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Engineering Mathematics

Book Description

Engineering Mathematics covers the four mathematics papers that are offered to undergraduate students of engineering. With an emphasis on problem-solving techniques and engineering applications, as well as detailed explanations of the mathematical concepts, this book will give the students a complete grasp of the mathematical skills that are needed by engineers.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Dedication
  5. Preface
  6. Symbols and Basic Formulae
  7. Part I
    1. 1. Sequences and Series
      1. 1.1 Sequences
      2. 1.2 Convergence of Sequences
      3. 1.3 The Upper and Lower Limits of a Sequence
      4. 1.4 Cauchy's Principle of Convergence
      5. 1.5 Monotonic Sequence
      6. 1.6 Theorems on Limits
      7. 1.7 Subsequence
      8. 1.8 Series
      9. 1.9 Comparison Tests
      10. 1.10 D’ Alembert's Ratio Test
      11. 1.11 Cauchy's Root Test
      12. 1.12 Raabe's Test
      13. 1.13 Logarithmic Test
      14. 1.14 De Morgan – Bertrand Test
      15. 1.15 Gauss's Test
      16. 1.16 Cauchy's Integral Test
      17. 1.17 Cauchy's Condensation Test
      18. 1.18 Kummer's Test
      19. 1.19 Alternating Series
      20. 1.20 Absolute Convergence of a Series
      21. 1.21 Convergence of the Series of the Type
      22. 1.22 Derangement of Series
      23. 1.23 Nature of Non-absolutely Convergent Series
      24. 1.24 Effect of Derangement of Non-absolutely Convergent Series
      25. 1.25 Uniform Convergence
      26. 1.26 Uniform Convergence of a Series of Functions
      27. 1.27 Properties of Uniformly Convergent Series
      28. Exercises
    2. 2. Mean Value Theorems and Expansion of Functions
      1. 2.1 Leibnitz's Theorem and Its Applications
      2. 2.2 General Theorems
      3. 2.3 Taylor's Infinite Series and Power Series Expansion
      4. 2.4 Maclaurin's Infinite Series
      5. 2.5 Expansion of Functions
      6. 2.6 Indeterminate forms
      7. Exercises
    3. 3. Curvature
      1. 3.1 Radius of Curvature of Intrinsic Curves
      2. 3.2 Radius of Curvature for Cartesian Curves
      3. 3.3 Radius of Curvature for Parametric Curves
      4. 3.4 Radius of Curvature for Pedal Curves
      5. 3.5 Radius of Curvature for Polar Curves
      6. 3.6 Radius of Curvature at the Origin
      7. 3.7 Centre of Curvature
      8. 3.8 Evolutes and Involutes
      9. 3.9 Equation of the Circle of Curvature
      10. 3.10 Chords of Curvature Parallel to the Coordinate Axes
      11. 3.11 Chord of Curvature in Polar Coordinates
      12. Exercises
    4. 4. Asymptotes and Curve Tracing
      1. 4.1 Determination of Asymptotes When the Equation of the Curve in Cartesian form is given
      2. 4.2 The Asymptotes of the General Rational Algebraic Curve
      3. 4.3 Asymptotes parallel to the Coordinate Axes
      4. 4.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve
      5. 4.5 Intersection of a Curve and its Asymptotes
      6. 4.6 Asymptotes by Expansion
      7. 4.7 Asymptotes of the Polar Curves
      8. 4.8 Circular Asymptotes
      9. 4.9 Curve Tracing (Cartesian Equations)
      10. 4.10 Curve Tracing (Polar Equations)
      11. 4.11 Curve Tracing (Parametric Equations)
      12. Exercises
    5. 5. Partial Differentiation
      1. 5.1 Continuity of a Function of Two Variables
      2. 5.2 Differentiability of a Function of Two Variables
      3. 5.3 The Differential Coefficients
      4. 5.4 Distinction Between Derivatives and Differential Coefficients
