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Engineering Mathematics, Volume II, Second Edition

Book Description

Engineering Mathematics - II is meant for undergraduate engineering students. Considering the vast coverage of the subject, usually this paper is taught in three to four semesters. The two volumes in Engineering Mathematics by Babu Ram offer a complete solution to these papers.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Dedication
  5. Preface to the Revised Edition
  6. 1. Preliminaries
    1. 1.1 - Sets and Functions
    2. 1.2 - Continuous and Piecewise Continuous Functions
    3. 1.3 - Derivability of a Function and Piecwise Smooth Functions
    4. 1.4 - The Riemann Integral
    5. 1.5 - The Causal and Null Function
    6. 1.6 - Functions of Exponential Order
    7. 1.7 - Periodic Functions
    8. 1.8 - Even and Odd Functions
    9. 1.9 - Sequence and Series
    10. 1.10 - Series of Functions
    11. 1.11 - Partial Fraction Expansion of a Rational Function
    12. 1.12 - Special Functions
    13. 1.13 - The Integral Transforms
  7. 2. Linear Algebra
    1. 2.1 - Concepts of Group, Ring, and Field
    2. 2.2 - Vector Space
    3. 2.3 - Linear Transformation
    4. 2.4 - Linear Algebra
    5. 2.5 - Rank and Nullity of a Linear Transformation
    6. 2.6 - Matrix of a Linear Transformation
    7. 2.7 - Change-of-basis Matrix (Transforming Matrix or Transition Matrix)
    8. 2.8 - Relation Between Matrices of a Linear Transformation in Different Bases
    9. 2.9 - Normed Linear Space
    10. 2.10 - Inner Product Space
    11. 2.11 - Least Square Line Approximation
    12. 2.12 - Minimal Solution to a System of Equations
    13. 2.13 - Matrices
    14. 2.14 - Algebra of Matrices
    15. 2.15 - Multiplication of Matrices
    16. 2.16 - Associative Law for Matrix Multiplication
    17. 2.17 - Distributive Law for Matrix Multiplication
    18. 2.18 - Transpose of a Matrix
    19. 2.19 - Symmetric, Skew-Symmetric, and Hermitian Matrices
    20. 2.20 - Lower and Upper Triangular Matrices
    21. 2.21 - Determinants
    22. 2.22 - Adjoint of a Matrix
    23. 2.23 - The Inverse of a Matrix
    24. 2.24 - Methods of Computing Inverse of a Matrix
    25. 2.25 - Rank of a Matrix
    26. 2.26 - Elementary Matrices
    27. 2.27 - Row Reduced Echelon Form and Normal Form of Matrices
    28. 2.28 - Equivalence of Matrices
    29. 2.29 - Row Rank and Column Rank of a Matrix
    30. 2.30 - Solution of System of Linear Equations
    31. 2.31 - Solution of Non-Homogeneous Linear System of Equations
    32. 2.32 - Consistency Theorem
    33. 2.33 - Homogeneous Linear Equations
    34. 2.34 - Characteristic Roots and Characteristic Vectors
    35. 2.35 - The Cayley-Hamilton Theorem
    36. 2.36 - Algebraic and Geometric Multiplicity of an Eigen Value
    37. 2.37 - Minimal Polynomial of a Matrix
    38. 2.38 - Orthogonal, Normal and Unitary Matrices
    39. 2.39 - Similarity of Matrices
    40. 2.40 - Diagonalization of a Matrix
    41. 2.41 - Triangularization of an Arbitrary Matrix
    42. 2.42 - Quadratic Forms
    43. 2.43 - Diagonalization of Quadratic Forms
    44. 2.44 - Miscellaneous Examples
    45. Exercises
  8. 3. Functions of Complex Variables
    1. 3.1 - Basic Concepts
    2. 3.2 - De-Moivre's Theorem
    3. 3.3 - Logarithms of Complex Numbers
    4. 3.4 - Hyperbolic Functions
    5. 3.5 - Relations Between Hyperbolic and Circular Functions
    6. 3.6 - Periodicity of Hyperbolic Functions
    7. 3.7 - Some Basic Concepts
    8. 3.8 - Analytic Functions
