Advanced engineering mathematics

Summary of contents
Contents
Preface to the first edition
Preface to the sixth edition
	New to this edition
	Acknowledgements
Hints on using the book
Useful background information
Programme 1 Numerical solutions of equations and interpolation
	Learning outcomes
		Introduction
		The Fundamental Theorem of Algebra
			The Fundamental Theorem of Algebra
			Relations between the coefficients and the roots of a polynomial equation
		Cubic equations
			Transforming a cubic to reduced form
			Tartaglia’s solution for a real root
		Numerical methods
			Bisection
		Numerical solution of equations by iteration
			Using a spreadsheet
			Relative addresses
		Newton–Raphson iterative method
			Tabular display of results
			Modified Newton–Raphson method
			And now . . .
		Interpolation
			Linear interpolation
			Graphical interpolation
			Gregory–Newton interpolation formula using forward finite differences
			Central differences
			Gregory–Newton backward differences
			Lagrange interpolation
	Review summary 1
	Can you?
		Checklist 1
	Test exercise 1
	Further problems 1
Programme 2 Laplace transforms 1
	Learning outcomes
		Introduction
		Laplace transforms
		Differentiating and integrating a transform
			Theorem 1 The first shift theorem
			Theorem 2 Multiplying by t and tn
			Theorem 3 Dividing by t
		Inverse transforms
			Rules of partial fractions
			The ‘cover up’ rule
			Table of inverse transforms
		Solution of differential equations by Laplace transforms
			Transforms of derivatives
			Solution of first-order differential equations
			Solution of second-order differential equations
			Simultaneous differential equations
	Review summary 2
	Can you?
		Checklist 2
	Test exercise 2
	Further problems 2
Programme 3 Laplace transforms 2
	Learning outcomes
		Introduction
		Heaviside unit step function
		Unit step at the origin
			Effect of the unit step function
			Laplace transform of u(t – c)
			Laplace transform of u(t – c).f(t – c ) (the second shift theorem)
		Differential equations involving the unit step function
		Convolution
		The convolution theorem
	Review summary 3
	Can you?
		Checklist 3
	Test exercise 3
	Further problems 3
Programme 4 Laplace transforms 3
	Learning outcomes
		Laplace transforms of periodic functions
			Periodic functions
			Inverse transforms
		The Dirac delta – the unit impulse
			Graphical representation
			Laplace transform of δ(t – a)
			The derivative of the unit step function
		Differential equations involving the unit impulse
		Harmonic oscillators
			Damped motion
			Forced harmonic motion with damping
			Resonance
	Review summary 4
	Can you?
		Checklist 4
	Test exercise 4
	Further problems 4
Programme 5 Difference equations and the Z transform
	Learning outcomes
		Introduction
		Sequences
		Difference equations
		Solving difference equations
			Solution by inspection
			The particular solution
		The Z transform
		Table of Z transforms
		Properties of Z transforms
			Linearity
			First shift theorem (shifting to the left)
			Second shift theorem (shifting to the right)
			Scaling
			Final value theorem
			The initial value theorem
			The derivative of the transform
			Summary
		Inverse transforms
		Solving difference equations
		Sampling
	Review summary 5
	Can you?
		Checklist 5
	Test exercise 5
	Further problems 5
Programme 6 Introduction to invariant linear systems
	Learning outcomes
		Invariant linear systems
			Systems
			Input-response relationships
			Linear systems
			Time-invariance of a continuous system
			Shift-invariance of a discrete system
		Differential equations
			The general nth order equation
			Zero-input response and zero-state response
			Zero-input, zero-response
			Time-invariance
		Responses of a continuous system
			Impulse response
			Arbitrary input
			Convolution
			Exponential response
			The transfer function H(s)
			Differential equations
		Responses of a discrete system
			The discrete unit impulse
			Arbitrary input
			Exponential response
			Transfer function
			Difference equations
	Review summary 6
	Can you?
