Engineering Mathematics

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Cover Contents Preface Symbols and Basic Formulae Part I Chapter 1: Sequences and Series 1.1 Sequences 1.2 Convergence of Sequences 1.3 The Upper and Lower Limits of a Sequence 1.4 Cauchy’s Principle of Convergence 1.5 Monotonic Sequence 1.6 Theorems on Limits 1.7 Subsequence 1.8 Series 1.9 Comparison Tests 1.10 D’ Alembert’s Ratio Test 1.11 Cauchy’s Root Test 1.12 Raabe’s Test 1.13 Logarithmic Test 1.14 De Morgan – Bertrand Test 1.15 Gauss’s Test 1.16 Cauchy’s Integral Test 1.17 Cauchy’s Condensation Test 1.18 Kummer’s Test 1.19 Alternating Series 1.20 Absolute Convergence of a Series 1.21 Convergence of the Series of the Type 1.22 Derangement of Series 1.23 Nature of Non-absolutely Convergent Series 1.24 Effect of Derangement of Non-absolutely Convergent Series 1.25 Uniform Convergence 1.26 Uniform Convergence of a Series of Functions 1.27 Properties of Uniformly Convergent Series Exercises Chapter 2: Mean Value Theorems and Expansion of Function 2.1 Leibnitz’s Theorem and its Applications 2.2 General Theorems 2.3 Taylor’s Infinite Series and Power Series Expansion 2.4 Maclaurin’s Infinite Series 2.5 Expansion of Functions 2.6 Indeterminate forms Exercises Chapter 3: Curvature 3.1 Radius of Curvature of Intrinsic Curves 3.2 Radius of Curvature for Cartesian Curves 3.3 Radius of Curvature for Parametric Curves 3.4 Radius of Curvature for Pedal Curves 3.5 Radius of Curvature for Polar Curves 3.5.1 Second Method 3.6 Radius of Curvature at the Origin 1. Newton’s Method 2. Method of Expansion 3.7 Centre of Curvature 3.8 Evolutes and Involutes 3.9 Equation of the Circle of Curvature 3.10 Chords of Curvature Parallel to the Coordinate Axes 3.11 Chord of Curvature in Polar Coordinates 1. Chord of Curvature Through the Pole (Origin) 2. Chord of Curvature Perpendicular to the Radius Vector Exercises Chapter 4: Asymptotes and Curve Tracing 4.1 Determination of Asymptotes when the Equation of the Curve in Cartesian form is Given 4.2 The Asymptotes of the General Rational Algebraic Curve 4.3 Asymptotes parallel to the Coordinate Axes (i) Asymptotes Parallel to y-axis of a Rational Algebraic Curve (ii) Asymptotes Parallel to the x-axis of a Rational Algebraic Curve 4.4 Working Rule for Finding Asymptotes of Rational Algebraic Curve 4.5 Intersection of a Curve and its Asymptotes 4.6 Asymptotes by Expansion 4.7 Asymptotes of the Polar Curves 4.8 Circular Asymptotes 4.9 Curve Tracing (Cartesian Equations) 4.10 Curve Tracing (Polar Equations) 4.11 Curve Tracing (Parametric Equations) Exercises Chapter 5: Partial Differentiation 5.1 Continuity of a Function of Two Variables 5.2 Differentiability of a Function of Two Variables 5.3 The Differential Coefficients 5.4 Distinction Between Derivatives and Differential Coefficients 5.5 Higher-Order Partial Derivatives 5.6 Envelopes and Evolutes 5.7 Homogeneous Functions and Euler’s Theorem 5.8 Differentiation of Composite Functions 5.9 Transformation From Cartesian to Polar Coordinates and Vice Versa 5.10 Taylor’s Theorem for Functions of Several Variables 5.11 Extreme Values 5.12 Lagrange’s Method of Undetermined Multipliers 5.13 Jacobians 5.14 Properties of Jacobian 5.15 Necessary and Sufficient Conditions for Jacobian to Vanish 5.16 Differentiation Under the Integral Sign Exercises Chapter 6: Beta and Gamma Functions 6.1 Beta Function 6.2 Properties of Beta Function 6.3 Gamma Function 6.4 Properties of Gamma Function 6.5 Relation Between Beta and Gamma Functions 6.6 Dirichlet’s and Liouville’s Theorems Exercises Chapter 7: Reduction Formulas 7.1 Reduction Formulas for R R sinn xdx and Cosn xdx 7.2 Reduction Formulas for R sinm x cosn x dx 7.3 Reduction Formulas for R R tann xdx and Secn xdx 7.4 Reduction Formulas for R R xn sin mxdx and xn cos mxdx 7.5 Reduction Formulas for R R xneax dx and xm ðlog xÞn dx 7.6 Reduction Formula for R cosm x sin nxdx Exercises Chapter 8: Volumes and Surfaces of Solids of Revolution 8.1 Volume of the solid of