Engineering Mathematics 8th Edition by K.A. Stroud, ISBN-13: 978-1352010275

Description

Trustpilot

Engineering Mathematics 8th Edition by K.A. Stroud, ISBN-13: 978-1352010275

[PDF eBook eTextbook] – Available Instantly

• Publisher: ‎ Bloomsbury Academic; 8th edition (March 27, 2020)
• Language: ‎ English
• 1180 pages
• ISBN-10: ‎ 1352010275
• ISBN-13: ‎ 978-1352010275

The best-selling introductory mathematics textbook for students on engineering and science degree and pre-degree courses. Sales stand at more than half a million copies world-wide.

Its unique programmed approach really works! Many thousands of students have found that they understand and excel through using this book. It takes you through the mathematics in a step-by-step fashion with a wealth of examples and exercises. The text demands that you engage with it by asking you to complete steps that you should be able to manage from previous examples or knowledge you have acquired, while carefully introducing new steps. By working with the authors through the examples, you become proficient as you go. By the time you come to trying examples on your own, confidence is high.

Aimed at undergraduates on Foundation and First Year degree programmes in all Engineering disciplines and Science. The Foundation section covers mathematics from GCSE onwards to allow for revision and gap-filling, and so means the book can be used for a range of abilities and all levels of access.

New to this Edition:

– A general revision of the entire contents

– In Matrices an emphasis on eigenvalues and eigenvectors and the introduction of the Cayley–Hamilton theorem

– New review summaries plus a new easy reference to help check back when you need more help

– Key chapters improved yet further as a result of detailed student feedback

Table of Contents:

Summary of contents

Contents

Preface to the first edition

Preface to the second edition

Preface to the third edition

Preface to the fourth edition

Preface to the eighth edition

How to use this book

The companion website – www.macmillanihe.com/stroud

Student review panel

The Personal Tutor

Useful background information

Symbols used in the text

Useful mathematical information

1 Algebraic identities

2 Trigonometrical identities

3 Standard curves

4 Laws of mathematics

PART I Foundation topics

Programme F.1 Arithmetic

Learning outcomes

Quiz F.1

Types of number

The natural numbers

Numerals and place value

Points on a line and order

The integers

Brackets

Addition and subtraction

Multiplication and division

Brackets and precedence rules

Rounding

Basic laws of arithmetic

1 Commutativity

2 Associativity

3 Distributivity

Estimating

Review summary

Review exercise

Factors and prime numbers

Factors

Prime numbers

Prime factorization

Fundamental theorem of arithmetic

Highest common factor (HCF)

Lowest common multiple (LCM)

Review summary

Review exercise

Fractions, ratios and percentages

Division of integers

Multiplying fractions

Of

Equivalent fractions

Dividing fractions

Adding and subtracting fractions

Fractions on a calculator

Ratios

Percentages

Review summary

Review exercise

Decimal numbers

Division of integers

Rounding

Significant figures

Decimal places

Trailing zeros

Fractions as decimals

Decimals as fractions

Unending decimals

Unending decimals as fractions

Rational, irrational and real numbers

Review summary

Review exercise

Powers

Raising a number to a power

The laws of powers

Powers on a calculator

Fractional powers and roots

Surds

Multiplication and division by integer powers of 10

Precedence rules

Standard form

Working in standard form

Using a calculator

Preferred standard form

Checking calculations

Accuracy

Review summary

Review exercise

Number systems

Denary (or decimal) system

Binary system

Octal system (base 8)

Duodecimal system (base 12)

Hexadecimal system (base 16)

An alternative method

Review summary

Review exercise

Change of base from denary to a new base

Binary form

Octal form

Duodecimal form

A denary decimal in octal form

Use of octals as an intermediate step

Reverse method

Review summary

Review exercise

Can you?

Checklist F.1

Test exercise F.1

Further problems F.1

Programme F.2 Introduction to algebra

Learning outcomes

Quiz F.2

Algebraic expressions

Symbols other than numerals

Constants

Variables

Rules of algebra

Commutativity

Associativity

Distributivity

Rules of precedence

Terms and coefficients

Collecting like terms

Similar terms

Expanding brackets

Nested brackets

Review summary

Review exercise

Powers and logarithms

Powers

Rules of indices

Logarithms

Rules of logarithms

Base 10 and base

Change of base

Logarithmic equations

Review summary

Review exercise

Algebraic multiplication and division

Multiplication

Division

Review summary

Review exercise

Algebraic fractions

Addition and subtraction

Multiplication and division

Review summary

Review exercise

Factorization of algebraic expressions

Common factors

Common factors by grouping

Useful products of two simple factors

Quadratic expressions as the product of two factors

Review summary

Review exercise

Can you?

