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## About this book

A long-standing, best-selling, comprehensive textbook covering all the mathematics required on upper level engineering mathematics undergraduate courses. Its unique programmed approach takes students through the mathematics they need in a step-by-step fashion with a wealth of examples and exercises. The text demands that students engage with it by asking them to complete steps that they should be able to manage from previous examples or knowledge they have acquired, while carefully introducing new steps. By working with the authors through the examples, students become proficient as they go. By the time they come to trying examples on their own, confidence is high.

This textbook is ideal for undergraduates on upper level courses in all Engineering disciplines and Science.

## Table of Contents

### Programme 1. Numerical solutions of equations and interpolation

Abstract
Appreciate the Fundamental Theorem of Algebra Find the two roots of a quadratic equation and recognise that for polynomial equations with real coefficients complex roots exist in complex conjugate pairs Use the relationships between the coefficients and the roots of a polynomial equation to find the roots of the polynomial Transform a cubic equation to its reduced form Use Tartaglia solution to find the roots of a cubic equation Find the solution of the equation the method of bisection Solve equations involving a single real variable by iteration and use a spreadsheet for efficiency Solve equations using the Newton-Raphson iterative method Use the modified Newton-Raphson method to find the first approximation when the derivative is small Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 2. Laplace transforms 1

Abstract
Obtain the Laplace transforms of simple standard expressions Use the first shift theorem to find the Laplace transform of a simple expression multiplied by an exponential Find the Laplace transform of a simple expression multiplied or divided by a variable Use partial fractions to find the inverse Laplace transform Use the cover up rule Use the Laplace transforms of derivatives to solve differential equations Use the Laplace transform to solve simultaneous differential equations The solution of a linear, ordinary differential equation with constant coefficients such as the second-order equation
K.A. Stroud, Dexter Booth

### Programme 3. Laplace transforms 2

Abstract
When you have completed this Programme you will be able to Use the Heaviside unit step function to switch expressions on and off Obtain the Laplace transform of expressions involving the Heaviside unit step function Solve linear, constant coefficient ordinary differential equations with piecewise continuous right-hand sides Understand what is meant by the convolution of two functions and use the convolution theorem to find the inverse transform of a product of transforms In the previous Programme, we dealt with the Laplace transforms of continuous functions of t. In practical applications, it is convenient to have a function which, in effect, switches on or switches off a given term at predescribed values of t. This we can do with the Heaviside unit step function.
K.A. Stroud, Dexter Booth

### Programme 4. Laplace transforms 3

Abstract
When you have completed this Programme you will be able to Find the Laplace transforms of periodic functions Obtain the inverse Laplace transforms of transforms of periodic functions Describe and use the unit impulse to evaluate integrals Obtain the Laplace transform of the unit impulse Use the Laplace transform to solve differential equations involving the unit impulse Solve the equation and describe the behaviour of an harmonic oscillator
K.A. Stroud, Dexter Booth

### Programme 5. Difference equations and the Z transform

Abstract
Convert the descriptive prescription of the output form of a sequence into a recursive description and recognise the importance of initial terms Recognise a difference equation, determine its order and generate its terms from a recursive description Obtain the solution to a difference equation as a sum of the homogeneous solution and the particular solution Define the Z transform of a sequence and derive transforms of specified sequences Make reference to a table of standard Z transforms Recognise the Z transform as being a linear transform and so obtain the transform of linear combinations of standard sequences Apply the first and second shift theorems, the translation theorem, the initial and final value theorems and the derivative theorem Use partial fractions to derive the inverse transforms Use the Z transform to solve linear, constant coefficient difference equations Create a sequence by sampling a continuous function and demonstrate the relationship between the Laplace and the Z transform
K.A. Stroud, Dexter Booth

### Programme 6. Introduction to invariant linear systems

Abstract
Recognise a system as a process whereby an input (either continuous or discrete) is converted to an output, also called the response of the system Distinguish between linear and non-linear systems and recognise time-invariant and shift-invariant systems Determine the zero-input response and the zero-state response Appreciate why zero valued boundary conditions give rise to a time-invariant system Demonstrate that the response of a continuous, linear, time-invariant system to an arbitrary input is the convolution of the input with response of the system to a unit impulse Understand the role of the exponential function with respect to a linear, time-invariant system Use the convolution theorem to find the response of a continuous, linear, time-invariant system to an arbitrary input Derive the system transfer function of a constant coefficient linear differential equation and use it to solve the equation Demonstrate that the response of a discrete, linear, shift-invariant system to an arbitrary input is the convolution sum of the input with response of the system to a unit impulse Understand the role of the exponential function with respect to a discrete linear, shift-invariant system Derive the system transfer function of a constant coefficient linear difference equation and use it to solve the equation Derive the constant coefficient difference equation from knowledge of its unit impulse response.
K.A. Stroud, Dexter Booth