    9. 3.9 - Integration of Complex-valued Functions
    10. 3.10 - Power Series Representation of an Analytic Function
    11. 3.11 - Zeros and Poles
    12. 3.12 - Residues and Cauchy's Residue Theorem
    13. 3.13 - Evaluation of Real Definite Integrals
    14. 3.14 - Conformal Mapping
    15. 3.15 - Miscellaneous Examples
    16. Exercises
  9. 4. Ordinary Differential Equations
    1. 4.1 - Definitions and Examples
    2. 4.2 - Formulation of Differential Equation
    3. 4.3 - Solution of Differential Equation
    4. 4.4 - Differential Equations of First Order
    5. 4.5 - Separable Equations
    6. 4.6 - Homogeneous Equations
    7. 4.7 - Equations Reducible to Homogeneous Form
    8. 4.8 - Linear Differential Equations of First Order and First Degree
    9. 4.9 - Equations Reducible to Linear Differential Equations
    10. 4.10 - Exact Differential Equation
    11. 4.11 - The Solution of Exact Differential Equation
    12. 4.12 - Equations Reducible to Exact Equation
    13. 4.13 - Applications of First Order and First Degree Equations
    14. 4.14 - Equations of First Order and Higher Degree
    15. 4.15 - Equations which can be Factorized into Factors of First Degree
    16. 4.16 - Equations which cannot be Factorized into Factors of First Degree
    17. 4.17 - Clairaut's Equation
    18. 4.18 - Higher Order Linear Differential Equations
    19. 4.19 - Solution of Homogeneous Linear Differential Equation with Constant Coefficients
    20. 4.20 - Complete Solution of Linear Differential Equation with Constant Coefficients
    21. 4.21 - Application of Linear Differential Equation
    22. 4.22 - Mass-Spring System
    23. 4.23 - Simple Pendulum
    24. 4.24 - Differential Equation with Variable Coefficients
    25. 4.25 - Method of Solution by Changing the Independent Variable
    26. 4.26 - Method of Solution by Changing the Dependent Variable
    27. 4.27 - Method of Undetermined Coefficients
    28. 4.28 - Method of Reduction of Order
    29. 4.29 - The Cauchy-Euler Homogeneous Linear Equation
    30. 4.30 - Legendre's Linear Equation
    31. 4.31 - Method of Variation of Parameters to Find Particular Integral
    32. 4.32 - Solution in Series
    33. 4.33 - Bessel's Equation and Bessel's Function
    34. 4.34 - Fourier-Bessel Expansion of a Continuous Function
    35. 4.35 - Legendre's Equation and Legendre's Polynomial
    36. 4.36 - Fourier-Legendre Expansion of a Function
    37. 4.37 - Miscellaneous Examples
    38. 4.38 - Simultaneous Linear Differential Equations with Constant Coefficient
    39. Exercises
  10. 5. Partial Differential Equations
    1. 5.1 - Formulation of Partial Differential Equation
    2. 5.2 - Solutions of a Partial Differential Equation
    3. 5.3 - Non-linear Partial Differential Equations of the First Order
    4. 5.4 - Charpit's Method
    5. 5.5 - Some Standard Forms of Non-linear Equations
    6. 5.6 - Linear Partial Differential Equations with Constant Coefficients
    7. 5.7 - Equations Reducible to Homogeneous Linear Form
    8. 5.8 - Classification of Second Order Linear Partial Differential Equations
    9. 5.9 - The Method of Separation of Variables
    10. 5.10 - Classical Partial Differential Equations
    11. 5.11 - Solutions of Laplace Equation
    12. 5.12 - Telephone Equations of a Transmission Line
    13. 5.13 - Miscellaneous Examples
    14. Exercises
  11. 6. Fourier Series
    1. 6.1 - Trigonometric Series
    2. 6.2 - Fourier (or Euler) Formulae
    3. 6.3 - Periodic Extension of a Function
    4. 6.4 - Fourier Cosine and Sine Series