Revolution (Cartesian Equations) 8.2 Volume of the Solid of Revolution (Parametric Equations) 8.3 Volume of the Solid of Revolution (Polar Curves) 8.4 Surface of the Solid of Revolution (Cartesian Equations) 8.5 Surface of the Solid of Revolution (Parametric Equations) 8.6 Surface of the Solid of Revolution (Polar Curves) Exercises Chapter 9: Multiple Integrals 9.1 Double Integrals 9.2 Properties of a Double Integral 9.3 Evaluation of Double Integrals (Cartesian Coordinates) 9.4 Evaluation of Double Integrals (Polar Coordinates) 9.5 Change of Variables in a Double Integral 9.6 Change of Order of Integration 9.7 Area Enclosed by Plane Curves (Cartesian and Polar Coordinates) 9.8 Volume and Surface Area as Double Integrals 9.9 Triple Integrals and their Evaluation Evaluation of Triple Integrals 9.10 Change to Spherical Polar Coordinates from Cartesian Coordinates in a Triple Integral 9.11 Volume as a Triple Integral Exercises Chapter 10: Vector Calculus 10.1 Differentiation of a Vector Unit Tangent Vector to a Curve The Condition is Necessary The Condition is Sufficient Velocity and Acceleration Tangential and Normal Acceleration Radial and Transverse Acceleration of a Moving Particle 10.2 Partial Derivatives of a Vector Function 10.3 Gradient of a Scalar Field 10.4 Geometrical Interpretation of a Gradient 10.5 Properties of a Gradient 10.6 Directional Derivatives 10.6.1 Directional Derivatives Along Coordinate Axes 10.7 Divergence of a Vector-Point Function 10.8 Physical Interpretation of Divergence 10.9 Curl of a Vector-Point Function 10.10 Physical Interpretation of Curl 10.11 The Laplacian Operator r2 10.12 Properties of Divergence and Curl 10.13 Integration of Vector Functions 10.14 Line Integral 10.15 Work Done by a Force 10.16 Surface Integral 10.17 Volume Integral 10.18 Gauss’s Divergence Theorem 10.19 Green’s Theorem in a Plane 10.20 Stoke’s Theorem Exercises Chapter 11: Three-Dimensional Geometry 11.1 Coordinate Planes 11.2 Distance Between Two Points 11.3 Direction Ratios and Direction Cosines of a Line 11.4 Section Formulae—Internal Division of a Line by a Point on the Line 11.4.1 External Division of a Line by a Point on the Extended Line 11.5 Straight Line in Three Dimensions 11.5.1 Collinearity of Three Points in Space 11.6 Angle Between Two Lines 11.7 Shortest Distance Between Two Skew Lines 11.8 Equation of a Plane 11.9 Equation of a Plane Passing Through a Given Point and Perpendicular to a Given Direction 11.10 Equation of a Plane Passing Through Three Points 11.11 Equation of a Plane Passing Through a Point and Parallel to Two Given Vectors 11.12 Equation of a Plane Passing Through Two Points and Parallel to a Line 11.13 Angle Between Two Planes 11.14 Angle Between a Line and a Plane 11.15 Perpendicular Distance of a Point From a Plane 11.16 Planes Bisecting the Angles Between Two Planes 11.17 Intersection of Planes 11.18 Planes Passing Through the Intersection of Two Given Planes 11.19 Sphere 11.20 Equation of a Sphere Whose Diameter is the Line Joining Two Given Points 11.21 Equation of a Sphere Passing Through Four Points 11.22 Equation of the Tangent Plane to a Sphere 11.23 Condition of Tangency 11.24 Angle of Intersection of Two Spheres 11.25 Condition of Orthogonality of Two Spheres 11.26 Cylinder 11.27 Equation of a Cylinder with given Axis and Guiding Curves 11.28 Right Circular Cylinder 11.29 Cone 11.30 Equation of a Cone with Vertex at the Origin 11.31 Equation of a Cone with Given Vertex and Guiding Curve 11.32 Right Circular Cone 11.33 Right Circular Cone with Vertex ða; b; cÞ, Semi Vertical Angle h, and ðl;m; nÞ the Direction Cosines of the Axis 11.34 Conicoids 11.35 Shape of an Ellipsoid 11.36 Shape of the Hyperboloid of One Sheet 11.37 Shape of the Hyperboloid of Two Sheets 11.38 Shape of the Elliptic Cone 11.39 Intersection of a Conicoid and a Line 11.40 Tangent Plane at a Point of Central Conicoid 11.41 Condition of Tangency 11.42 Equation of Normal to the Central Conicoid at Any Point (a; b; c) On It Exercises Part