Checklist F.2

Test exercise F.2

Further problems F.2

Programme F.3 Expressions and equations

Learning outcomes

Quiz F.3

Expressions and equations

Evaluating expressions

Equations

Conditional equation

Identity

Defining equation

Assigning equation

Formula

Evaluating independent variables

Transposition of formulas

The evaluation process

Review summary

Polynomials

Polynomial expressions

Review exercise

Evaluation of polynomials

Evaluation of a polynomial by nesting

Remainder theorem

Factor theorem

The general quadratic equation

Factorization of fourth-order polynomials

Review summary

Review exercise

Can you?

Checklist F.3

Test exercise F.3

Further problems F.3

Programme F.4 Graphs

Learning outcomes

Quiz F.4

Graphs of equations

Equations

Ordered pairs of numbers

Cartesian axes

Drawing a graph

Review summary

Review exercise

Using a spreadsheet

Spreadsheets

Rows and columns

Text and number entry

Formulas

Clearing entries

Construction of a Cartesian graph

Displays

Review summary

Review exercise

Inequalities

Less than or greater than

Review summary

Review exercise

Absolute values

Modulus

Graphs

Inequalities

Less-than inequalities

Greater-than inequalities

Interaction

Review summary

Review exercise

Can you?

Checklist F.4

Test exercise F.4

Further problems F.4

Programme F.5 Linear equations

Quiz F.5

Linear equations

Solution of simple equations

Simultaneous linear equations with two unknowns

1 Solution by substitution

2 Solution by elimination

Simultaneous linear equations with three unknowns

Pre-simplification

Review summary

Review exercise

Can you?

Checklist F.5

Test exercise F.5

Further problems F.5

Programme F.6 Polynomial equations

Learning outcomes

Quiz F.6

Polynomial equations

Quadratic equations

Cubic equations having at least one simple linear factor

Fourth-order equations having at least two linear factors

Review summary

Review exercise

Can you?

Checklist F.6

Test exercise F.6

Further problems F.6

Programme F.7 Binomials

Learning outcomes

Quiz F.7

Factorials and combinations

Factorials

Combinations

Three properties of combinatorial coefficients

Review summary

Review exercise

Binomial expansions

Pascal’s triangle

Binomial expansions

The general term of the binomial expansion

Review summary

Review exercise

The Σ (sigma) notation

General terms

The sum of the first n natural numbers

Rules for manipulating sums

The exponential number e

Review summary

Review exercise

Can you?

Checklist F.7

Test exercise F.7

Further problems F.7

Programme F.8 Partial fractions

Learning outcomes

Quiz F.8

Partial fractions

Review summary

Review exercise

Denominators with repeated and quadratic factors

Review summary

Review exercise

Can you?

Checklist F.8

Test exercise F.8

Further problems F.8

Programme F.9

Trigonometry

Learning outcomes

Quiz F.9

Angles

Rotation

Radians

Triangles

Trigonometric ratios

Reciprocal ratios

Pythagoras’ theorem

Special triangles

Half equilateral

Review summary

Review exercise

Trigonometric identities

The fundamental identity

Two more identities

Identities for compound angles

Trigonometric formulas

Sums and differences of angles

Double angles

Sums and differences of ratios

Products of ratios

Review summary

Review exercise

Can you?

Checklist F.9

Test exercise F.9

Further problems F.9

Programme F.10 Functions

Learning outcomes

Quiz F.10

Processing numbers

Functions are rules but not all rules are functions

Functions and the arithmetic operations

Inverses of functions

Graphs of inverses

The graph of y = x3

The graph of y = x1/3

The graphs of y = x3 and y = x1/3 plotted together

Review summary

Review exercise

Composition

Function of a function

Inverses of compositions

Review summary

Can you?