    5. 6.5 - Complex Fourier Series
    6. 6.6 - Spectrum of Periodic Functions
    7. 6.7 - Properties of Fourier Coeffcients
    8. 6.8 - Dirichlet's Kernel
    9. 6.9 - Integral Expression for Partial Sums of a Fourier Series
    10. 6.10 - Fundamental Theorem (Convergence Theorem) of Fourier Series
    11. 6.11 - Applications of Fundamental Theorem of Fourier Series
    12. 6.12 - Convolution Theorem for Fourier Series
    13. 6.13 - Integration of Fourier Series
    14. 6.14 - Differentiation of Fourier Series
    15. 6.15 - Examples of Expansions of Functions in Fourier Series
    16. 6.16 - Method to Find Harmonics of Fourier Series of a Function from Tabular Values
    17. 6.17 - Signals and Systems
    18. 6.18 - Classification of Signals
    19. 6.19 - Classification of Systems
    20. 6.20 - Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input
    21. 6.21 - Application to Differential Equations
    22. 6.22 - Application to Partial Differential Equations
    23. 6.23 - Miscellaneous Examples
    24. Exercises
  12. 7. Fourier Transform
    1. 7.1 - Fourier Integral Theorem
    2. 7.2 - Fourier Transforms
    3. 7.3 - Fourier Cosine and Sine Transforms
    4. 7.4 - Properties of Fourier Transforms
    5. 7.5 - Solved Examples
    6. 7.6 - Complex Fourier Transforms
    7. 7.7 - Convolution Theorem
    8. 7.8 - Parseval's Identities
    9. 7.9 - Fourier Integral Representation of a Function
    10. 7.10 - Finite Fourier Transforms
    11. 7.11 - Applications of Fourier Transforms
    12. 7.12 - Application to Differential Equations
    13. 7.13 - Application to Partial Differential Equations
    14. Exercises
  13. 8. Discrete Fourier Transform
    1. 8.1 - Approximation of Fourier Coefficients of a Periodic Function
    2. 8.2 - Definition and Examples of DFT
    3. 8.3 - Inverse DFT
    4. 8.4 - Properties of DFT
    5. 8.5 - Cyclical Convolution and Convolution Theorem for DFT
    6. 8.6 - Parseval's Theorem for the DFT
    7. 8.7 - Matrix Form of the DFT
    8. 8.8 - N-point Inverse DFT
    9. 8.9 - Fast Fourier Transform (FFT)
    10. Exercises
  14. 9. Laplace Transform
    1. 9.1 - Definition and Examples of Laplace Transform
    2. 9.2 - Properties of Laplace Transforms
    3. 9.3 - Limiting Theorems
    4. 9.4 - Miscellaneous Examples
    5. Exercises
  15. 10. Inverse Laplace Transform
    1. 10.1 - Definition and Examples of Inverse Laplace Transform
    2. 10.2 - Properties of Inverse Laplace Transform
    3. 10.3 - Partial Fractions Method to Find Inverse Laplace Transform
    4. 10.4 - Heaviside's Expansion Theorem
    5. 10.5 - Series Method to Determine Inverse Laplace Transform
    6. 10.6 - Convolution Theorem
    7. 10.7 - Complex Inversion Formula
    8. 10.8 - Miscellaneous Examples
    9. Exercises
  16. 11. Applications of Laplace Transform
    1. 11.1 - Ordinary Differential Equations
    2. 11.2 - Simultaneous Differential Equations
    3. 11.3 - Difference Equations
    4. 11.4 - Integral Equations
    5. 11.5 - Integro-differential Equations
    6. 11.6 - Solution of Partial Differential Equation
    7. Exercises
  17. 12. The Z-transform
    1. 12.1 - Some Elementary Concepts
    2. 12.2 - Definition of Z -transform
    3. 12.3 - Convergence of Z-transform
    4. 12.4 - Examples of Z-transform
    5. 12.5 - Properties of the Z-transform
    6. 12.6 - Inverse Z-transform
    7. 12.7 - Convolution Theorem
    8. 12.8 - The Transfer Function (or System Function)
    9. 12.9 - Systems Described by Difference Equations
