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Engineering Mathematics, Volume II, Second Edition

Name: Engineering Mathematics, Volume II, Second Edition
Author: Babu Ram
ISBN: 9788131785034

by Babu Ram

May 2012

Intermediate to advanced

954 pages

36h 6m

English

Pearson India

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Cover
Title Page
Contents
Dedication
Preface to the Revised Edition
1. Preliminaries
1.1 - Sets and Functions1.2 - Continuous and Piecewise Continuous Functions1.3 - Derivability of a Function and Piecwise Smooth Functions1.4 - The Riemann Integral1.5 - The Causal and Null Function1.6 - Functions of Exponential Order1.7 - Periodic Functions1.8 - Even and Odd Functions1.9 - Sequence and Series1.10 - Series of Functions1.11 - Partial Fraction Expansion of a Rational Function1.12 - Special Functions1.13 - The Integral Transforms
2. Linear Algebra
2.1 - Concepts of Group, Ring, and Field2.2 - Vector Space2.3 - Linear Transformation2.4 - Linear Algebra2.5 - Rank and Nullity of a Linear Transformation2.6 - Matrix of a Linear Transformation2.7 - Change-of-basis Matrix (Transforming Matrix or Transition Matrix)2.8 - Relation Between Matrices of a Linear Transformation in Different Bases2.9 - Normed Linear Space2.10 - Inner Product Space2.11 - Least Square Line Approximation
2.12 - Minimal Solution to a System of Equations
2.13 - Matrices2.14 - Algebra of Matrices2.15 - Multiplication of Matrices2.16 - Associative Law for Matrix Multiplication2.17 - Distributive Law for Matrix Multiplication2.18 - Transpose of a Matrix2.19 - Symmetric, Skew-Symmetric, and Hermitian Matrices2.20 - Lower and Upper Triangular Matrices2.21 - Determinants2.22 - Adjoint of a Matrix2.23 - The Inverse of a Matrix2.24 - Methods of Computing Inverse of a Matrix2.25 - Rank of a Matrix2.26 - Elementary Matrices2.27 - Row Reduced Echelon Form and Normal Form of Matrices2.28 - Equivalence of Matrices2.29 - Row Rank and Column Rank of a Matrix2.30 - Solution of System of Linear Equations2.31 - Solution of Non-Homogeneous Linear System of Equations2.32 - Consistency Theorem2.33 - Homogeneous Linear Equations2.34 - Characteristic Roots and Characteristic Vectors2.35 - The Cayley-Hamilton Theorem
2.36 - Algebraic and Geometric Multiplicity of an Eigen Value
2.37 - Minimal Polynomial of a Matrix2.38 - Orthogonal, Normal and Unitary Matrices2.39 - Similarity of Matrices2.40 - Diagonalization of a Matrix2.41 - Triangularization of an Arbitrary Matrix2.42 - Quadratic Forms2.43 - Diagonalization of Quadratic Forms2.44 - Miscellaneous ExamplesExercises
3. Functions of Complex Variables
3.1 - Basic Concepts3.2 - De-Moivre's Theorem3.3 - Logarithms of Complex Numbers3.4 - Hyperbolic Functions3.5 - Relations Between Hyperbolic and Circular Functions3.6 - Periodicity of Hyperbolic Functions3.7 - Some Basic Concepts3.8 - Analytic Functions3.9 - Integration of Complex-valued Functions3.10 - Power Series Representation of an Analytic Function