		Checklist 6
	Test exercise 6
	Further problems
Programme 7 Fourier series 1
	Learning outcomes
		Introduction
			Periodic functions
			Graphs of y = Asin nx
			Harmonics
			Non-sinusoidal periodic functions
			Analytic description of a periodic function
		Integrals of periodic functions
			Summary
			Orthogonal functions
			Fourier series
			Summary
		Dirichlet conditions
			Effect of harmonics
			Gibbs’ phenomenon
			Sum of a Fourier series at a point of discontinuity
	Review summary 7
	Can you?
		Checklist 7
	Test exercise 7
	Further problems 7
Programme 8 Fourier series 2
	Learning outcomes
		Odd and even functions and half-range series
			Odd and even functions
			Products of odd and even functions
			Half-range series
			Series containing only odd harmonics or only even harmonics
			Significance of the constant term 1/2 a0
		Functions with periods other than 2π
			Functions with period T
			Fourier coefficients
			Half-range series with arbitrary period
	Review summary 8
	Can you?
		Checklist 8
	Test exercise 8
	Further problems 8
Programme 9 Introduction to the Fourier transform
	Learning outcomes
		Complex Fourier series
			Introduction
			Complex exponentials
		Complex spectra
		The two domains
		Continuous spectra
		Fourier’s integral theorem
		Some special functions and their transforms
			Even functions
			Odd functions
			Top-hat function
			The Dirac delta (refer to Programme 4, Frames 29ff)
			The triangle function
		Alternative forms
		Properties of the Fourier transform
			Linearity
			Time shifting
			Frequency shifting
			Time scaling
			Symmetry
			Differentiation
		The Heaviside unit step function
		Convolution
		The convolution theorem
		Fourier cosine and sine transforms
		Table of transforms
	Review summary 9
	Can you?
		Checklist 9
	Test exercise 9
	Further problems 9
Programme 10 Power series solutions of ordinary differential equations 1
	Learning outcomes
		Higher derivatives
			Leibnitz theorem – nth derivative of a product of two functions
			Choice of function for u and v
		Power series solutions
			Leibnitz–Maclaurin method
			Cauchy–Euler equi-dimensional equations
	Review summary 10
	Can you?
		Checklist 10
	Test exercise 10
	Further problems 10
Programme 11 Power series solutions of ordinary differential equations 2
	Learning outcomes
		Introduction
			Solution of differential equations by the method of Frobenius
			The indicial equation
	Review summary 11
	Can you?
		Checklist 11
	Test exercise 11
	Further problems 11
Programme 12 Power series solutions of ordinary differential equations 3
	Learning outcomes
		Introduction
			Bessel’s equation
			Gamma and Bessel functions
			Graphs of Bessel functions J0(x) and J1(x)
		Legendre’s equation
			Legendre polynomials
			Rodrigue’s formula and the generating function
		Sturm–Liouville systems
			Orthogonality
			Summary
			Legendre’s equation revisited
			Polynomials as a finite series of Legendre polynomials
	Review summary 12
	Can you?
		Checklist 12
	Test exercise 12
	Further problems 12
Programme 13 Numerical solutions of ordinary differential equations
	Learning outcomes
		Introduction
		Taylor’s series
			Function increment
		First-order differential equations
			Euler’s method
			The exact value and the errors
			Graphical interpretation of Euler’s method
			The Euler–Cauchy method – or the improved Euler method
			Euler–Cauchy calculations
			Runge–Kutta method
		Second-order differential equations
			Euler second-order method
			Runge–Kutta method for second-order differential equations
		Predictor–corrector methods
	Review summary 13
	Can you?
		Checklist 13
	Test exercise 13
	Further problems 13
Programme 14 Matrix algebra
	Learning outcomes
		Singular and non-singular matrices
			Rank of a matrix
		Elementary operations and equivalent matrices
		Consistency of a set of linear equations
			Uniqueness of solutions
		Solution of sets of linear equations
			Inverse method
			Row transformation method
			Gaussian elimination method
			Triangular decomposition method
			Using an electronic spreadsheet
			Comparison of methods
		Matrix transformation
			Rotation of axes
	Review summary 14
	Can you?