      5. 5.5 Higher-Order Partial Derivatives
      6. 5.6 Envelopes and Evolutes
      7. 5.7 Homogeneous Functions and Euler's Theorem
      8. 5.8 Differentiation of Composite Functions
      9. 5.9 Transformation From Cartesian to Polar Coordinates and Vice Versa
      10. 5.10 Taylor's Theorem For Functions of Several Variables
      11. 5.11 Extreme Values
      12. 5.12 Lagrange's Method of Undetermined Multipliers
      13. 5.13 Jacobians
      14. 5.14 Properties of Jacobian
      15. 5.15 Necessary and Sufficient Conditions for Jacobian to Vanish
      16. 5.16 Differentiation Under the Integral Sign
      17. Exercises
    6. 6. Beta and Gamma Functions
      1. 6.1 Beta Function
      2. 6.2 Properties of Beta Function
      3. 6.3 Gamma Function
      4. 6.4 Properties of Gamma Function
      5. 6.5 Relation Between Beta and Gamma Functions
      6. 6.6 Dirichlet's and Liouville's Theorems
      7. Exercises
    7. 7. Reduction Formulas
      1. 7.1 Reduction Formulas for ∫sinn xdx and ∫cosn xdx
      2. 7.2 Reduction Formulas for ∫sinm x cosn x dx
      3. 7.3 Reduction Formulas for ∫tann xdx and ∫secn xdx
      4. 7.4 Reduction Formulas for ∫xn and ∫xn cos mxdx
      5. 7.5 Reduction Formulas for ∫ xn eax dx and ∫ xm (log x)n dx
      6. 7.6 Reduction Formula for ∫cosm x sin nxdx
      7. Exercises
    8. 8. Volumes and Surfaces of Solids of Revolution
      1. 8.1 Volume of the solid of Revolution (Cartesian Equations)
      2. 8.2 Volume of the Solid of Revolution (Parametric Equations)
      3. 8.3 Volume of the Solid of Revolution (Polar Curves)
      4. 8.4 Surface of the Solid of Revolution (Cartesian Equations)
      5. 8.5 Surface of the Solid of Revolution (Parametric Equations)
      6. 8.6 Surface of the Solid of Revolution (Polar Curves)
      7. Exercises
    9. 9. Multiple Integrals
      1. 9.1 Double Integrals
      2. 9.2 Properties of a Double Integral
      3. 9.3 Evaluation of Double Integrals (Cartesian Coordinates)
      4. 9.4 Evaluation of Double Integrals (Polar Coordinates)
      5. 9.5 Change of Variables in a Double Integral
      6. 9.6 Change of Order of Integration
      7. 9.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates)
      8. 9.8 Volume and Surface Area as Double Integrals
      9. 9.9 Triple Integrals and their Evaluation
      10. 9.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral
      11. 9.11 Volume as a Triple Integral
      12. Exercises
    10. 10. Vector Calculus
      1. 10.1 Differentiation of a Vector
      2. 10.2 Partial Derivatives of a Vector Function
      3. 10.3 Gradient of a Scalar Field
      4. 10.4 Geometrical Interpretation of a Gradient
      5. 10.5 Properties of a Gradient
      6. 10.6 Directional Derivatives
      7. 10.7 Divergence of a Vector-Point Function
      8. 10.8 Physical Interpretation of Divergence
      9. 10.9 Curl of a Vector-Point Function
      10. 10.10 Physical Interpretation of Curl
      11. 10.11 The Laplacian Operator ∇2
      12. 10.12 Properties of Divergence and Curl
      13. 10.13 Integration of Vector Functions
      14. 10.14 Line Integral
      15. 10.15 Work Done by a Force
      16. 10.16 Surface Integral
      17. 10.17 Volume Integral
      18. 10.18 Gauss's Divergence Theorem
      19. 10.19 Green's Theorem in a Plane
      20. 10.20 Stoke's Theorem
      21. Exercises
    11. 11. Three-Dimensional Geometry
      1. 11.1 Coordinate Planes
      2. 11.2 Distance Between Two Points
      3. 11.3 Direction Ratios and Direction Cosines of a Line