II Chapter 12: Preliminaries 12.1 Sets and Functions 12.2 Continuous and Piecewise Continuous Functions 12.3 Derivability of a Function and Piecewise Smooth Functions 12.4 The Riemann Integral 12.5 The Causal and Null Functions 12.6 Functions of Exponential Order 12.7 Periodic Functions 12.8 Even and Odd Functions 12.9 Sequence and Series 12.10 Series of Functions 12.11 Partial Fraction Expansion of a Rational Function 12.12 Special Functions 12.13 The Integral Transforms Chapter 13: Linear Algebra 13.1 Concepts of Group, Ring, and Field 13.2 Vector Space 13.3 Linear Transformation 13.4 Linear Algebra 13.5 Rank and Nullity of a Linear Transformation 13.6 Matrix of a Linear Transformation 13.7 Normed Linear Space 13.8 Inner Product Space 13.9 Matrices 13.10 Algebra of Matrices 13.11 Multiplication of Matrices 13.12 Associtative Law for Matrix Multiplication 13.13 Distributive Law for Matrix Multiplication 13.14 Transpose of a Matrix 13.15 Symmetric, Skew-symmetric, and Hermitian Matrices Properties of Symmetric and Skew-Symmetric Matrices 13.16 Lower and Upper Triangular Matrices 13.17 Adjoint of a Matrix 13.18 The Inverse of a Matrix 13.19 Methods of Computing Inverse of a Matrix 1. Method of an Adjoint Matrix 2. Method Using Definition of Inverse 3. Method of Matrix Equation 4. Method of Elementary Transformation (Gauss-Jordan Method) 13.20 Rank of a Matrix 13.21 Elementary Matrices 13.22 Equivalence of Matrices 13.23 Row and Column Equivalence of Matrices 13.24 Row Rank and Column Rank of a Matrix 13.25 Solution of System of Linear Equations 13.26 Solution of Non-homogeneous Linear System of Equations (A) Matrix Inversion Method B. Cramer’s Rule 13.27 Consistency Theorem 13.28 Homogeneous Linear Equations 13.29 Characteristic Roots and Vectors 13.30 The Cayley-Hamilton Theorem 13.31 Algebraic and Geometric Multiplicity of an Eigenvalue 13.32 Minimal Polynomial of a Matrix 13.33 Orthogonal, Normal, and Unitary Matrices 13.34 Similarity of Matrices 13.35 Triangularization of an Arbitrary Matrix 13.36 Quadratic Forms 13.37 Diagonalization of Quadratic Forms Exercises Chapter 14: Functions of Complex Variables 14.1 Basic Concepts 14.1.1 Logarithms of Complex Numbers 14.1.2 Real and Imaginary Parts of Log (x + iy) 14.1.3 Hyperbolic Functions 14.1.4 Relations Between Hyperbolicand Circular Functions 14.1.5 Periodicity of Hyperbolic Function 14.2 Analytic Functions 14.3 Integration of Complex-Valued Functions 14.4 Power Series Representation of an Analytic Function 14.5 Zeros and Poles 14.6 Residues and Cauchy’s Residue Theorem 14.7 Evaluation of Real Definite Integrals 14.8 Conformal Mapping Exercises Chapter 15: Differential Equations 15.1 Definitions and Examples 15.2 Formulation of Differential Equation 15.3 Solution of Differential Equation 15.4 Differential Equations of First order 15.5 Separable Equations 15.6 Homogeneous Equations 15.7 Equations Reducible to Homogeneous Form 15.8 Linear Differential Equations 15.9 Equations Reducible to Linear Differential Equations 15.10 Exact Differential Equation 15.11 The Solution of Exact Differential Equation 15.12 Equations Reducible to Exact Equation 15.13 Applications of First Order and First Degree Equations (A) Problems Related to Electric Circuits (B) Problems Related to Newton’s Law of Cooling (C) Problems Relating to Heat Flow (D) Rate Problems (E) Falling Body Problems (F) Orthogonal Trajectories 15.14 Linear Differential Equations 15.15 Solution of Homogeneous Linear Differential Equation with Constant Coefficients Case I. Distinct Real Roots Case II. Repeated Real Roots Case III. Conjugate Complex Roots 15.16 Complete Solution of Linear Differential Equation with Constant Coefficients 15.16.1 Standard Cases of Particular Integrals 15.17 Method of Variation of Parameters to Find Particular Integral 15.18 Differential Equations with Variable Coefficients 15.19 Simultaneous Linear Differential Equations with Constant Coefficients 15.20 Applications of Linear Differential Equations 15.21 Mass-Spring