		Checklist 14
	Test exercise 14
	Further problems 14
Programme 15 Systems of ordinary differential equations
	Learning outcomes
		Eigenvalues of 2 x 2 matrices
			Characteristic equation
			Sum and product of eigenvalues
			Eigenvectors
		Systems of linear, first-order ordinary differential equations
			Summary
			Repeated eigenvalues
		Diagonalization of a matrix
			Modal matrix
			Spectral matrix
		Systems of linear, second-order differential equations
			Summary
	Review summary 15
	Can you?
		Checklist 15
	Test exercise 15
	Further problems 15
Programme 16 Direction fields
	Learning outcomes
		Differential equations
			Introduction
			Family of solutions
			Direction fields
		DFIELD
			Introduction
			A specific solution
			Family of solutions
			Autonomous differential equations
			Equilibrium solutions
			The phase line
			Summary
			Semi-stable solution
		Non-autonomous equations
			Introduction
	Review summary 16
	Can you?
		Checklist 16
	Test exercise 16
	Further problems 16
Programme 17 Phase plane analysis
	Learning outcomes
		Phase plane analysis
			Introduction
			Mass-spring system
			PPLANE
			Phase plane analysis
			Eigenvalues and the phase plane
			Imaginary eigenvalues
			Two complex eigenvalues
			Behaviour around the critical point
			Two real and negative eigenvalues
			Behaviour around the critical point
			Two real and positive eigenvalues
			Two real eigenvalues of different signs
			Two identical eigenvalues
			Star node
			Singular coefficient matrix
			The inhomogeneous case
			Critical point moved to the origin
	Review summary 17
	Can you?
		Checklist 17
	Test exercise 17
	Further problems 17
Programme 18 Nonlinear systems
	Learning outcomes
	Multiple critical points
		Introduction
		Linearization
		Problems with linearization
	Review summary
	Can you?
		Checklist 18
	Test exercise 18
	Further problems 18
Programme 19 Dynamical systems
	Learning outcomes
		Dynamical systems
			Introduction
			Predator-prey problems
			Competition within a single population
			Two non-interacting populations
			Two interacting populations
		Second-order differential equations
			Undamped pendulum: small oscillations
			Undamped pendulum: no approximation
			Damped pendulum
		Bifurcation
			First-order equations
			Second-order equations
		Limit cycles
		The Van der Pol equation
	Review summary 19
	Can you?
		Checklist 19
	Test exercise 19
	Further problems 19
Programme 20 Partial differentiation
	Learning outcomes
		Small increments
			Taylor’s theorem for one independent variable
			Taylor0s theorem for two independent variables
			Small increments
			Rates of change
			Implicit functions
			Change of variables
		Inverse functions
			General case
			Summary
		Stationary values of a function
			Maximum and minimum values
			Saddle point
		Lagrange undetermined multipliers
			Functions with three independent variables
	Review summary 20
	Can you?
		Checklist 20
	Test exercise 20
	Further problems 20
Programme 21 Partial differential equations
	Learning outcomes
		Introduction
		Partial differential equations
			Solution by direct integration
			Initial conditions and boundary conditions
			The wave equation
			Solution of the wave equation
			Solution by separating the variables
		The heat conduction equation for a uniform finite bar
			Solutions of the heat conduction equation
		Laplace’s equation
			Solution of the Laplace equation
		Laplace’s equation in plane polar coordinates
			The problem
			Separating the variables
			Summary
			The n = 0 case
	Revision summary 21
	Can you?
		Checklist 21
	Test exercise 21
	Further problems 21
Programme 22 Numerical solutions of partial differential equations
	Learning outcomes
		Introduction
		Numerical approximation to derivatives
		Functions of two real variables
			Grid values
			Computational molecules
			Summary of procedures
		Derivative boundary conditions
		Second-order partial differential equations
			Elliptic equations
			Hyperbolic equations
			Parabolic equations
		Second partial derivatives
		Time-dependent equations
		The Crank–Nicolson procedure
		Dimensional analysis
	Review summary 22
	Can you?
