# Vectors in Physics and Engineering

## Preview

## Book Description

This text is an introduction to the use of vectors in a wide range of undergraduate disciplines. It is written specifically to match the level of experience and mathematical qualifications of students entering undergraduate and Higher National programmes and it assumes only a minimum of mathematical background on the part of the reader. Basic mathematics underlying the use of vectors is covered, and the text goes from fundamental concepts up to the level of first-year examination questions in engineering and physics. The material treated includes electromagnetic waves, alternating current, rotating fields, mechanisms, simple harmonic motion and vibrating systems. There are examples and exercises and the book contains many clear diagrams to complement the text. The provision of examples allows the student to become proficient in problem solving and the application of the material to a range of applications from science and engineering demonstrates the versatility of vector algebra as an analytical tool.

## Table of Contents

Preface

1 Vector algebra I: Scaling and adding vectors

- INTRODUCTION TO SCALARS, NUMBERS AND VECTORS
- Scalars and numbers
- Introducing vectors
- Displacements and arrows
- Vector notation
- SCALING VECTORS AND UNIT VECTORS
- Scaling a vector or multiplication of a vector by a number
- Unit Vectors
- VECTOR ADDITION-THE TRIANGLE ADDITION RULE
- LINEAR COMBINATIONS OF VECTORS
- CARTESIAN VECTORS
- Cartesian coordinates of a point-a review
- Cartesian unit vectors and cartesian components of a vector
- MAGNITUDES AND DIRECTIONS OF CARTESIAN VECTORS
- SCALING AND ADDING CARTESIANVECTORS
- VECTORS IN SCIENCE AND ENGINEERING
- Definition of a vector and evidence for vector behavior
- Vector problems in science and engineering

- Vector algebra II: Scalar products and vector products
- THE SCALAR PRODUCT
- Definition of the scalar product and projections
- The scalar product in vector algebra
- CARTESIAN FORM OF THE SCALAR PRODUCT
- THE ANGLE BETWEEN TWO VECTORS
- THE VECTOR PRODUCT
- Definition of the vector product
- The vector product in vector algebra
- CARTESIAN FORM OF THE VECTOR PRODUCT
- TRIPLE PRODUCTS OF VECTORS
- The scalar triple product
- The vector triple product
- SCALAR AND BECTOR PRODUCTS IN SCIENCE AND ENGINEERING
- Background summary: Forces, torque and equilibrium
- Background summary: work and energy
- Background summary: Engergy and torque on dipoles in electric and magnetic fields
- Time-dependent vectors
- INTRODUCTING VECTOR FUNCTIONS
- Scalar functions – a review
- Vector functions of time
- DIFFERENTIATING VECTOR FUNCTIONS – DEFINITIONS OF VELOCITY AND ACCELERATION
- Differentiation of a scalar function – a review
- Differentiation of a vector function
- Definition of velocity and acceleration
- RULES OF DIFFERENTIATION OF VECTOR FUNCTIONS
- ROTATIONAL MOTION- THE ANGULAR VELOCITY VECTOR
- ROTATING VECTORS OF CONSTANT MAGNITUDE
- APPLICATION TO RELATIVE MOTION AND INTERIAL FORCES
- Relative translational motion and inertial forces
- Relative rotational motion and inertial forces
- Scalar and vector fields
- Pictorial representations of fields
- Scalar field contours
- Vector field lines
- SCALAR FIELD FUNCTIONS
- Specifying scalar field functions
- Cartesian scalar fields
- Graphs and contours
- VECTOR FIELD FUNCTIONS
- Specifying vector field functions
- Cartesian vector fields
- Equation of a field line
- POLAR COORDINATE SYSTEMS
- Symmetries and coordinate systems
- Cylindrical polar coordinate systems
- Spherical polar coordinate systems
- INTRODUCING FLUX AND CIRCULATION
- Flux of a vector field
- Circulation of a vector field
- Differentiating fields
- Directional Derivatives and Partial Derivatives
- GRADIENT OF A SCALAR FIELD
- Introducing gradient
- Calculating gradients
- Gradient and physical law
- DIVERGENCE OF A VECTOR FIELD
- Introducing divergence
- Calculating divergence
- Divergence and physical law
- CURL OF A VECTOR FIELD
- Introducing curl
- Calculating curl
- Curl and physical law
- THE VECTOR DIFFERENTIAL OPERATOR "DEL"
- Introducing differential operators
- The "del" operator
- The Laplacian operator
- Vector-field identities
- Integrating fields
- DEFINITE INTEGRALS – A REVIEW
- LINE INTEGRALS
- Defining the scalar line integral
- Evaluating simple line integrals
- LINE INTEGRALS ALONG PARAMETERISED CURVES
- Parameterisation of a curve
- A systematic technique for evaluating line integrals
- CONSERVATIVE FIELDS
- SURFACE INTEGRALS
- Introducing surface integrals
- Expressing surface integrals as double integrals and evaluating them
- STOKES’S THEOREM
- An integral form of curl
- Deriving Stokes’s theorem
- Using Stokes’s theorem
- VOLUME INTEGRALS
- GAUSS’S THEOREM (THE DIVERGENE THEOREM)

Appendix A SI units and physical constants

Appendix B Mathematical conventions and useful results

Answers to selected Problems

Index