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Engineering Mathematics, Volume 2

Book Description

Designed for the core papers Engineering Mathematics II and III, which students take up across the second and third semesters, Engineering Mathematics Volume-II offers detailed theory with a wide variety of solved examples with reference to engineering applications, along with over 1,000 objective-type questions that include multiple choice questions, fill in the blanks, match the following and true or false statements.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. About the Author
  5. Dedication
  6. Preface
  7. 1. Eigenvalues and Eigenvectors
    1. 1.1 Introduction
    2. 1.2 Linear Transformation
    3. 1.3 Characteristic Value Problem
      1. Exercise 1.1
    4. 1.4 Properties of Eigenvalues and Eigenvectors
    5. 1.5 Cayley-Hamilton Theorem
      1. Exercise 1.2
    6. 1.6 Reduction of a Square Matrix to Diagonal Form
    7. 1.7 Powers of a Square Matrix A— Finding of Modal Matrix P and Inverse Matrix A-1
      1. Exercise 1.3
  8. 2. Quadratic Forms
    1. 2.1 Introduction
    2. 2.2 Quadratic Forms
    3. 2.3 Canonical Form (or) Sum of the Squares Form
    4. 2.4 Nature of Real Quadratic Forms
    5. 2.5 Reduction of a Quadratic Form to Canonical Form
    6. 2.6 Sylvestor’s Law of Inertia
    7. 2.7 Methods of Reduction of a Quadratic Form to a Canonical Form
    8. 2.8 Singular Value Decomposition of a Matrix
      1. Exercise 2.1
  9. 3. Solution of Algebraic and Transcendental Equations
    1. 3.1 Introduction to Numerical Methods
    2. 3.2 Errors and their Computation
    3. 3.3 Formulas for Errors
    4. 3.4 Mathematical Pre-requisites
    5. 3.5 Solution of Algebraic and Transcendental Equations
    6. 3.6 Direct Methods of Solution
    7. 3.7 Numerical Methods of Solution of Equations of the Form f(x) = 0
      1. Exercise 3.1
  10. 4. Interpolation
    1. 4.1 Introduction
    2. 4.2 Interpolation with Equal Intervals
    3. 4.3 Symbolic Relations and Separation of Symbols
      1. Exercise 4.1
    4. 4.4 Interpolation
    5. 4.5 Interpolation Formulas for Equal Intervals
      1. Exercise 4.2
    6. 4.6 Interpolation with Unequal Intervals
    7. 4.7 Properties Satisfied by Δ ′
    8. 4.8 Divided Difference Interpolation Formula
    9. 4.9 Inverse Interpolation Using Lagrange’s Interpolation Formula
    10. 4.10 Central Difference Formulas
      1. Exercise 4.3
  11. 5. Curve Fitting
    1. 5.1 Introduction
    2. 5.2 Curve Fitting by the Method of Least Squares
    3. 5.3 Curvilinear (or Nonlinear) Regression
    4. 5.4 Curve Fitting by a Sum of Exponentials
    5. 5.5 Weighted Least Squares Approximation
      1. Exercise 5.1
  12. 6. Numerical Differentiation and Integration
    1. 6.1 Introduction
    2. 6.2 Errors in Numerical Differentiation
    3. 6.3 Maximum and Minimum Values of a Tabulated Function
      1. Exercise 6.1
    4. 6.4 Numerical Integration: Introduction
      1. Exercise 6.2
    5. 6.5 Cubic Splines
    6. 6.6 Gaussian Integration
      1. Exercise 6.3
  13. 7. Numerical Solution of Ordinary Differential Equations
    1. 7.1 Introduction
    2. 7.2 Methods of Solution
    3. 7.3 Predictor-Corrector Methods
      1. Exercise 7.1
  14. 8. Fourier Series
    1. 8.1 Introduction
    2. 8.2 Periodic Functions, Properties
    3. 8.3 Classifiable Functions—Even and Odd Functions
    4. 8.4 Fourier Series, Fourier Coefficients and Euler’s Formulae in (a, a +2 π)
    5. 8.5 Dirichlet’s Conditions for Fourier Series Expansion of a Function
    6. 8.6 Fourier Series Expansions: Even/Odd Functions
    7. 8.7 Simply-Defined and Multiply-(Piecewise) Defined Functions
      1. Exercise 8.1
    8. 8.8 Change of Interval: Fourier Series in Interval (a, a + 2l) :
      1. Exercise 8.2
    9. 8.9 Fourier Series Expansions of Even and Odd Functions in (−l, l )
      1. Exercise 8.3
    10. 8.10 Half-Range Fourier Sine/Cosine Series: Odd and Even Periodic Continuations
      1. Exercise 8.4
    11. 8.11 Root Mean Square (RMS) Value of a Function
      1. Exercise 8.5
  15. 9. Fourier Integral Transforms
    1. 9.1 Introduction
    2. 9.2 Integral Transforms
    3. 9.3 Fourier Integral Theorem
    4. 9.4 Fourier Integral in Complex Form
    5. 9.5 Fourier Transform of f (x)
    6. 9.6 Finite Fourier Sine Transform and Finite Fourier Cosine Transform (FFCT)
    7. 9.7 Convolution Theorem for Fourier Transforms
    8. 9.8 Properties of Fourier Transform
      1. Exercise 9.1
    9. 9.9 Parseval’s Identity for Fourier Transforms
    10. 9.10 Parseval’s Identities for Fourier Sine and Cosine Transforms