      4. 11.4 Section Formulae—Internal division of a line by a point on the line
      5. 11.5 Straight Line in Three Dimensions
      6. 11.6 Angle Between Two Lines
      7. 11.7 Shortest Distance Between Two Skew Lines
      8. 11.8 Equation of a Plane
      9. 11.9 Equation of a Plane Passing Through a Given Point and Perpendicular to a Given Direction
      10. 11.10 Equation of a Plane Passing Through Three Points
      11. 11.11 Equation of a Plane Passing Through a Point and Parallel to Two Given Vectors
      12. 11.12 Equation of a Plane Passing Through Two Points and Parallel to a Line
      13. 11.13 Angle Between Two Planes
      14. 11.14 Angle Between a Line and a Plane
      15. 11.15 Perpendicular Distance of a Point From a Plane
      16. 11.16 Planes Bisecting the Angles Between Two Planes
      17. 11.17 Intersection of Planes
      18. 11.18 Planes Passing Through the Intersection of Two Given Planes
      19. 11.19 Sphere
      20. 11.20 Equation of a Sphere Whose Diameter is the Line Joining Two Given Points
      21. 11.21 Equation of a Sphere Passing Through Four Points
      22. 11.22 Equation of the Tangent Plane to a Sphere
      23. 11.23 Condition of Tangency
      24. 11.24 Angle of Intersection of Two Spheres
      25. 11.25 Condition of Orthogonality of Two Spheres
      26. 11.26 Cylinder
      27. 11.27 Equation of a Cylinder with given Axis and Guiding Curves
      28. 11.28 Right Circular Cylinder
      29. 11.29 Cone
      30. 11.30 Equation of a Cone with Vertex at the Origin
      31. 11.31 Equation of a Cone with Given Vertex and Guiding Curve
      32. 11.32 Right Circular Cone
      33. 11.33 Right Circular Cone with Vertex (α, β, γ), Semi Vertical Angle θ, and (l, m, n) the Direction Cosines of the Axis
      34. 11.34 Conicoids
      35. 11.35 Shape of an Ellipsoid
      36. 11.36 Shape of the Hyperboloid of One Sheet
      37. 11.37 Shape of the Hyperboloid of Two Sheets
      38. 11.38 Shape of the Elliptic Cone
      39. 11.39 Intersection of a Conicoid and a Line
      40. 11.40 Tangent Plane at a Point of Central Conicoid
      41. 11.41 Condition of Tangency
      42. 11.42 Equation of Normal to the Central Conicoid at Any Point (α, β, γ) On It
      43. Exercises
  8. Part II
    1. 12. Preliminaries
      1. 12.1 Sets and Functions
      2. 12.2 Continuous and Piecewise Continuous Functions
      3. 12.3 Derivability of a Function and Piecewise Smooth Functions
      4. 12.4 The Riemann Integral
      5. 12.5 The Causal and Null Functions
      6. 12.6 Functions of Exponential Order
      7. 12.7 Periodic Functions
      8. 12.8 Even and Odd Functions
      9. 12.9 Sequence and Series
      10. 12.10 Series of Functions
      11. 12.11 Partial Fraction Expansion of a Rational Function
      12. 12.12 Special Functions
      13. 12.13 The Integral Transforms
    2. 13. Linear Algebra
      1. 13.1 Concepts of Group, Ring, and Field
      2. 13.2 Vector Space
      3. 13.3 Linear Transformation
      4. 13.4 Linear Algebra
      5. 13.5 Rank and Nullity of a Linear Transformation
      6. 13.6 Matrix of a Linear Transformation
      7. 13.7 Normed Linear Space
      8. 13.8 Inner Product Space
      9. 13.9 Matrices
      10. 13.10 Algebra of Matrices
      11. 13.11 Multiplication of Matrices
      12. 13.12 Associtative Law for Matrix Multiplication
      13. 13.13 Distributive Law for Matrix Multiplication
      14. 13.14 Transpose of a Matrix
      15. 13.15 Symmetric, Skew-symmetric, and Hermitian Matrices
      16. 13.16 Lower and Upper Triangular Matrices