Checklist F.10

Review exercise

Test exercise F.10

Further problems F.10

Programme F.11 Trigonometric and exponential functions

Learning outcomes

Quiz F.11

Introduction

Trigonometric functions

Rotation

The tangent

Period

Amplitude

Phase difference

Inverse trigonometric functions

Trigonometric equations

Equations of the form acosx + bsinx = c

Review summary

Review exercise

Exponential and logarithmic functions

Exponential functions

Review summary

Review exercise

Indicial equations

Odd and even functions

Odd and even parts

Odd and even parts of the exponential function

Limits of functions

The rules of limits

Review summary

Review exercise

Can you?

Checklist F.11

Test exercise F.11

Further problems F.11

Programme F.12 Differentiation

Learning outcomes

Quiz F.12

Gradients

The gradient of a straight-line

The gradient of a curve at a given point

Algebraic determination of the gradient of a curve

Derivatives of powers of x

Two straight lines

Two curves

Differentiation of polynomials

Second and higher derivatives and an alternative notation

Review summary

Review exercise

Standard derivatives and rules

Limiting value of

Standard derivatives

Derivative of y = sinx

Derivative of y = cosx

Derivative of y = ex

Derivative of a product of functions

Derivative of a quotient of functions

Derivative of a function of a function

Derivative of y = ax

Review summary

Review exercise

Newton–Raphson iterative method

Notation

Tabular display of results

First approximations

Can you?

Checklist F.12

Review summary

Review exercise

Test exercise F.12

Further problems F.12

Programme F.13 Integration

Learning outcomes

Quiz F.13

Integration

Constant of integration

Standard integrals

Review summary

Review exercise

Integration of polynomial expressions

Functions of a linear function of x

Review summary

Review exercise

Integration by partial fractions

Review summary

Review exercise

Areas under curves

Review summary

Review exercise

Integration as a summation

The area between a curve and an intersecting line

Review summary

Review exercise

Can you?

Checklist F.13

Test exercise F.13

Further problems F.13

PART II

Programme 1 Complex numbers 1

Learning outcomes

Introduction

Ideas and symbols

The symbol j

Quadratic equations

Powers of j

Positive integer powers

Negative integer powers

Complex numbers

Addition and subtraction

Multiplication

Division

Equal complex numbers

Review exercise

Graphical representation of a complex number

Argand diagram

Graphical addition of complex numbers

Polar form of a complex number

Exponential form of a complex number

Review summary

Can you?

Checklist 1

Test exercise 1

Further problems 1

Programme 2 Complex numbers 2

Learning outcomes

Polar-form calculations

Review exercise

Roots of a complex number

Expansions

Expansions of sinnθ nd cosnθ

Expansions for cosnθ and sinnθ

Loci problems

Review summary

Can you?

Checklist 2

Test exercise 2

Further problems 2

Programme 3 Hyperbolic functions

Learning outcomes

Introduction

Graphs of hyperbolic functions

Review exercise

Evaluation of hyperbolic functions

Inverse hyperbolic functions

Log form of the inverse hyperbolic functions

Hyperbolic identities

Relationship between trigonometric and hyperbolic functions

Review summary

Can you?

Checklist 3

Determinants of the third order

Test exercise 3

Further problems 3

Programme 4 Determinants

Learning outcomes

Determinants

Evaluation of a third-order determinant

Simultaneous equations in three unknowns

Review exercise

Consistency of a set of equations

Properties of determinants

Review summary

Can you?

Checklist 4

Test exercise 4

Further problems 4

Programme 5 Matrices

Learning outcomes

Matrices – definitions

Matrix notation

Equal matrices

Addition and subtraction of matrices

Multiplication of matrices

Scalar multiplication

Multiplication of two matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Cofactors

Adjoint of a square matrix

Inverse of a square matrix

Product of a square matrix and its inverse

Solution of a set of linear equations

Gaussian elimination method for solving a set of linear equations

Eigenvectors and eigenvalues

Eigenvalues

Eigenvectors

Cayley–Hamilton theorem

Inverse matrices

Raising a matrix to a whole number power

Review summary

Can you?

Checklist 5

Test exercise 5

Further problems 5

Programme 6 Vectors

Learning outcomes

Introduction: scalar and vector quantities

Vector representation

Types of vector

Addition of vectors

The sum of a number of vectors a+b+c+d + . . .