		Checklist 22
	Test exercise 22
	Further problems 22
Programme 23 Multiple integration 1
	Learning outcomes
		Introduction
		Differentials
			Exact differential
			Integration of exact differentials
		Area enclosed by a closed curve
		Line integrals
			Alternative form of a line integral
			Properties of line integrals
			Regions enclosed by closed curves
			Line integrals round a closed curve
			Line integral with respect to arc length
			Parametric equations
			Dependence of the line integral on the path of integration
			Exact differentials in three independent variables
		Green’s theorem
	Review summary 23
	Can you?
		Checklist 23
	Test exercise 23
	Further problems 23
Programme 24 Multiple integration 2
	Learning outcomes
		Double integrals
			Surface integrals
		Three dimensional coordinate systems
			Cartesian coordinates
			Cylindrical coordinates
			Spherical coordinates
			Element of volume in the three coordinate systems
		Volume integrals
			Change of variables in multiple integrals
		Curvilinear coordinates
			Transformation in three dimensions
	Review summary 24
	Can you?
		Checklist 24
	Test exercise 24
	Further problems 24
Programme 25 Integral functions
	Learning outcomes
		Gamma and beta functions
			The gamma function
			Review
			The beta function
			Reduction formulas
			Review
			Relationship between the gamma and beta functions
			Application of gamma and beta functions
			Duplication formula for gamma functions
		The error function
			The graph of erf (x)
			The complementary error function erfc (x)
		Elliptic functions
			Standard forms of elliptic functions
			Complete elliptic functions
			Alternative forms of elliptic functions
	Review summary 25
	Can you?
		Checklist 25
	Test exercise 25
	Further problems 25
Programme 26 Vector analysis 1
	Learning outcomes
		Introduction
		Triple products
			Scalar triple product of three vectors
			Properties of scalar triple products
			Coplanar vectors
			Vector triple products of three vectors
		Differentiation of vectors
			Differentiation of sums and products of vectors
			Unit tangent vectors
		Partial differentiation of vectors
			Integration of vector functions
		Scalar and vector fields
			grad (gradient of a scalar function)
			Directional derivatives
			Unit normal vectors
			grad of sums and products of scalars
			div (divergence of a vector function)
			curl (curl of a vector function)
		Summary of grad, div and curl
			Multiple operations
	Review summary 26
	Can you?
		Checklist 26
	Test exercise 26
	Further problems 26
Programme 27 Vector analysis 2
	Learning outcomes
		Line integrals
			Scalar field
			Vector field
		Volume integrals
		Surface integrals
			Scalar fields
			Vector fields
		Conservative vector fields
		Divergence theorem
		Stokes’ theorem
		Direction of unit normal vectors to a surface S
		Green’s theorem
	Review summary 27
	Can you?
		Checklist 27
	Test exercise 27
	Further problems 27
Programme 28 Vector analysis 3
	Learning outcomes
		Curvilinear coordinates
			Orthogonal curvilinear coordinates
		Orthogonal coordinate systems in space
		Scale factors
			Scale factors for coordinate systems
		General curvilinear coordinate system (u,v,w)
		Transformation equations
		Element of arc ds and element of volume dV in orthogonal curvilinear coordinates
		grad, div and curl in orthogonal curvilinear coordinates
		Particular orthogonal systems
	Review summary 28
	Can you?
		Checklist 28
	Test exercise 28
	Further problems 28
Programme 29 Complex analysis 1
	Learning outcomes
		Functions of a complex variable
		Complex mapping
		Complex mapping
			Mapping of a straight line in the z-plane onto the w-plane under the transformation w = f(z)
			Types of transformation of the form w = az + b
		Nonlinear transformations
			Mapping of regions
	Review summary 29
	Can you?
