You are previewing Mathematical Methods.
O'Reilly logo
Mathematical Methods

Book Description

Mathematics lays the basic foundation for engineering students to pursue their core subjects. Mathematical Methodscovers topics on matrices, linear systems of equations, eigen values, eigenvectors, quadratic forms, Fourier series, partial differential equations, Z-transforms, numerical methods of solutions of equation, differentiation, integration and numerical solutions of ordinary differential equations. The book features numerical solutions of algebraic and transcendental equations by iteration, bisection, Newton - Raphson methods; the numerical methods include cubic spline method, Runge-Kutta methods and Adams-Bashforth - Moulton methods; applications to one-dimensional heat equations, wave equations and Laplace equations; clear concepts of classifiable functions—even and odd functions—in Fourier series; exhaustive coverage of LU decomposition—tridiagonal systems in solutions of linear systems of equations; over 900 objective-type questions that include multiple choice questions fill in the blanks match the following and true or false statements and the atest University model question papers with solutions.

Table of Contents

  1. Cover
  2. Title page
  3. Contents
  4. About the Author
  5. Dedication
  6. Preface
  7. Chapter 1. Matrices and Linear Systems of Equations
    1. 1.1 Introduction
    2. 1.2 Algebra of Matrices
    3. 1.3 Matrix Multiplication
    4. 1.4 Determinant of a Square Matrix
    5. 1.5 Related Matrices
    6. 1.6 Determinant-Related Matrices
    7. 1.7 Special Matrices
    8. Exercise 1.1
    9. 1.8 Linear Systems of Equations
    10. 1.9 Homogeneous (H) and Nonhomogeneous (NH) Systems of Equations
    11. 1.10 Elementary Row and Column Operations (Transformations) for Matrices
    12. Exercise 1.2
    13. 1.11 Inversion of a Nonsingular Matrix
    14. Exercise 1.3
    15. 1.12 Rank of a Matrix
    16. 1.13 Methods for Finding the Rank of a Matrix
    17. Exercise 1.4
    18. 1.14 Existence and Uniqueness of Solutions of a System of Linear Equations
    19. 1.15 Methods of Solution of NH and H Equations
    20. 1.16 Homogeneous System of Equations (H)
    21. Exercise 1.5
  8. Chapter 2. Eigenvalues and Eigenvectors
    1. 2.1 Introduction
    2. 2.2 Linear Transformation
    3. 2.3 Characteristic Value Problem
    4. Exercise 2.1
    5. 2.4 Properties of Eigenvalues and Eigenvectors
    6. 2.5 Cayley–Hamilton Theorem
    7. Exercise 2.2
    8. 2.6 Reduction of a Square Matrix to Diagonal Form
    9. 2.7 Powers of a Square Matrix A—Finding of Modal Matrix P and Inverse Matrix A−1
    10. Exercise 2.3
  9. Chapter 3. Real and Complex Matrices
    1. 3.1 Introduction
    2. 3.2 Orthogonal/Orthonormal System of Vectors
    3. 3.3 Real Matrices
    4. Exercise 3.1
    5. 3.4 Complex Matrices
    6. 3.5 Properties of Hermitian, Skew-Hermitian and Unitary Matrices
    7. Exercise 3.2
  10. Chapter 4. Quadratic Forms
    1. 4.1 Introduction
    2. 4.2 Quadratic Forms
    3. 4.3 Canonical Form (or) Sum of the Squares Form
    4. 4.4 Nature of Real Quadratic Forms
    5. 4.5 Reduction of a Quadratic Form to Canonical Form
    6. 4.6 Sylvestor's Law of Inertia
    7. 4.7 Methods of Reduction of a Quadratic Form to a Canonical Form
    8. Exercise 4.1
  11. Chapter 5. Solution of Algebraic and Transcendental Equations
    1. 5.1 Introduction to Numerical Methods
    2. 5.2 Errors and their Computation
    3. 5.3 Formulas for Errors
    4. 5.4 Mathematical Pre-Requisites
    5. 5.5 Solution of Algebraic and Transcendental Equations
    6. 5.6 Direct Methods of Solution
    7. 5.7 Numerical Methods of Solution of Equations of the Form f(x) = 0
    8. Exercise 5.1
  12. Chapter 6. Interpolation
    1. 6.1 Introduction
    2. 6.2 Interpolation with Equal Intervals
    3. 6.3 Symbolic Relations and Separation of Symbols
    4. Exercise 6.1
    5. 6.4 Interpolation
    6. 6.5 Interpolation Formulas For Equal Intervals
    7. Exercise 6.2
    8. 6.6 Interpolation with Unequal Intervals
    9. 6.7 Properties Satisfied by ∆'
    10. 6.8 Divided Difference Interpolation Formula
    11. 6.9 Inverse Interpolation Using Lagrange's Interpolation Formula
    12. 6.10 Central Difference Formulas
    13. Exercise 6.3