Components of a given vector

Components of a vector in terms of unit vectors

Vectors in space

Direction cosines

Scalar product of two vectors

Vector product of two vectors

Angle between two vectors

Direction ratios

Review summary

Can you?

Checklist 6

Test exercise 6

Further problems 6

Programme 7 Differentiation

Learning outcomes

Standard derivatives

Functions of a function

Products

Quotients

Logarithmic differentiation

Review exercise

Implicit functions

Parametric equations

Review summary

Can you?

Checklist 7

Test exercise 7

Further problems 7

Programme 8 Differentiation applications

Learning outcomes

Differentiation of inverse trigonometric functions

Review exercise

Derivatives of inverse hyperbolic functions

Review exercise

Maximum and minimum values

Points of inflexion

Review summary

Can you?

Checklist 8

Test exercise 8

Further problems 8

Programme 9 Tangents, normals and curvature

Learning outcomes

Equation of a straight line

Tangents and normals to a curve at a given point

Curvature

Centre of curvature

Review summary

Can you?

Checklist 9

Test exercise 9

Further problems 9

Programme 10 Sequences

Learning outcomes

Functions with integer input

Sequences

Graphs of sequences

Arithmetic sequence

Geometric sequence

Harmonic sequence

Recursive prescriptions

Other sequences

Review summary

Review exercise

Difference equations

Solving difference equations

Second-order, homogeneous equations

Equal roots of the characteristic equation

Review summary

Review exercise

Limits of sequences

Infinity

Limits

Infinite limits

Rules of limits

Multiplication by a constant

Sums and differences

Products and quotients

Indeterminate limits

Review summary

Review exercise

Can you?

Checklist 10

Test exercise 10

Further problems 10

Programme 11 Series 1

Learning outcomes

Series

Arithmetic series

Arithmetic mean

Geometric series

Geometric mean

Series of powers of the natural numbers

Sum of natural numbers

Sum of squares

Sum of cubes

Infinite series

Limiting values

Convergent and divergent series

Tests for convergence

Test 1. A series

Test 2. The comparison test

Test 3: D’Alembert’s ratio test for positive terms

Absolute convergence

Review summary

Can you?

Checklist 11

Test exercise 11

Further problems 11

Programme 12 Series 2

Learning outcomes

Power series

Introduction

Maclaurin’s series

Series for tan x

Standard series

Review summary

The binomial series

Taylor’s series

Limiting values – indeterminate forms

L’Hȏpital’s rule for finding limiting values

Approximate values

Review summary

Can you?

Checklist 12

Test exercise 12

Further problems 12

Programme 13 Curves and curve fitting

Learning outcomes

Introduction

Standard curves

Straight line

Second-degree curves

Third-degree curves

Circle

Ellipse

Hyperbola

Logarithmic curves

Exponential curves

Hyperbolic curves

Trigonometrical curves

Tangent curve

Asymptotes

Determination of an asymptote

Asymptotes parallel to the x- and y-axes

Systematic curve sketching, given the equation of the curve

Symmetry

Intersection with the axes

Change of origin

Asymptotes

Large and small values of x and y

Stationary points

Limitations

Curve fitting

Straight-line law

Graphs of the form y = axn, where a and n are constants

Graphs of the form y = aenx

Method of least squares

Fitting a straight-line graph

Using a spreadsheet

Correlation

Correlation

Measures of correlation

The Pearson product-moment correlation coefficient

Spearman’s rank correlation coefficient

Review summary

Can you?

Checklist 13

Test exercise 13

Further problems 13

Programme 14 Partial differentiation 1

Learning outcomes

Partial differentiation

Review summary

Review exercise

Small increments

Can you?

Checklist 14

Test exercise 14

Review summary

Further problems 14

Programme 15 Partial differentiation 2

Learning outcomes

Partial differentiation

Rate-of-change problems

Change of variables

Can you?

Checklist 15

Review summary

Test exercise 15

Further problems 15

Programme 16 Integration 1

Learning outcomes

Introduction

Standard integrals

Functions of a linear function of x

Integrals of the forms

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Review summary

Can you?

Checklist 16

Test exercise 16

Further problems 16

Programme 17 Integration 2

Learning outcomes

Review summary

Can you?

Checklist 17

Test exercise 17

Further problems 17

Programme 18 Reduction formulas

Learning outcomes

Review summary

Can You?