      17. 13.17 Adjoint of a Matrix
      18. 13.18 The Inverse of a Matrix
      19. 13.19 Methods of Computing Inverse of a Matrix
      20. 13.20 Rank of a Matrix
      21. 13.21 Elementary Matrices
      22. 13.22 Equivalence of Matrices
      23. 13.23 Row and Column Equivalence of Matrices
      24. 13.24 Row Rank and Column Rank of a Matrix
      25. 13.25 Solution of System of Linear Equations
      26. 13.26 Solution of Non-homogeneous Linear System of Equations
      27. 13.27 Consistency Theorem
      28. 13.28 Homogeneous Linear Equations
      29. 13.29 Characteristic Roots and Vectors
      30. 13.30 The Cayley-Hamilton Theorem
      31. 13.31 Algebraic and Geometric Multiplicity of an Eigenvalue
      32. 13.32 Minimal Polynomial of a Matrix
      33. 13.33 Orthogonal, Normal, and Unitary Matrices
      34. 13.34 Similarity of Matrices
      35. 13.35 Triangularization of an Arbitrary Matrix
      36. 13.36 Quadratic Forms
      37. 13.37 Diagonalization of Quadratic Forms
      38. Exercises
    3. 14. Functions of Complex Variables
      1. 14.1 Basic Concepts
      2. 14.2 Analytic Functions
      3. 14.3 Integration of Complex-Valued Functions
      4. 14.4 Power Series Representation of an Analytic Function
      5. 14.5 Zeros and Poles
      6. 14.6 Residues and Cauchy's Residue Theorem
      7. 14.7 Evaluation of Real Definite Integrals
      8. 14.8 Conformal Mapping
      9. Exercises
    4. 15. Differential Equations
      1. 15.1 Definitions and Examples
      2. 15.2 Formulation of Differential Equation
      3. 15.3 Solution of Differential Equation
      4. 15.4 Differential Equations of First order
      5. 15.5 Separable Equations
      6. 15.6 Homogeneous Equations
      7. 15.7 Equations Reducible to Homogeneous Form
      8. 15.8 Linear Differential Equations
      9. 15.9 Equations Reducible to Linear Differential Equations
      10. 15.10 Exact Differential Equation
      11. 15.11 The Solution of Exact Differential Equation
      12. 15.12 Equations Reducible to Exact Equation
      13. 15.13 Applications of First Order and First Degree Equations
      14. 15.14 Linear Differential Equations
      15. 15.15 Solution of Homogeneous Linear Differential Equation with Constant Coefficients
      16. 15.16 Complete Solution of Linear Differential Equation with Constant Coefficients
      17. 15.17 Method of Variation of Parameters to Find Particular Integral
      18. 15.18 Differential Equations with Variable Coefficients
      19. 15.19 Simultaneous Linear Differential Equations with Constant Coefficients
      20. 15.20 Applications of Linear Differential Equations
      21. 15.21 Mass-Spring System
      22. 15.22 Simple Pendulum
      23. 15.23 Solution in Series
      24. 15.24 Bessel's Equation and Bessel's Function
      25. 15.25 Legendre's Equation and Legendre's Polynomial
      26. 15.26 Fourier–Legendre Expansion of a Function
      27. Exercises
    5. 16. Partial Differential Equations
      1. 16.1 Formulation of Partial Differential Equation
      2. 16.2 Solutions of a Partial Differential Equation
      3. 16.3 Non-linear Partial Differential Equations of the First Order
      4. 16.4 Charpit's Method
      5. 16.5 Some Standard forms of Non-linear Equations
      6. 16.6 The Method of Separation of Variables
      7. 16.7 One-Dimensional Heat Equation
      8. 16.8 One-DimensionalWave Equation
      9. 16.9 Two-Dimensional Heat Equation
      10. Exercises
    6. 17. Fourier Series
      1. 17.1 Trigonometric Series
      2. 17.2 Fourier (or Euler) Formulae