### Programme 7. Fourier series 1

Abstract
When you have completed this Programme you will be able to: Determine the period and amplitude of a periodic function Write down the harmonics of a periodic trigonometric function Give an analytic description of a non-sinusoidal periodic function Evaluate integrals with periodic integrands Demonstrate the orthogonality of the trigonometric functions sin nx and cos nx for n 0, 1, 2,.. Describe a periodic function as a Fourier series subject to Dirichlet conditions Obtain the Fourier coefficients and hence the Fourier series of a periodic function Describe the effects of the harmonics in the construction of the Fourier series Find the value of the Fourier series at a point of discontinuity of the periodic function
K.A. Stroud, Dexter Booth

### Programme 8. Fourier series 2

Abstract
When you have completed this Programme you will be able to Obtain the Fourier coefficients of a function with arbitrary period T. Recognise even and odd functions and their products. Derive the Fourier series of even and odd functions. Derive half-range Fourier series . Recognise the conditions for the Fourier series to contain only odd or only even harmonics. Explain the geometric significance of the constant term a0=2. Derive half-range Fourier series with arbitrary period
K.A. Stroud, Dexter Booth

### Programme 9. Introduction to the Fourier transform

Abstract
When you have completed this Programme you will be able to Convert a trigonometric Fourier series into a doubly infinite sum of complex exponentials Derive the complex Fourier series of a function that satisfies Dirichlet’s conditions Recognise the function sinc Separate a discrete complex spectrum into an amplitude spectrum and a phase spectrum State Fourier’s integral theorem in terms of complex exponentials Define and derive the Fourier transform of a function satisfying Dirichle conditions Separate a continuous complex spectrum into an amplitude spectrum and a phase spectrum Recognise the functions and derive their Fourier transforms along with those of the Dirac delta and the Heaviside unit step Recognise alternative forms of the function-transform pair Reproduce a collection of properties of the Fourier transform Evaluate the convolution of two functions and describe its Fourier transform Derive the Fourier sine and cosine transformations.
K.A. Stroud, Dexter Booth

### Programme 10. Power series solutions of ordinary differential equations 1

Abstract
When you have completed this Programme you will be able to Obtain the nth derivative of the exponential, circular and hyperbolic functions Apply the Leibnitz theorem to derive the nth derivative of a product of expressions Use the Leibnitz-Maclaurin method of obtaining a series solution to a second-order homogeneous differential equation with constant coefficients Solve Cauchy-Euler equi-dimensional equations.
K.A. Stroud, Dexter Booth

### Programme 11. Power series solutions of ordinary differential equations 2

Abstract
Apply Frobenius method of obtaining a series solution to a second-order homogeneous ordinary differential equation by first differentiating the assumed series several times Substitute into the differential equation and equate coefficients of corresponding powers Derive the indicial equation Distinguish between the possible four distinct outcomes arising from the indicial equation In the previous Programme we established the solutions of second order ordinary differential equations as power series in integer powers of x. Such solutions are not always possible and a more general method is to assume a trial solution of the formAppreciate the Fundamental Theorem of Algebra Find the two roots of a quadratic equation and recognise that for polynomial equations with real coefficients complex roots exist in complex conjugate pairs Use the relationships between the coefficients and the roots of a polynomial equation to find the roots of the polynomial Transform.
K.A. Stroud, Dexter Booth

### Programme 12. Power series solutions of ordinary differential equations 3

Abstract
Apply Frobenius method to Bessel equation to derive Bessel functions of the first kind Apply Frobenius method to Legendre equation to derive Legendre polynomials Use Rodrigue formula to derive Legendre polynomials and the generating function to obtain some of their properties Recognise a Sturm-Liouville system and the orthogonality properties of its eigenfunctions Write a polynomial in x as a finite series of Legendre polynomials A common feature of certain differential equations is that they appear in a multiplicity of guises in the application of mathematics to problems in physics and engineering. For example, Bessel’s equation appears in the study of electromagnetic radiation, heat conduction, vibrational modes of a membrane and signal processing to name but a few. Many of these equations have solutions (called special functions) in the form of infinite series that are accessible by the method of Frobenius and in this Programme we shall consider two of these equations, namely Bessels equation and Legendres equation.
K.A. Stroud, Dexter Booth

### Programme 13. Numerical solutions of equations and interpolation

Abstract
The range of differential equations that can be solved by straightforward analytical methods is relatively restricted. Even solution in series may not always be satisfactory, either because of the slow convergence of the resulting series or because of the involved manipulation in repeated stages of differentiation. In such cases, where a differential equation and known boundary conditions are given, an approximate solution is often obtainable by the application of numerical methods, where a numerical solution is obtained at discrete values of the independent variable. The solution of differential equations by numerical methods is a wide subject. The present Programme introduces some of the simpler methods, which nevertheless are of practical use.
K.A. Stroud, Dexter Booth