System 15.22 Simple Pendulum 15.23 Solution in Series 15.23.1 Solution About Ordinary Point 15.23.2 Solution About Singular Point (Forbenious Method) 15.24 Bessel’s Equation and Bessel’s Function 15.25 Legendre’s Equation and Legendre’s Polynomial 15.26 Fourier–Legendre Expansion of a Function Exercises Chapter 16: Partial Differential Equations 16.1 Formulation of Partial Differential Equation 16.2 Solutions of a Partial Differential Equation (A) Direct Integration Method (B) Lagrange’s Method 16.3 Non-linear Partial Differential Equations of the First Order 16.4 Charpit’s Method 16.5 Some Standard forms of Non-linear Equations (A) Equations of the Form f ( p, q) = 0 (B) Equation of the form f (z, p, q) = 0 (C) Separable Equations (D) Clairut’s Equation 16.6 The Method of Separation of Variables 16.7 One-Dimensional Heat Equation 16.8 One-DimensionalWave Equation 16.9 Two-Dimensional Heat Equation Exercises Chapter 17: Fourier Series 17.1 Trigonometric Series 17.2 Fourier (or Euler) Formulae 17.3 Periodic Extension of a Function 17.4 Fourier Cosine and Sine Series 17.5 Complex Fourier Series 17.6 Spectrum of Periodic Functions 17.7 Properties of Fourier Coefficients 17.8 Dirichlet’s Kernel 17.9 Integral Expression for Partial Sums of a Fourier Series 17.10 Fundamental Theorem (Convergence Theorem) of Fourier Series 17.11 Applications of Fundamental Theorem of Fourier Series 17.12 Convolution Theorem for Fourier Series 17.13 Integration of Fourier Series 17.14 Differentiation of Fourier Series 17.15 Examples of Expansions of Functions in Fourier Series 17.16 Signals and Systems 17.17 Classification of Signals 17.18 Classification of Systems 17.19 Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input 17.20 Application to Differential Equations 7.21 Application to Partial Differential Exercises Chapter 18: Fourier Transform 18.1 Fourier Integral Theorem 18.2 Fourier Transforms 18.3 Fourier Cosine and Sine Transforms 18.4 Properties of Fourier Transforms 18.5 Solved Examples 18.6 Complex Fourier Transforms 18.7 Convolution Theorem 18.8 Parseval’s Identities 18.9 Fourier Integral Representation of a Function 18.10 Finite Fourier Transforms 18.11 Applications of Fourier Transforms 18.12 Applicatio 18.13 Application to Partial Differential Equations Exercises Chapter 19: Discrete Fourier Transform 19.1 Approximation of Fourier Coefficients of a Periodic Function 19.2 Definition and Examples of DFT 19.3 Inverse DFT 19.4 Properties of DFT 19.5 Cyclical Convolution and Convolution Theorem for DFT 19.6 Parseval’s Theorem for the DFT 19.7 Matrix form of the DFT 19.8 N-Point Inverse DFT 19.9 Fast Fourier Transform (FFT) Exercises Chapter 20: Laplace Transform 20.1 Definition and Examples of Laplace Transform 20.2 Properties of Laplace Transforms 20.3 Limiting Theorems Exercises Chapter 21: Inverse Laplace Transform 21.1 Definition and Examples of Inverse Laplace Transform 21.2 Properties of Inverse Laplace Transform 21.3 Partial Fractions Method to Find Inverse Laplace Transform 21.4 Heaviside’s Expansion Theorem 21.5 Series Method to Determine Inverse Laplace Transform 21.6 Convolution Theorem 21.7 Complex Inversion Formula Exercises Chapter 22: Applications of Laplace Transform 22.1 Ordinary Differential Equations (a) Ordinary Differential Equations with Constant Solution (b) Problems Related to Electrical Circuits (c) Mechanical System (Mass-Spring System) (d) Ordinary Differential Equations with Polynomial (Variable) Coefficients 22.2 Simultaneous Differential Equations 22.3 Difference Equations 22.4 Integral Equations 22.5 Integro-Differential Equations 22.6 Solution of Partial Differential Equation 22.7 Evaluation of Integrals Exercises Chapter 23: The z-transform 23.1 Some Elementary Concepts 23.2 Definition of z-transform 23.3 Convergence of z-transform 23.4 Examples of z-transform 23.5 Properties of the z-transform 23.5.1. Table of z-transforms 23.6 Inverse z-transform (A) Contour Integration Method (B) Partial Fractions Method (C) Power Series