONS OF DIFFERENTIAL CALCULUS 371-459 3.1 Introduction377 3.1.1 Geometrical Applications 377 3.1.2 Related Rates and Indeterminate Forms 394 Exercise 3.1406 3.2 Maxima, Minima and Points of Inflexion411 Exercise 3.2 421 3.3 Expansions of Functions 422 Exercise 3.3431 3.4 Curvature, Envelopes and Evolutes 432 3.4.1 Curvature 432 3.4.2 Envelopes442 3.4.3 Evolutes and Involutes 448 Exercise 3.4455 Answers to Selected Exercises457 4. INTEGRATION 460-523 4.1 Introduction460 4.1.1 Theory of the Integral460 4.1.2 Properties of the Definite Integral465 4.1.3 Methods of Integration471 4.1.4 Reduction Formulae and Definite Integrals 480 Exercise 4.1 490 4.2 Applications of the Definite Integral492 4.2.1 Volumes and Surfaces of Solids of Revolution499 4.2.2 Length of an Arc of a Curve 505 4.2.3 Centroid of Area and Volume of Revolution 508 4.2.4 Theorems of Pappus 570 4.2.5 Areas and Volumes in Polar Coordinates 514 4.2.6 Mean Values and Root-mean-square Values 577 Exercise 4.2518 Answers to Selected Exercises521 5. ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 524-583 5.1 Introduction524 5.1.1 Analytical Methods of Solution 529 5.1.2 Method of Variation of Parameters 540 5.1.3 Equations of Higher Degree 543 Exercise 5.1549 5.2 Applications involving First-Order Equations 552 5.2.1 Trajectories 552 5.2.2 Flow of Electricity 556 5.2.3 Newton's Law of Cooling 560 5.2.4 Laws of Decay and Growth 562 5.2.5 Hooke's Law 564 5.2.6 Body Falling in a Resisting Medium 567 5.2.7 Satellite Launch Vehicles 575 Exercise 5.2 579 Answers to Selected Exercises581 6. NUMERICAL METHODS 584-669 6.1 Interpolation and Least Squares 584 6.1.1 Lagrange Interpolation 584 6.1.2 Newton's Divided Difference Formula 593 6.1.3 Finite Differences: Newton's Formulae 597 6.1.4 Central Differences: Stirling's, Bessel's and Everett's Formulae 603 6.1.5 Empirical Laws and Least Squares Curve Fitting 608 Exercise 6.1613 6.2 Solution of Nonlinear Equations 616 6.2.1 The Bisection Method617 6.2.2 The Regula-falsi Method 677 6.2.3 Newton-Raphson Method 619 6.2.4 Generalized Newton's Method622 6.2.5 Muller's Method623 6.2.6 Solution of Systems of Nonlinear Equations 625 Exercise 6.2 626 6.3 Numerical Differentiation627 6.3.1 Numerical Differentiation Formulae628 6.3.2 Maximum and Minimum Values of a Tabulated Function 633 Exercise 6.3 635 6.4 Numerical Integration 636 6.4.1Trapezoidal Rule637 6.4.2 Simpson's 1/3-rule638 6.4.3 Simpson's 3/8-rule 638 6.4.4 Romberg Integration 639 Exercise 6.4 644 6.5 Numerical Methods for First-Order Differential Equations 646 6.5.1 Solution by Taylor's Series647 6.5.2 Picard's Iterative Method 649 6.5.3 Euler's Method652 6.5.4 Runge-Kutta Methods 656 6.5.5 Milne's Method 662 Exercise 6.5 665 Answers to Selected Exercises 668 BIBLIOGRAPHY 671 INDEX 673-676
812033616X
510 / SAS