    10. Exercises
  18. 13. Elements of Statistics and Probability
    1. 13.1 - Introduction
    2. 13.2 - Measures of Central Tendency
    3. 13.3 - Measures of Variability (Dispersion)
    4. 13.4 - Measure of Skewness
    5. 13.5 - Measures of Kurtosis
    6. 13.6 - Covariance
    7. 13.7 - Correlation and Coefficient of Correlation
    8. 13.8 - Regression
    9. 13.9 - Angle Between the Regression Lines
    10. 13.10 - Probability
    11. 13.11 - Conditional Probability
    12. 13.12 - Independent Events
    13. 13.13 - Probability Distribution
    14. 13.14 - Mean and Variance of a Random Variable
    15. 13.15 - Binomial Distribution
    16. 13.16 - Pearson's Constants for Binomial Distribution
    17. 13.17 - Poisson Distribution
    18. 13.18 - Constants of the Poisson Distribution
    19. 13.19 - Normal Distribution
    20. 13.20 - Characteristics of the Normal Distribution
    21. 13.21 - Normal Probability Integral
    22. 13.22 - Areas Under the Standard Normal Curve
    23. 13.23 - Fitting of Normal Distribution to a Given Data
    24. 13.24 - Sampling
    25. 13.25 - Level of Significance and Critical Region
    26. 13.26 - Test of Significance for Large Samples
    27. 13.27 - Confidence Interval for the Mean
    28. 13.28 - Test of Significance for Single Proportion
    29. 13.29 - Test of Significance for Difference of Proportion
    30. 13.30 - Test of Significance for Difference of Means
    31. 13.31 - Test of Significance for the Difference of Standard Deviations
    32. 13.32 - Sampling with Small Samples
    33. 13.33 - Significance Test of Difference Between Sample Means
    34. 13.34 - Chi-square Distribution
    35. 13.35 - X2-Test as a Test of Goodness-of-fit
    36. 13.36 - Snedecor's F-distribution
    37. 13.37 - Fisher's Z-distribution
    38. 13.38 - Miscellaneous Examples
    39. Exercises
  19. 14. Linear Programming
    1. 14.1 - Linear Programming Problems
    2. 14.2 - Formulation of an LPP
    3. 14.3 - Graphical Method to Solve LPP
    4. 14.4 - Canonical and Standard Forms of LPP
    5. 14.5 - Basic Feasible Solution of an LPP
    6. 14.6 - Simplex Method
    7. 14.7 - Tabular Form of the Solution
    8. 14.8 - Generalization of Simplex Algorithm
    9. 14.9 - Two-phase Method
    10. 14.10 - Duality Property
    11. 14.11 - Dual Simplex Method
    12. 14.12 - Transportation Problems
    13. 14.13 - Matrix Form of the Transportation Problem
    14. 14.14 - Transportation Problem Table
    15. 14.15 - Basic Initial Feasible Solution of Transportation Problem
    16. 14.16 - Test for the Optimality of Basic Feasible Solution
    17. 14.17 - Degeneracy in Transportation Problem
    18. 14.18 - Unbalanced Transportation Problems
    19. Exercises
  20. 15. Basic Numerical Methods
    1. 15.1 - Approximate Numbers and Significant Figures
    2. 15.2 - Classical Theorems used in Numerical Methods
    3. 15.3 - Types of Errors
    4. 15.4 - General Formula for Errors
    5. 15.5 - Solution of Non-linear Equations
    6. 15.6 - Linear System of Equations
    7. 15.7 - Finite Differences
    8. 15.8 - Error Propagation
    9. 15.9 - Interpolation
    10. 15.10 - Interpolation with Unequal Spaced Points
    11. 15.11 - Newton's Fundamental (Divided Difference) Formula
    12. 15.12 - Lagrange's Interpolation Formula
    13. 15.13 - Curve Fitting
    14. 15.14 - Numerical Quadrature
    15. 15.15 - Ordinary Differential Equations
    16. 15.16 - Numerical Solution of Partial Differential Equations
    17. Exercises
  21. Statistical Tables
  22. Copyright