  13. Chapter 7. Curve Fitting
    1. 7.1 Introduction
    2. 7.2 Curve Fitting by the Method of Least Squares
    3. 7.3 Curvilinear (or Nonlinear) Regression
    4. 7.4 Curve fitting by a Sum of Exponentials
    5. 7.5 Weighted Least Squares Approximation
    6. Exercise 7.1
  14. Chapter 8. Numerical Differentiation and Integration
    1. 8.1 Introduction
    2. 8.2 Errors in Numerical Differentiation
    3. 8.3 Maximum and Minimum Values of a Tabulated Function
    4. Exercise 8.1
    5. 8.4 Numerical Integration: Introduction
    6. Exercise 8.2
    7. 8.5 Cubic Splines
    8. Exercise 8.3
  15. Chapter 9. Numerical Solution of Ordinary Differential Equations
    1. 9.1 Introduction
    2. 9.2 Methods of Solution
    3. 9.3 Predictor–Corrector Methods
    4. Exercise 9.1
  16. Chapter 10. Fourier Series
    1. 10.1 Introduction
    2. 10.2 Periodic Functions, Properties
    3. 10.3 Classifiable Functions—Even and Odd Functions
    4. 10.4 Fourier Series, Fourier Coefficients and Euler's Formulae in (α, α + 2π)
    5. 10.5 Dirichlet's Conditions for Fourier Series Expansion of a Function
    6. 10.6 Fourier Series Expansions: Even/Odd Functions
    7. 10.7 Simply-Defined and Multiply-(Piecewise) Defined Functions
    8. Exercise 10.1
    9. 10.8 Change of Interval: Fourier Series in Interval (α, α + 2l)
    10. Exercise 10.2
    11. 10.9 Fourier Series Expansions of Even and Odd Functions in (–l, l)
    12. Exercise 10.3
    13. 10.10 Half-Range Fourier Sine/Cosine Series: Odd and Even Periodic Continuations
    14. Exercise 10.4
    15. 10.11 Root Mean Square (RMS) Value of a Function
    16. Exercise 10.5
  17. Chapter 11. Fourier Integral Transforms
    1. 11.1 Introduction
    2. 11.2 Integral Transforms
    3. 11.3 Fourier Integral Theorem
    4. 11.4 Fourier Integral in Complex Form
    5. 11.5 Fourier Transform of f(x)
    6. 11.6 Finite Fourier Sine Transform and Finite Fourier Cosine Transform (FFCT)
    7. 11.7 Convolution Theorem for Fourier Transforms
    8. 11.8 Properties of Fourier Transform
    9. Exercise 11.1
    10. 11.9 Parseval's Identity for Fourier Transforms
    11. 11.10 Parseval's Identities for Fourier Sine and Cosine Transforms
    12. Exercise 11.2
  18. Chapter 12. Partial Differential Equations
    1. 12.1 Introduction
    2. 12.2 Order, Linearity and Homogeneity of a Partial Differential Equation
    3. 12.3 Origin of Partial Differential Equation
    4. 12.4 Formation of Partial Differential Equation by Elimination of Two Arbitrary Constants
    5. Exercise 12.1
    6. 12.5 Formation of Partial Differential Equations by Elimination of Arbitrary Functions
    7. Exercise 12.2
    8. 12.6 Classification of First-Order Partial Differential Equations
    9. 12.7 Classification of Solutions of First-Order Partial Differential Equation
    10. 12.8 Equations Solvable by Direct Integration
    11. Exercise 12.3
    12. 12.9 Quasi-Linear Equations of First Order
    13. 12.10 Solution of Linear, Semi-Linear and Quasi-Linear Equations
    14. Exercise 12.4
    15. 12.11 Nonlinear Equations of First Order
    16. Exercise 12.5
    17. 12.12 Euler's Method of Separation of Variables
    18. Exercise 12.6
    19. 12.13 Classification of Second-order Partial Differential Equations
    20. Exercise 12.7
    21. Exercise 12.8
    22. Exercise 12.9
  19. Chapter 13. Z-Transforms and Solution of Difference Equations
    1. 13.1 Introduction
    2. 13.2 Z-Transform: Definition
    3. 13.3 Z-Transforms of Some Standard Functions (Special Sequences)
    4. 13.4 Recurrence Formula for the Sequence of a Power of Natural Numbers
    5. 13.5 Properties of Z-Transforms
    6. Exercise 13.1
    7. 13.6 Inverse Z-Transform
    8. Exercise 13.2
    9. 13.7 Application of Z-Transforms: Solution of a Difference Equation; by Z-Transform
    10. 13.8 Method for Solving a Linear Difference Equation with Constant Coefficients
    11. Exercise 13.3
  20. Question Bank
    1. Multiple Choice Questions
    2. Fill in the Blanks
    3. Match the Following
    4. True or False Statements
  21. Solved Question Papers
  22. Bibliography
  23. Notes
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 5
    5. Chapter 6
    6. Chapter 7
    7. Chapter 8
    8. Chapter 9
    9. Chapter 10
    10. Chapter 12
  24. Acknowledgements
  25. Copyright