### Programme 14. Partial differentiation

Abstract
Derive a form of Taylor’s series from Maclaurin’s series and from it describe a function increment as a series of first and higher-order derivatives of the function Describe and apply by means of a spreadsheet the Euler method, the Euler-Cauchy method and the Runge-Kutta method for first-order differential equations . Describe and apply by means of a spreadsheet the Euler second-order method and the Runge-Kutta method for second-order ordinary differential equations Describe and apply by means of a spreadsheet a simple predictor-corrector method. The range of differential equations that can be solved by straightforward analytical methods is relatively restricted. Even solution in series may not always be satisfactory, either because of the slow convergence of the resulting series or because of the involved manipulation in repeated stages of differentiation. In such cases, where a differential equation and known boundary conditions are given, an approximate solution is often obtainable by the application of numerical methods, where a numerical solution is obtained at discrete values of the independent variable. The solution of differential equations by numerical methods is a wide subject. The present Programme introduces some of the simpler methods, which nevertheless are of practical use.
K.A. Stroud, Dexter Booth

### Programme 15. Partial differential equations

Abstract
Summarise the introductory methods of solving ordinary differential equations Solve partial differential equations that are amenable to solution by direct integration Apply initial and boundary conditions Solve the one-dimensional wave and heat equations by separating the variables and obtaining eigenfunctions and corresponding eigenvalues Solve the two-dimensional Laplace equation in Cartesian coordinates Recognise the need for alternative coordinate systems and solve the two-dimensional Laplace equation in plane polar coordinatesUse the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 16. Matrix algebra

Abstract
Determine whether a matrix is singular or non-singular Determine the rank of a matrix Determine the consistency of a set of linear equations and hence demonstrate the uniqueness of their solution Obtain the solution of a set of simultaneous linear equations by using matrix inversion, by row transformation, by Gaussian elimination, by triangular decomposition and by using an electronic spreadsheet Use matrices to represent transformations between coordinate systems Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 17. Systems of ordinary differential equations

Abstract
Obtain the eigenvalues and corresponding eigenvectors of a square matrix Demonstrate the validity of the Cayley-Hamilton theorem Solve systems of first-order ordinary differential equations using eigenvalue and eigenvector methods Construct the modal matrix from the eigenvectors of a matrix and the spectral matrix from the eigenvalues Solve systems of second-order ordinary differential equations using diagonalisation.
K.A. Stroud, Dexter Booth

### Programme 18. Numerical solutions of partial differential equations

Abstract
Derive the finite difference formulas for the first partial derivatives of a function of two real variables and construct the central finite difference formula to represent a first-order partial differential equation Draw a rectangular grid of points overlaid on the domain of a function of two real variables and evaluate the function at the boundary grid points Construct the computational molecule for a first-order partial differential equation in two real variables and use the molecule to evaluate the solutions to the equation at the grid points interior to the boundary Describe the solution as a set of simultaneous linear equations and use matrices to represent them Invert the coefficient matrix and thereby represent the solution to the partial differential equation as a column matrix Take account of a boundary condition in the form of the derivative normal to the boundary Obtain the central finite difference formulas for the second derivatives of a function of two real variables and construct finite difference formulas for second-order partial differential equations Use the forward difference formula for the first time derivatives in partial differential equations involving time and distance Use the Crank-Nicolson procedure for a partial differential equation involving a first time derivative Appreciate the use of dimensional analysis in the conversion of a partial differential equation modelling a physical system into a dimensionless equation
K.A. Stroud, Dexter Booth

### Programme 19. Multiple integration 1

Abstract
Evaluate double and triple integrals and apply them to the determination of the areas of plane figures and the volumes of solids Understand the role of the differential of a function of two or more real variables Determine exact differentials in two real variables and their integrals Evaluate the area enclosed by a closed curve by contour integration Evaluate line integrals and appreciate their properties Evaluate line integrals around closed curves within a simply connected region Link line integrals to integrals along the x-axis Link line integrals to integrals along a contour given in parametric form Discuss the dependence of a line integral between two points on the path of integration Determine exact differentials in three real variables and their integrals Demonstrate the validity and use of Green’s theorem
K.A. Stroud, Dexter Booth

### Programme 20. Multiple integration 2

Abstract
Evaluate double integrals and surface integrals Relate three-dimensional Cartesian coordinates to cylindrical and spherical polar forms Evaluate volume integrals in Cartesian coordinates and in cylindrical and spherical polar coordinates Use the Jacobian to convert integrals given in Cartesian coordinates into general curvilinear coordinates in two and three dimensions Use the modified Newton-Raphson method to find the first approximation when the derivative is small Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced.
K.A. Stroud, Dexter Booth