Checklist 18

Test exercise 18

Further problems 18

Programme 19 Integration applications 1

Learning outcomes

Basic applications

Areas under curves

Definite integrals

Parametric equations

Mean values

Root mean square (rms) value

Review summary

Can you?

Checklist 19

Test exercise 19

Further problems 19

Programme 20 Integration applications 2

Learning outcomes

Introduction

Volume of a solid of revolution

Centroid of a plane figure

Centre of gravity of a solid of revolution

Length of a curve

Parametric equations

Surface of revolution

Parametric equations

Rules of Pappus

Review summary

Can you?

Checklist 20

Test exercise 20

Further problems 20

Programme 21 Integration applications 3

Learning outcomes

Moments of inertia

Radius of gyration

Parallel axes theorem

Perpendicular axes theorem (for thin plates)

Useful standard results

Second moment of area

Composite figures

Centre of pressure

Pressure at a point P, depth z below the surface of a liquid

Total thrust on a vertical plate immersed in liquid

Depth of the centre of pressure

Can you?

Checklist 21

Review summary

Test exercise 21

Further problems 21

Programme 22 Approximate integration

Learning outcomes

Introduction

Approximate integration

Method 1: By series

Method 2: By Simpson’s rule

Proof of Simpson’s rule

Review summary

Can you?

Checklist 22

Test exercise 22

Further problems 22

Programme 23 Polar coordinate systems

Learning outcomes

Introduction to polar coordinates

Polar curves

Standard polar curves

Applications

Review summary

Can you?

Checklist 23

Test exercise 23

Further problems 23

Programme 24 Multiple integrals

Learning outcomes

Summation in two directions

Double integrals

Triple integrals

Applications

Review exercise

Alternative notation

Determination of areas by multiple integrals

Determination of volumes by multiple integrals

Review summary

Can you?

Checklist 24

Test exercise 24

Further problems 24

Programme 25 First-order differential equations

Learning outcomes

Introduction

Formation of differential equations

Solution of differential equations

Method 1: By direct integration

Method 2: By separating the variables

Review exercise

Method 3: Homogeneous equations – by substituting y = vx

Review exercise

Method 4: Linear equations – use of integrating factor

Review exercise

Bernoulli’s equation

Review summary

Can you?

Checklist 25

Further problems 25

Programme 26 Second-order differential equations

Learning outcomes

Homogeneous equations

Test exercise 25

Review exercise

Inhomogeneous equations

Particular solution

Can you?

Checklist 26

Review summary

Test exercise 26

Further problems 26

Programme 27 Introduction to Laplace transforms

Learning outcomes

The Laplace transform

The inverse Laplace transform

Table of Laplace transforms

Review summary

Review exercise

Laplace transform of a derivative

Two properties of Laplace transforms

Table of Laplace transforms

Review summary

Review exercise

Generating new transforms

Laplace transforms of higher derivatives

Table of Laplace transforms

Linear, constant-coefficient, inhomogeneous differential equations

Review summary

Review exercise

Can you?

Checklist 27

Test exercise 27

Further problems 27

Programme 28 Data handling and statistics

Learning outcomes

Introduction

Arrangement of data

Tally diagram

Grouped data

Grouping with continuous data

Relative frequency

Rounding off data

Class boundaries

Histograms

Frequency histogram

Relative frequency histogram

Measures of central tendency

Mean

Coding for calculating the mean

Decoding

Coding with a grouped frequency distribution

Mode

Mode with grouped data

Median

Median with grouped data

Measures of dispersion

Mean deviation

Range

Standard deviation

Alternative formula for the standard deviation

Distribution curves

Frequency polygons

Frequency curves

Normal distribution curve

Standardized normal curve

Review summary

Can you?

Checklist 28

Test exercise 28

Further problems 28

Programme 29 Probability

Learning outcomes

Probability

Random experiments

Events

Sequences of random experiments

Combining events

Events and probabilities

Probability

Assigning probabilities

Review summary

Probabilities of combined events

Or

Non-mutually exclusive events

And – multiplication rule

Dependent events

Independent events

Probability trees

Conditional probability

Review summary

Probability distributions

Random variables

Expectation

Variance and standard deviation

Bernoulli trials

Binomial probability distribution

Expectation and standard deviation

The Poisson probability distribution

Binomial and Poisson compared

Continuous probability distributions

Normal distribution curve (or normal curve)

Standard normal curve

Review summary

Can you?