		Checklist 29
	Test exercise 29
	Further problems 29
Programme 30 Complex analysis 2
	Learning outcomes
		Differentiation of a complex function
			Regular function
			Cauchy–Riemann equations
		Harmonic functions
		Complex integration
			Contour integration – line integrals in the z-plane
			Cauchy’s theorem
			Deformation of contours at singularities
		Conformal transformation (conformal mapping)
			Conditions for conformal transformation
			Critical points
			Schwarz–Christoffel transformation
			Open polygons
	Review summary 30
	Can you?
		Checklist 30
	Text exercise 30
	Further problems 30
Programme 31 Complex analysis 3
	Learning outcomes
		Maclaurin series
		Radius of convergence
		Singular points
			Poles
			Removable singularities
		Circle of convergence
		Taylor’s series
		Laurent’s series
		Residues
		Calculating residues
		Integrals of real functions
			Integrals of the form
	Review summary 31
	Can you?
		Checklist 31
	Test exercise 31
	Further problems 31
Programme 32 Optimization and linear programming
	Learning outcomes
		Optimization
			Linear programming (or linear optimization)
			Linear inequalities
			Graphical representation of linear inequalities
			Solver
			Solver parameters
		Applications
		Nonlinear programming
	Review summary 32
	Can you?
		Checklist 32
	Test exercise 32
	Further problems 32
Appendix
	1 Green’s theorem
		Proof of Green’s theorem
	2 Proof that sec
	3 Vector triple products
	4 Divergence theorem (Gauss’ theorem)
	5 Stokes’ theorem
		Proof of Stokes’ theorem
Answers
	Test exercise 1 (page 42)
	Further problems 1 (page 43)
	Test exercise 2 (page 90)
	Further problems 2 (page 91)
	Test exercise 3 (page 121)
	Further problems 3 (page 122)
	Test exercise 4 (page 154)
	Further problems 4 (page 155)
	Test exercise 5 (page 191)
	Further problems 5 (page 191)
	Test exercise 6 (page 236)
	Further problems 6 (page 237)
	Test exercise 7 (page 266)
	Further problems 7 (page 267)
	Test exercise 8 (page 297)
	Further problems 8 (page 298)
	Test exercise 9 (page 334)
	Further problems 9 (page 335)
	Test exercise 10 (page 357)
	Further problems 10 (page 357)
	Test exercise 11 (page 376)
	Further problems 11 (page 376)
	Test exercise 12 (page 394)
	Further problems 12 (page 395)
	Text exercise 13 (page 434)
	Further problems 13 (page 435)
	Test exercise 14 (page 478)
	Further problems 14 (page 479)
	Test exercise 15 (page 510)
	Further problems 15 (page 510)
	Test exercise 16 (page 536)
	Further problems 16 (page 536)
	Test exercise 17 (page 577)
	Further problems 17 (page 578)
	Test exercise 18 (page 600)
	Further problems 18 (page 600)
	Test exercise 19 (page 635)
	Further problems 19 (page 635)
	Test exercise 20 (page 680)
	Further problems 20 (page 680)
	Test exercise 21 (page 717)
	Further problems 21 (page 718)
	Test exercise 22 (page 761)
	Further problems 22 (page 762)
	Test exercise 23 (page 815)
	Further problems 23 (page 816)
	Test exercise 24 (page 858)
	Further problems 24 (page 858)
	Test exercise 25 (page 895)
	Further problems 25 (page 895)
	Text exercise 26 (page 941)
	Further problems 26 (page 941)
	Test exercise 27 (page 991)
	Further problems 27 (page 992)
	Test exercise 28 (page 1019)
	Further problems 28 (page 1019)
	Test exercise 29 (page 1058)
	Further problems 29 (page 1059)
	Test exercise 30 (page 1106)
	Further problems 30 (page 1107)
	Test exercise 31 (page 1136)
	Further problems 31 (page 1137)
	Test exercise 32 (page 1160)
	Further problems 32 (page 1161)
Index