3.11 - Zeros and Poles
3.12 - Residues and Cauchy's Residue Theorem3.13 - Evaluation of Real Definite Integrals3.14 - Conformal Mapping3.15 - Miscellaneous ExamplesExercises
4. Ordinary Differential Equations
4.1 - Definitions and Examples4.2 - Formulation of Differential Equation4.3 - Solution of Differential Equation4.4 - Differential Equations of First Order4.5 - Separable Equations4.6 - Homogeneous Equations4.7 - Equations Reducible to Homogeneous Form4.8 - Linear Differential Equations of First Order and First Degree4.9 - Equations Reducible to Linear Differential Equations4.10 - Exact Differential Equation4.11 - The Solution of Exact Differential Equation4.12 - Equations Reducible to Exact Equation4.13 - Applications of First Order and First Degree Equations
4.14 - Equations of First Order and Higher Degree
4.15 - Equations which can be Factorized into Factors of First Degree4.16 - Equations which cannot be Factorized into Factors of First Degree4.17 - Clairaut's Equation4.18 - Higher Order Linear Differential Equations4.19 - Solution of Homogeneous Linear Differential Equation with Constant Coefficients4.20 - Complete Solution of Linear Differential Equation with Constant Coefficients4.21 - Application of Linear Differential Equation4.22 - Mass-Spring System4.23 - Simple Pendulum4.24 - Differential Equation with Variable Coefficients4.25 - Method of Solution by Changing the Independent Variable4.26 - Method of Solution by Changing the Dependent Variable4.27 - Method of Undetermined Coefficients4.28 - Method of Reduction of Order
4.29 - The Cauchy-Euler Homogeneous Linear Equation
4.30 - Legendre's Linear Equation4.31 - Method of Variation of Parameters to Find Particular Integral4.32 - Solution in Series4.33 - Bessel's Equation and Bessel's Function4.34 - Fourier-Bessel Expansion of a Continuous Function4.35 - Legendre's Equation and Legendre's Polynomial4.36 - Fourier-Legendre Expansion of a Function4.37 - Miscellaneous Examples4.38 - Simultaneous Linear Differential Equations with Constant CoefficientExercises
5. Partial Differential Equations
5.1 - Formulation of Partial Differential Equation5.2 - Solutions of a Partial Differential Equation5.3 - Non-linear Partial Differential Equations of the First Order5.4 - Charpit's Method5.5 - Some Standard Forms of Non-linear Equations5.6 - Linear Partial Differential Equations with Constant Coefficients
5.7 - Equations Reducible to Homogeneous Linear Form
5.8 - Classification of Second Order Linear Partial Differential Equations5.9 - The Method of Separation of Variables5.10 - Classical Partial Differential Equations5.11 - Solutions of Laplace Equation5.12 - Telephone Equations of a Transmission Line5.13 - Miscellaneous ExamplesExercises
6. Fourier Series
6.1 - Trigonometric Series6.2 - Fourier (or Euler) Formulae6.3 - Periodic Extension of a Function6.4 - Fourier Cosine and Sine Series6.5 - Complex Fourier Series6.6 - Spectrum of Periodic Functions6.7 - Properties of Fourier Coeffcients6.8 - Dirichlet's Kernel6.9 - Integral Expression for Partial Sums of a Fourier Series6.10 - Fundamental Theorem (Convergence Theorem) of Fourier Series6.11 - Applications of Fundamental Theorem of Fourier Series6.12 - Convolution Theorem for Fourier Series6.13 - Integration of Fourier Series6.14 - Differentiation of Fourier Series6.15 - Examples of Expansions of Functions in Fourier Series6.16 - Method to Find Harmonics of Fourier Series of a Function from Tabular Values6.17 - Signals and Systems6.18 - Classification of Signals6.19 - Classification of Systems6.20 - Response of a Stable Linear Time Invariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input6.21 - Application to Differential Equations6.22 - Application to Partial Differential Equations6.23 - Miscellaneous ExamplesExercises
7. Fourier Transform
7.1 - Fourier Integral Theorem7.2 - Fourier Transforms7.3 - Fourier Cosine and Sine Transforms7.4 - Properties of Fourier Transforms7.5 - Solved Examples7.6 - Complex Fourier Transforms7.7 - Convolution Theorem7.8 - Parseval's Identities7.9 - Fourier Integral Representation of a Function7.10 - Finite Fourier Transforms7.11 - Applications of Fourier Transforms7.12 - Application to Differential Equations7.13 - Application to Partial Differential EquationsExercises
8. Discrete Fourier Transform
8.1 - Approximation of Fourier Coefficients of a Periodic Function8.2 - Definition and Examples of DFT8.3 - Inverse DFT8.4 - Properties of DFT8.5 - Cyclical Convolution and Convolution Theorem for DFT8.6 - Parseval's Theorem for the DFT8.7 - Matrix Form of the DFT8.8 - N-point Inverse DFT8.9 - Fast Fourier Transform (FFT)Exercises
9. Laplace Transform