Checklist 29

Test exercise 29

Further problems 29

Answers

Part I

Quiz F.1 (page 4)

Test exercise F.1 (page 59)

Further problems F.1 (page 60)

Quiz F.2 (page 64)

Test exercise F.2 (page 94)

Further problems F.2 (page 95)

Quiz F.3 (page 98)

Test exercise F.3 (page 121)

Further problems F.3 (page 121)

Quiz F.4 (page 124)

Test exercise F.4 (page 154)

Further problems F.4 (page 154)

Quiz F.5 (page 158)

Test exercise F.5 (page 1704)

Further problems F.5 (page 170)

Quiz F.6 (page 174)

Test exercise F.6 (page 185)

Further problems F.6 (page 185)

Quiz F.7 (page 188)

Test exercise F.7 (page 212)

Further problems F.7 (page 212)

Quiz F.8 (page 216)

Test exercise F.8 (page 232)

Further problems F.8 (page 232)

Quiz F.9 (page 236)

Test exercise F.9 (page 256)

Further problems F.9 (page 257)

Quiz F.10 (page 260)

Test exercise F.10 (page 277)

Further problems F.10 (page 277)

Quiz F.11 (page 280)

Test exercise F.11 (page 305)

Further problems F.11 (page 306)

Quiz F.12 (page 310)

Test exercise F.12 (page 345)

Further problems F.12 (page 346)

Quiz F.13 (page 348)

Test exercise F.13 (page 374)

Further problems F.13 (page 375)

Part II

Test exercise 1 (page 403)

Further problems 1 (page 404)

Test exercise 2 (page 428)

Further problems 2 (page 429)

Test exercise 3 (page 451)

Further problems 3 (page 452)

Test exercise 4 (page 480)

Further problems 4 (page 481)

Test exercise 5 (page 515)

Further problems 5 (page 516)

Test exercise 6 (page 541)

Further problems 6 (page 542)

Test exercise 7 (page 560)

Further problems 7 (page 561)

Test exercise 8 (page 582)

Further problems 8 (page 582)

Test exercise 9 (page 604)

Further problems 9 (page 605)

Test exercise 10 (page 638)

Further problems 10 (page 639)

Test exercise 11 (page 663)

Further problems 11 (page 664)

Test exercise 12 (page 689)

Further problems 12 (page 689)

Test exercise 13 (page 732)

Further problems 13 (page 733)

Test exercise 14 (page 754)

Further problems 14 (page 755)

Test exercise 15 (page 771)

Further problems 15 (page 771)

Test exercise 16 (page 797)

Further problems 16 (page 797)

Test exercise 17 (page 826)

Further problems 17 (page 826)

Test exercise 18 (page 838)

Further problems 18 (page 839)

Test exercise 19 (page 856)

Further problems 19 (page 857)

Test exercise 20 (page 878)

Further problems 20 (page 878)

Test exercise 21 (page 907)

Further problems 21 (page 908)

Test exercise 22 (page 926)

Further problems 22 (page 927)

Test exercise 23 (page 949)

Further problems 23 (page 950)

Test exercise 24 (page 974)

Further problems 24 (page 975)

Test exercise 25 (page 1009)

Further problems 25 (page 1009)

Test exercise 26 (page 1034)

Further problems 26 (page 1034)

Test exercise 27 (page 1052)

Further problems 27 (page 1052)

Test exercise 28 (page 1081)

Further problems 28 (page 1082)

Test exercise 29 (page 1122)

Further problems 29 (page 1123)

Index

K.A. Stroud was formerly Principal Lecturer in the Department of Mathematics at Coventry University, UK. He is also the author of Foundation Mathematics and Advanced Engineering Mathematics, companion volumes to this book.

Dexter J. Booth was formerly Principal Lecturer in the School of Computing and Engineering at the University of Huddersfield, UK. He is the author of several mathematics textbooks and is co-author of Foundation Mathematics and Advanced Engineering Mathematics.

What makes us different?

• Instant Download

• Always Competitive Pricing

• 100% Privacy

• FREE Sample Available

• 24-7 LIVE Customer Support