      1. Exercise 9.2
  16. 10. Partial Differential Equations
    1. 10.1 Introduction
    2. 10.2 Order, Linearity and Homogeneity of a Partial Differential Equation
    3. 10.3 Origin of Partial Differential Equation
    4. 10.4 Formation of Partial Differential Equation by Elimination of Two Arbitrary Constants
      1. Exercise 10.1
    5. 10.5 Formation of Partial Differential Equations by Elimination of Arbitrary Functions
      1. Exercise 10.2
    6. 10.6 Classification of First-Order Partial Differential Equations
    7. 10.7 Classification of Solutions of First-Order Partial Differential Equation
    8. 10.8 Equations Solvable by Direct Integration
      1. Exercise 10.3
    9. 10.9 Quasi-Linear Equations of First Order 0.10 Solution of Linear, Semi-Linear and Quasi-Linear Equations
      1. Exercise 10.4
    10. 10.11 Nonlinear Equations of First Order
      1. Exercise 10.5
    11. 10.12 Euler’s Method of Separation of Variables
      1. Exercise 10.6
    12. 10.13 Classification of Second-Order Partial Differential Equations
      1. Exercise 10.7
      2. Exercise 10.8
    13. 10.14 Two-dimensional Wave Equation
      1. Exercise 10.9
  17. 11. Z-Transforms and Solution of Difference Equations
    1. 11.1 Introduction
    2. 11.2 Z-Transform: Definition
    3. 11.3 Z-Transforms of Some Standard Functions (Special Sequences)
    4. 11.4 Recurrence Formula for the Sequence of a Power of Natural Numbers
    5. 11.5 Properties of Z-Transforms
      1. Exercise 11.1
    6. 11.6 Inverse Z-Transform
      1. Exercise 11.2
    7. 11.7 Application of Z-Transforms: Solution of a Difference Equations by Z-Transform
    8. 11.8 Method for Solving a Linear Difference Equation with Constant Coefficients
      1. Exercise 11.3
  18. 12. Special Functions
    1. 12.1 Introduction
    2. 12.2 Gamma Function
    3. 12.3 Recurrence Relation or Reduction Formula
    4. 12.4 Various Integral Forms of Gamma Function
      1. Exercise 12.1
    5. 12.5 Beta Function
    6. 12.6 Various Integral Forms of Beta Function
    7. 12.7 Relation Between Beta and Gamma Functions
    8. 12.8 Multiplication Formula
    9. 12.9 Legendre’s Duplication Formula
      1. Exercise 12.2
    10. 12.10 Legendre Functions
      1. Exercise 12.3
    11. 12.11 Bessel Functions
      1. Exercise 12.4
      2. Exercise 12.5
  19. 13. Functions of a Complex Variable
    1. 13.1 Introduction
    2. 13.2 Complex Numbers-Complex Plane
      1. Exercise 13.1
      2. Exercise 13.2
    3. 13.3 Laplace’s Equation: Harmonic and Conjugate Harmonic Functions
      1. Exercise 13.3
  20. 14. Elementary Functions
    1. 14.1 Introduction
    2. 14.2 Elementary Functions of a Complex Variable
      1. Exercise 14.1
  21. 15. Complex Integration
    1. 15.1 Introduction
    2. 15.2 Basic Concepts
    3. 15.3 Complex Line Integral
    4. 15.4 Cauchy-Goursat Theorem
    5. 15.5 Cauchy’s Theorem for Multiply-Connected Domain
    6. 15.6 Cauchy’s Integral Formula (C.I.F.) or Cauchy’s Formula Theorem
    7. 15.7 Morera’s Theorem (Converse of Cauchy’s Theorem)
    8. 15.8 Cauchy’s Inequality
      1. Exercise 15.1
  22. 16. Complex Power Series
    1. 16.1 Introduction
    2. 16.2 Sequences and Series
    3. 16.3 Power Series
    4. 16.4 Series of Complex Functions
    5. 16.5 Uniform Convergence of a Series of Functions
    6. 16.6 Weierstrass’s M-Test
    7. 16.7 Taylor’s Theorem(Taylor Series)
    8. 16.8 Laurent’s Series
    9. 16.9 Higher Derivatives of Analytic Functions
      1. Exercise 16.1
  23. 17. Calculus of Residues
    1. 17.1 Evaluation of Real Integrals
      1. Exercise 17.1
      2. Exercise 17.2
      3. Exercise 17.3
  24. 18. Argument Principle and Rouche’s Theorem
    1. 18.1 Introduction
    2. 18.2 Meromorphic Function
    3. 18.3 Argument Principle (Repeated Single Pole/Zero)
    4. 18.4 Generalised Argument Theorem
    5. 18.5 Rouche’s Theorem
    6. 18.6 Liouville Theorem
    7. 18.7 Fundamental Theorem of Algebra
    8. 18.8 Maximum Modulus Theorem for Analytic Functions
      1. Exercise 18.1
  25. 19. Conformal Mapping
    1. 19.1 Introduction
    2. 19.2 Conformal Mapping: Conditions for Conformality
    3. 19.3 Conformal Mapping by Elementary Functions
    4. 19.4 Some Special Transformations
    5. 19.5 Bilinear or Mobius or Linear Fractional Transformations
    6. 19.6 Fixed Points of the Transformation
      1. Exercise 19.1
  26. Notes
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 10
    10. Chapter 12
    11. Chapter 13
    12. Chapter 15
    13. Chapter 16
    14. Chapter 18
    15. Chapter 19
  27. Question Bank
  28. Acknowldegements
  29. Copyright