9.1 - Definition and Examples of Laplace Transform9.2 - Properties of Laplace Transforms9.3 - Limiting Theorems9.4 - Miscellaneous ExamplesExercises
10. Inverse Laplace Transform
10.1 - Definition and Examples of Inverse Laplace Transform10.2 - Properties of Inverse Laplace Transform10.3 - Partial Fractions Method to Find Inverse Laplace Transform10.4 - Heaviside's Expansion Theorem10.5 - Series Method to Determine Inverse Laplace Transform10.6 - Convolution Theorem10.7 - Complex Inversion Formula10.8 - Miscellaneous ExamplesExercises
11. Applications of Laplace Transform
11.1 - Ordinary Differential Equations11.2 - Simultaneous Differential Equations11.3 - Difference Equations11.4 - Integral Equations11.5 - Integro-differential Equations11.6 - Solution of Partial Differential EquationExercises
12. The Z-transform
12.1 - Some Elementary Concepts12.2 - Definition of Z -transform12.3 - Convergence of Z-transform12.4 - Examples of Z-transform12.5 - Properties of the Z-transform12.6 - Inverse Z-transform12.7 - Convolution Theorem12.8 - The Transfer Function (or System Function)12.9 - Systems Described by Difference EquationsExercises
13. Elements of Statistics and Probability
13.1 - Introduction13.2 - Measures of Central Tendency13.3 - Measures of Variability (Dispersion)13.4 - Measure of Skewness13.5 - Measures of Kurtosis13.6 - Covariance13.7 - Correlation and Coefficient of Correlation13.8 - Regression13.9 - Angle Between the Regression Lines13.10 - Probability13.11 - Conditional Probability13.12 - Independent Events13.13 - Probability Distribution13.14 - Mean and Variance of a Random Variable13.15 - Binomial Distribution13.16 - Pearson's Constants for Binomial Distribution
13.17 - Poisson Distribution
13.18 - Constants of the Poisson Distribution13.19 - Normal Distribution13.20 - Characteristics of the Normal Distribution13.21 - Normal Probability Integral13.22 - Areas Under the Standard Normal Curve13.23 - Fitting of Normal Distribution to a Given Data13.24 - Sampling13.25 - Level of Significance and Critical Region13.26 - Test of Significance for Large Samples13.27 - Confidence Interval for the Mean13.28 - Test of Significance for Single Proportion13.29 - Test of Significance for Difference of Proportion13.30 - Test of Significance for Difference of Means13.31 - Test of Significance for the Difference of Standard Deviations13.32 - Sampling with Small Samples13.33 - Significance Test of Difference Between Sample Means13.34 - Chi-square Distribution13.35 - X2-Test as a Test of Goodness-of-fit13.36 - Snedecor's F-distribution13.37 - Fisher's Z-distribution13.38 - Miscellaneous ExamplesExercises
14. Linear Programming
14.1 - Linear Programming Problems14.2 - Formulation of an LPP14.3 - Graphical Method to Solve LPP14.4 - Canonical and Standard Forms of LPP14.5 - Basic Feasible Solution of an LPP14.6 - Simplex Method14.7 - Tabular Form of the Solution14.8 - Generalization of Simplex Algorithm14.9 - Two-phase Method14.10 - Duality Property14.11 - Dual Simplex Method14.12 - Transportation Problems14.13 - Matrix Form of the Transportation Problem14.14 - Transportation Problem Table14.15 - Basic Initial Feasible Solution of Transportation Problem14.16 - Test for the Optimality of Basic Feasible Solution14.17 - Degeneracy in Transportation Problem14.18 - Unbalanced Transportation ProblemsExercises
15. Basic Numerical Methods
15.1 - Approximate Numbers and Significant Figures15.2 - Classical Theorems used in Numerical Methods15.3 - Types of Errors15.4 - General Formula for Errors15.5 - Solution of Non-linear Equations15.6 - Linear System of Equations15.7 - Finite Differences15.8 - Error Propagation15.9 - Interpolation15.10 - Interpolation with Unequal Spaced Points15.11 - Newton's Fundamental (Divided Difference) Formula15.12 - Lagrange's Interpolation Formula15.13 - Curve Fitting
15.14 - Numerical Quadrature
15.15 - Ordinary Differential Equations15.16 - Numerical Solution of Partial Differential EquationsExercises
Statistical Tables
Copyright

Content preview from Engineering Mathematics, Volume II, Second Edition

Geometrically, if d_i is the vertical distance from the data point (x_i, y_i) to the point (x_i, a+b_xi) on the line, then d_i = y_i – a – bx_i (see Figure 26.4). We must minimize the sum of the squares of the vertical distances d_i, that is, the sum

Figure 15.4

To minimize e (a,b), we equate to zero the partial derivatives of (63) with respect to ‘a’ and ‘b’. Thus

which are known as normal equations. We write these equations in the form ...

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Engineering Mathematics, Volume II, Second Edition

by Babu Ram

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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