### Programme 21. Integral functions

Abstract
Derive the recurrence relation for the gamma function and evaluate the gamma function for certain rational arguments Evaluate integrals that require the use of the gamma function in their solution Identify the beta function and evaluate integrals that require the use of the beta function in their solution Derive the relationship between the gamma function and the beta function Use the duplication formula to evaluate the gamma function for half integer arguments Recognise the error function and its relation to the Gaussian probability distribution Recognise elliptic functions of the first and second kind Evaluate integrals that require the use of elliptic functions in their solution Use alternative forms of the elliptic functions Solve equations using the Newton-Raphson iterative method Use the modified Newton-Raphson method to find the first approximation when the derivative is small Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced.
K.A. Stroud, Dexter Booth

### Programme 22. Vector analysis 1

Abstract
Obtain the scalar and vector product of two vectors Reproduce the relationships between the scalar and vector products of the Cartesian coordinate unit vectors Obtain the scalar and vector triple products and appreciate their geometric significance Differentiate a vector field and derive a unit vector tangential to the vector field at a point Integrate a vector field Obtain the gradient of a scalar field, the directional derivative and a unit normal to a surface Obtain the divergence of a vector field and recognise a solenoidal vector field Obtain the curl of a vector field Obtain combinations of div, grad and curl acting on scalar and vector fields as appropriate
K.A. Stroud, Dexter Booth

### Programme 23. Vector analysis 2

Abstract
Evaluate the line integral of a scalar and a vector field in Cartesian coordinates Evaluate the volume integral of a vector field Evaluate the surface integral of a scalar and a vector field Determine whether or not a vector field is a conservative vector field Apply Gauss divergence theorem Apply Stokes theorem Determine the direction of unit normal vectors to a surface Apply Green theorem in the plane Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 24. Vector analysis 3

Abstract
Derive the family of curves of constant coordinates for curvilinear coordinates Derive unit base vectors and scale factors in orthogonal curvilinear coordinates Obtain the element of arc ds and the element of volume dV in orthogonal curvilinear coordinates Obtain expressions for the operators grad, div and curl in orthogonal curvilinear coordinates Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 25. Complex analysis 1

Abstract
The foundations of complex numbers and their application to hyperbolic functions were treated fully in Programmes 1, 2 and 3 of Engineering Mathematics (Sixth Edition) and these provide valuable revision should you feel it to be necessary before embarking on the new work. It will be assumed that you are already familiar with the material covered in those previous Programmes and it would be a wise move to work through the relevant Test exercises to refresh your memory on this all-important part of the course. Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 26. Complex analysis 2

Abstract
Appreciate when the derivative of a function of a complex variable exists Understand the notions of regular functions and singularities and be able to obtain the derivative of a regular function from first principles Derive the Cauchy-Riemann equations and apply them to find the derivative of a regular function Understand the notion of an harmonic function and derive a conjugate function Evaluate line and contour integrals in the complex plane Derive and apply Cauchy’s theorem Apply Cauchy’s theorem to contours around regions that contain singularities Define the essential characteristics of and conditions for a conformal mapping Locate critical points of a function of a complex variable Determine the image in the w-plane of a figure in the z-plane under a conformal transformation Describe and apply the Schwarz-Christoffel transformation
K.A. Stroud, Dexter Booth

### Programme 27. Complex analysis 3

Abstract
Expand a function of a complex variable about the origin in a Maclaurin seriesDetermine the circle and radius of convergence of a Maclaurin series expansionRecognise singular points in the form of poles of order n, removable and essential singularitiesExpand a function of a complex variable about a point in the complex plane in a Taylor series, transforming the coordinates with a shift of originExpand a function of a complex variable about a singular point in a Laurent series Recognise the principal and analytic parts of the Laurent series and link the form of the principal part to the type of singularityRecognise the residue of a Laurent series and state the Residue theoremCalculate the residues at the poles of an expression without resort to deriving the Laurent seriesEvaluate certain types of real integrals using the Residue theorem Understand the meaning of interpolation and use simple linear and graphical interpolation Use the Gregory-Newton interpolation formula with forward and backward differences for equally spaced domain points Use the Gauss interpolation formulas using central differences for equally spaced domain points Use Lagrange interpolation when the domain points are not equally spaced
K.A. Stroud, Dexter Booth

### Programme 28. Optimization and linear programming

Abstract
Describe an optimization problem in terms of the objective function and a set of constraints Algebraically manipulate and graphically describe inequalities Solve a linear programming problem in two real variables Use the simplex method to describe a linear programming problem in two real variables as a problem in two real variables with two slack variables Set up the simplex tableau and compute the simplex Use the simplex method to solve a linear programming problem in three real variables with three slack variables Introduce artificial variables into the solution method as and when the need arises Solve minimisation problems using the simplex method Construct the algebraic form of the objective function and the constraints for a problem stated in words.
K.A. Stroud, Dexter Booth
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