Engineering Mathematics II : For RGPV

Description

Cover Contents Preface Acknowledgements Roadmap to the Syllabus Symbols and Basic Formulae Chapter 1: Fourier Series 1.1 Trigonometric Series 1.2 Fourier (or Euler) Formulae 1.3 Periodic Extension of a Function 1.4 Fourier Cosine and Sine Series 1.5 Complex Fourier Series 1.6 Spectrum of Periodic Functions 1.7 Properties of Fourier Coefficients 1.8 Dirichlet’s Kernel 1.9 Integral Expression for Partial Sums of a Fourier Series 1.10 Fundamental Theorem (Convergence Theorem) of Fourier Series 1.11 Applications of Fundamental Theorem of Fourier Series 1.12 Convolution Theorem for Fourier Series 1.13 Integration of Fourier Series 1.14 Differentiation of Fourier Series 1.15 Examples of Expansions of Functions in Fourier Series 1.16 Method to Find Harmonics of Fourier Series of a Function from Tabular Values 1.17 Signals and Systems 1.18 Classifi cation of Signals 1.19 Classification of Systems 1.20 Response of a Stable Linear Timeinvariant Continuous Time System (LTC System) to a Piecewise Smooth and Periodic Input 1.21 Application to Differential Equations 1.22 Application to Partial Differential Equations 1.23 Miscellaneous Examples Exercises Chapter 2: Fourier Transform 2.1 Fourier Integral Theorem 2.2 Fourier Transforms 2.3 Fourier Cosine and Sine Transforms 2.4 Properties of Fourier Transforms 2.5 Solved Examples 2.6 Complex Fourier Transforms 2.7 Convolution Theorem 2.8 Parseval’s Identities 2.9 Fourier Integral Representation of a Function 2.10 Finite Fourier Transforms 2.11 Applications of Fourier Transforms 2.12 Application to Differential Equations 2.13 Application to Partial Differential Equations Exercises Chapter 3: Laplace Transform 3.1 Definition and Examples of Laplace Transform 3.2 Properties of Laplace Transforms 3.3 Limiting Theorems 3.4 Miscellaneous Examples Exercises Chapter 4: Inverse Laplace Transform 4.1 Definition and Examples of Inverse Laplace Transform 4.2 Properties of Inverse Laplace Transform 4.3 Partial Fractions Method to Find Inverse Laplace Transform 4.4 Heaviside’s Expansion Theorem 4.5 Series Method to Determine Inverse Laplace Transform 4.6 Convolution Theorem 4.7 Complex Inversion Formula 4.8 Miscellaneous Examples Exercises Chapter 5: Applications of Laplace Transform 5.1 Ordinary Differential Equations 5.2 Simultaneous Differential Equations 5.3 Difference Equations 5.4 Integral Equations 5.5 Integro-differential Equations 5.6 Solution of Partial Differential Equations 5.7 Evaluation of Integrals 5.8 Miscellaneous Examples Exercises Chapter 6: Second Order Differential Equation with Variable Coefficients 6.1 Method of Solution by Changing Independent Variable 6.2 Method of Solution by Changing the Dependent Variable 6.3 Method of Undetermined Coefficients 6.4 Method of Reduction of Order 6.5 Cauchy–euler Homogeneous Linear Equation 6.6 Legendre’s Linear Equation 6.7 Method of Variation of Parameters to Find Particular Integral Exercises Chapter 7: Series Solution of Ordinary Differential Equations 7.1 Solution in Series 7.2 Bessel’s Equation and Bessel’s Function 7.3 Fourier–bessel Expansion of a Continuous Function 7.4 Legendre’s Equation and Legendre’s Polynomial 7.5 Fourier–Legendre Expansion of a Function 7.6 Miscellaneous Examples Exercises Chapter 8: Partial Differential Equations 8.1 Formulation of Partial Differential Equation 8.2 Solutions of a Partial Differential Equation 8.3 Miscellaneous Examples Exercises Chapter 9: Non-Linear Partial Differential Equations 9.1 Non-linear Partial Differential Equations of the First Order 9.2 Charpit’s Method 9.3 Some Standard Forms of Non-linear Equations Exercises Chapter 10: Partial Differential Equationswith Constant Coefficient 10.1 Linear Partial Differential Equations with Constant Coefficients 10.2 Equations Reducible to Homogeneous Linear Form Exercises Chapter 11: Classical Partial Differential Equations 11.1 Classification of Second Order Linear Partial Differential Equations 11.2 The Method of Separation of Variables 11.3 Classical Partial Differential Equations 11.4 Solutions of Laplace Equation 11.5 Telephone Equations of a Transmission Line 11.6 Miscellaneous Examples Exercises Chapter 12: Vector Differentiation 12.1 Differentiation of a Vector 12.2 Partial Derivatives of a Vector Function 12.3 Gradient of a Scalar Field 12.4 Geometrical Interpretation of a Gradient 12.5 Properties of a Gradient 12.6 Directional Derivatives 12.7 Divergence of a Vector-point Function 12.8 Physical Interpretation of Divergence 12.9 Curl of a Vector-point Function 12.10 Physical Interpretation of Curl 12.11 The Laplacian Operator ∇2 12.12 Properties of Divergence and Curl 12.13 Miscellaneous Examples Exercises Chapter 13: Integration of Vector Functions 13.1 Integration of Vector Functions 13.2 Line Integral 13.3 Work Done by a Force 13.4 Surface Integral 13.5 Volume Integral 13.6 Gauss’s Divergence Theorem 13.7 Green’s Theorem in a Plane 13.8 Stoke’s Theorem 13.9 Miscellaneous Examples Exercises Solved Question Papers Index