Method for FindingInverse z-transform 23.7 Convolution Theorem 23.8 The Transfer Function (or System Function) 23.9 Systems Described by Difference Equations Exercises Chapter 24: Elements of Statistics and Probability 24.1 Measures of Central Tendency 24.2 Measures of Variability (Dispersion) 24.3 Measure of Skewness 24.4 Measures of Kurtosis 24.5 Covariance 24.6 Correlation and Coefficient of Correlation 24.7 Regression 24.8 Angle Between the Regression Lines 24.9 Probability 24.10 Conditional Probability 24.11 Independent Events 24.12 Probability Distribution 24.13 Mean and Variance of a Random Variable 24.14 Binomial Distribution 24.15 Pearson’s Constants for Binomial Distribution 24.16 Poisson Distribution 24.17 Constants of the PoissonDistribution 24.18 Normal Distribution 24.19 Characteristics of the Normal Distribution 24.20 Normal Probability Integral 24.21 Areas Under the Standard Normal Curve 24.22 Fitting of Normal Distribution to a Given Data 24.23 Sampling 24.24 Level of Significance and Critical Region 24.25 Test of Significance for Large Samples 24.26 Confidence Interval for the Mean 24.27 Test of significance for Single Proportion 24.28 Test of Significance for Difference of Proportion 24.29 Test of Significance for Difference of Means 24.30 Test of Significance for the Difference of Standard Deviations 24.31 Sampling with Small Samples 24.32 Significance Test of Difference Between Sample Means 24.33 Chi-square Distribution 24.34 x2-test as a Test of Goodness-of-Fit 24.35 Snedecor’s F-Distribution 24.36 Fisher’s Z-Distribution Exercises Chapter 25: Linear Programming 25.1 Linear Programming Problems 25.2 Formulation of a Linear Programming Problem (LPP) 25.3 Graphical Method to Solve Linear Programming Problem 25.4 Canonical and Standard forms of Linear Programming Problem 25.5 Basic Feasible Solution of an LPP 25.6 Simplex Method 25.7 Tabular form of the Solution 25.8 Generalization of Simplex Algorithm 25.9 Two-Phase Method 25.10 Duality Property 25.11 Dual Simplex Method 25.12 Transportation Problems 25.13 Matrix form of the Transportation Problem 25.14 Transportation Problem Table 25.15 Basic Initial Feasible Solution of Transportation Problem A. North-West Corner Method B. Matrix Minima or Least Cost Method 25.16 Test for the Optimality of Basic Feasible Solution 25.17 Degeneracy in Transportation Problem 25.18 Unbalanced Transportation Problems Exercises Chapter 26: Basic Numerical Methods 26.1 Approximate Numbers and Significant Figures 26.2 Classical Theorems Used in Numerical Methods 26.3 Types of Errors 26.4 General Formula for Errors 26.5 Solution of Non-Linear Equations 1. Bisection Method (Bolzano Method) 2. Regula-Falsi Method 3. Newton–Raphson Method 4. Fixed Point Iteration 5. Newton’s Method for Finding Multiple Roots 26.6 Linear System of Equations 1. Gauss’s Elimination Method 2. Jordan’s Modification to Gauss’s Method 3. Iterative Methods for Linear Systems 26.7 Finite Differences 26.8 Error Propagation 26.9 Interpolation (a) Newton’s Forward Difference Formula (b) Newton’s Backward Difference Formula 26.10 Interpolation With Unequal Spaced Points (a) Divided Differences 26.11 Newton’s Fundamental (Divided Difference) Formula 26.12 Lagrange’s Interpolation Formula 26.13 Curve Fitting (a) Least Square Line Approximation (B) The Power Fit (C) Least Square Parabola (Parabola of Beast Fit 26.14 Numerical Quadrature (Integration) 26.15 Ordinary Differential Equations Classification of Methods of Solution 1. Taylor Series Method 2. Euler’s Method 3. Picard’s Method of Successive Integration 4. Fourth Order Runge-Kutta Method 26.16 Numerical Solution of Partial Differential Equations 26.16.1 Formation of Difference Equation 26.16.2 Geometric Representation of Partial Difference Quotients 26.16.3 Standard Five Point Formula and Diagonal Five-Point Formula 26.16.4 Point Jacobi’s Method 26.16.5 Gauss–Seidel Method 26.16.6 Poisson’s Equation 26.16.7 Parabolic Equations 26.16.8 Hyperbolic Equations Exercises Bibliography Index