      3. 17.3 Periodic Extension of a Function
      4. 17.4 Fourier Cosine and Sine Series
      5. 17.5 Complex Fourier Series
      6. 17.6 Spectrum of Periodic Functions
      7. 17.7 Properties of Fourier Coefficients
      8. 17.8 Dirichlet's Kernel
      9. 17.9 Integral Expression for Partial Sums of a Fourier Series
      10. 17.10 Fundamental Theorem (Convergence Theorem) of Fourier Series
      11. 17.11 Applications of Fundamental Theorem of Fourier Series
      12. 17.12 Convolution Theorem for Fourier Series
      13. 17.13 Integration of Fourier Series
      14. 17.14 Differentiation of Fourier Series
      15. 17.15 Examples of Expansions of Functions in Fourier Series
      16. 17.16 Signals and Systems
      17. 17.17 Classification of Signals
      18. 17.18 Classification of Systems
      19. 17.19 Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input
      20. 17.20 Application to Differential Equations
      21. 17.21 Application to Partial Differential Equations
      22. Exercises
    7. 18. Fourier Transform
      1. 18.1 Fourier Integral Theorem
      2. 18.2 Fourier Transforms
      3. 18.3 Fourier Cosine and Sine Transforms
      4. 18.4 Properties of Fourier Transforms
      5. 18.5 Solved Examples
      6. 18.6 Complex Fourier Transforms
      7. 18.7 Convolution Theorem
      8. 18.8 Parseval's Identities
      9. 18.9 Fourier Integral Representation of a Function
      10. 18.10 Finite Fourier Transforms
      11. 18.11 Applications of Fourier Transforms
      12. 18.12 Application to Differential Equations
      13. 18.13 Application to Partial Differential Equations
      14. Exercises
    8. 19. Discrete Fourier Transform
      1. 19.1 Approximation of Fourier Coefficients of a Periodic Function
      2. 19.2 Definition and Examples of DFT
      3. 19.3 Inverse DFT
      4. 19.4 Properties of DFT
      5. 19.5 Cyclical Convolution and Convolution Theorem for DFT
      6. 19.6 Parseval's Theorem for the DFT
      7. 19.7 Matrix form of the DFT
      8. 19.8 N-Point Inverse DFT
      9. 19.9 Fast Fourier Transform (FFT)
      10. Exercises
    9. 20. Laplace Transform
      1. 20.1 Definition and Examples of Laplace Transform
      2. 20.2 Properties of Laplace Transforms
      3. 20.3 Limiting Theorems
      4. Exercises
    10. 21. Inverse Laplace Transform
      1. 21.1 Definition and Examples of Inverse Laplace Transform
      2. 21.2 Properties of Inverse Laplace Transform
      3. 21.3 Partial Fractions Method to Find Inverse Laplace Transform
      4. 21.4 Heaviside's Expansion Theorem
      5. 21.5 Series Method to Determine Inverse Laplace Transform
      6. 21.6 Convolution Theorem
      7. 21.7 Complex Inversion Formula
      8. Exercises
    11. 22. Applications of Laplace Transform
      1. 22.1 Ordinary Differential Equations
      2. 22.2 Simultaneous Differential Equations
      3. 22.3 Difference Equations
      4. 22.4 Integral Equations
      5. 22.5 Integro-Differential Equations
      6. 22.6 Solution of Partial Differential Equation
      7. 22.7 Evaluation of Integrals
      8. Exercises
    12. 23. The z-transform
      1. 23.1 Some Elementary Concepts
      2. 23.2 Definition of z-transform
      3. 23.3 Convergence of z-transform
      4. 23.4 Examples of z-transform
      5. 23.5 Properties of the z-transform
      6. 23.6 Inverse z-transform
      7. 23.7 Convolution Theorem
      8. 23.8 The Transfer Function (or System Function)
      9. 23.9 Systems Described by Difference Equations
      10. Exercises
    13. 24. Elements of Statistics and Probability
      1. 24.1 Measures of Central Tendency
      2. 24.2 Measures of Variability (Dispersion)
      3. 24.3 Measure of Skewness
      4. 24.4 Measures of Kurtosis
      5. 24.5 Covariance
      6. 24.6 Correlation and Coefficient of Correlation
      7. 24.7 Regression
      8. 24.8 Angle Between the Regression Lines
      9. 24.9 Probability
      10. 24.10 Conditional Probability
      11. 24.11 Independent Events
      12. 24.12 Probability Distribution
      13. 24.13 Mean and Variance of a Random Variable
      14. 24.14 Binomial Distribution
      15. 24.15 Pearson's Constants for Binomial Distribution
      16. 24.16 Poisson Distribution
      17. 24.17 Constants of the PoissonDistribution
      18. 24.18 Normal Distribution
      19. 24.19 Characteristics of the Normal Distribution
      20. 24.20 Normal Probability Integral
      21. 24.21 Areas Under the Standard Normal Curve
      22. 24.22 Fitting of Normal Distribution to a Given Data
      23. 24.23 Sampling
      24. 24.24 Level of Significance and Critical Region
      25. 24.25 Test of Significance for Large Samples
      26. 24.26 Confidence Interval for the Mean
      27. 24.27 Test of significance for Single Proportion
      28. 24.28 Test of Significance for Difference of Proportion
      29. 24.29 Test of Significance for Difference of Means
      30. 24.30 Test of Significance for the Difference of Standard Deviations
      31. 24.31 Sampling with Small Samples
      32. 24.32 Significance Test of Difference Between Sample Means
      33. 24.33 Chi-square Distribution
      34. 24.34 χ2-test as a Test of Goodness-of-Fit
      35. 24.35 Snedecor's F-Distribution
      36. 24.36 Fisher's Z-Distribution
      37. Exercises
    14. 25. Linear Programming
      1. 25.1 Linear Programming Problems
      2. 25.2 Formulation of a Linear Programming Problem (LPP)
      3. 25.3 Graphical Method to Solve Linear Programming Problem
      4. 25.4 Canonical and Standard forms of Linear Programming Problem
      5. 25.5 Basic Feasible Solution of an LPP
      6. 25.6 Simplex Method
      7. 25.7 Tabular form of the Solution
      8. 25.8 Generalization of Simplex Algorithm
      9. 25.9 Two-Phase Method
      10. 25.10 Duality Property
      11. 25.11 Dual Simplex Method
      12. 25.12 Transportation Problems
      13. 25.13 Matrix form of the Transportation Problem
      14. 25.14 Transportation Problem Table
      15. 25.15 Basic Initial Feasible Solution of Transportation Problem
      16. 25.16 Test for the Optimality of Basic Feasible Solution
      17. 25.17 Degeneracy in Transportation Problem
      18. 25.18 Unbalanced Transportation Problems
      19. Exercises
    15. 26. Basic Numerical Methods
      1. 26.1 Approximate Numbers and Significant Figures
      2. 26.2 Classical Theorems Used in Numerical Methods
      3. 26.3 Types of Errors
      4. 26.4 General Formula for Errors
      5. 26.5 Solution of Non-Linear Equations
      6. 26.6 Linear System of Equations
      7. 26.7 Finite Differences
      8. 26.8 Error Propagation
      9. 26.9 Interpolation
      10. 26.10 Interpolation With Unequal Spaced Points
      11. 26.11 Newton's Fundamental (Divided Difference) Formula
      12. 26.12 Lagrange's Interpolation Formula
      13. 26.13 Curve Fitting
      14. 26.14 Numerical Quadrature (Integration)
      15. 26.15 Ordinary Differential Equations
      16. 26.16 Numerical Solution of Partial Differential Equations
      17. Exercises
  9. Bibliography
  10. Acknowledgements
  11. Copyright