Vector calculus is the foundation stone on which a vast amount of applied mathematics is based. Topics such as fluid dynamics, solid mechanics and electromagnetism depend heavily on the calculus of vector quantities in three dimensions. This book covers the material in a comprehensive but concise manner, combining mathematical rigour with physical insight. There are many diagrams to illustrate the physical meaning of the mathematical concepts, which is essential for a full understanding of the subject. Each chapter concludes with a summary of the most important points, and there are worked examples that cover all of the material. The final chapter introduces some of the most important applications of vector calculus, including mechanics and electromagnetism.
Content Level » Lower undergraduate
Keywords » Algebra - Calculus - Vector calculus
Related subjects » Applications - Engineering - Theoretical, Mathematical & Computational Physics
Author(s): Paul C. Matthews
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer
Year: 1998
Language: English
Pages: 182
Front Matter
Cover
Springer Undergraduate Mathematics Series
Advisory Board & List of Publist Books
Vector Calculus
Copyright
©1998 Springer-Verlag London Limited
ISBN 3540761802
QA433.M38 1998 515'.63-dc21
LCCN 97-41191
Second Indian Reprint 2008
ISBN 978-81-8128-295-8
Preface
Table of Contents
1 Vector Algebra
1.1 Vectors and scalars
1.1.1 Definition of a vector and a scalar
1.1.2 Addition of vectors
1.1.3 Components of a vector
1.2 Dot product
1.2.1 Applications of the dot product
EXERCISES
1.3 Cross product
1.3.1 Applications of the cross product
1.4 Scalar triple product
1.5 Vector triple product
1.6 Scalar fields and vector fields
Summary of Chapter 1
EXERCISES
2 Line, Surface and Volume Integrals
2.1 Applications and methods of integration
2.1.1 Examples of the use of integration
2.1.2 Integration by substitution
2.1.3 Integration by parts
2.2 Line integrals
2.2.1 Introductory example: work done against a force
2.2.2 Evaluation of line integrals
2.2.3 Conservative vector fields
2.2.4 Other forms of line integrals
EXERCISES
2.3 Surface integrals
2.3.1 Introductory example: flow through a pipe
2.3.2 Evaluation of surface integrals
2.3.3 Other forms of surface integrals
2.4 Volume integrals
2.4.1 Introductory example: mass of an object with variable density
2.4.2 Evaluation of volume integrals
Summary of Chapter 2
EXERCISES
3 Gradient, Divergence and Curl
3.1 Partial differentiation and Taylor series
3.1.1 Partial differentiation
3.1.2 Taylor series in more than one variable
3.2 Gradient of a scalar field
3.2.1 Gradients, conservative fields and potentials
3.2.2 Physical applications of the gradient
EXERCISES
3.3 Divergence of a vector field
3.3.1 Physical interpretation of divergence
3.3.2 Laplacian of a scalar field
3.4 Curl of a vector field
3.4.1 Physical interpretation of curl
3.4.2 Relation between curl and rotation
3.4.3 Curl and conservative vector fields
Summary of Chapter 3
EXERCISES
4 Suffix Notation and its Applications
4.1 Introduction to suffix notation
4.2 The Kronecker delta \delta_ij
4.3 The alternating tensor \epsilon_ijk
4.4 Relation between \epsilon_ijk and \delta_ij
EXERCISES
4.5 Grad, div and curl in suffix notation
4.6 Combinations of grad, div and curl
4.7 Grad, div and curl applied to products of functions
Summary of Chapter 4
EXERCISES
5 Integral Theorems
5.1 Divergence theorem
5.1.1 Conservation of mass for a fluid
5.1.2 Applications of the divergence theorem
5.1.3 Related theorems linking surface and volume integrals
EXERCISES
5.2 Stokes's theorem
5.2.1 Applications of Stokes's theorem
5.2.2 Related theorems linking line and surface integrals
Summary of Chapter 5
EXERCISES
6 Curvilinear Coordinates
6.1 Orthogonal curvilinear coordinates
6.2 Grad, div and curl in orthogonal curvilinear coordinate systems
6.2.1 Gradient
6.2.2 Divergence
6.2.3 Curl
EXERCISES
6.3 Cylindrical polar coordinates
6.4 Spherical polar coordinates
Summary of Chapter 6
EXERCISES
7 Cartesian Tensors
7.1 Coordinate transformations
7.2 Vectors and scalars
7.3 Tensors
7.3.1 The quotient rule
EXERCISES
7.3.2 Symmetric and anti-symmetric tensors
7.3.3 Isotropic tensors
7.4 Physical examples of tensors
7.4.1 Ohm's law
7.4.2 The inertia tensor
Summary of Chapter 7
EXERCISES
8 Applications of Vector Calculus
8.1 Heat transfer
8.2 Electromagnetism
8.2.1 Electrostatics
8.2.2 Electromagnetic waves in a vacuum
EXERCISES
8.3 Continuum mechanics and the stress tensor
8.4 Solid mechanics
8.5 Fluid mechanics
8.5.1 Equation of motion for a fluid
8.5.2 The vorticity equation
8.5.3 Bernoulli's equation
Summary of Chapter 8
EXERCISES
Solutions
Solutions to Exercises for Chapter 1
Solutions to Exercises for Chapter 2
Solutions to Exercises for Chapter 3
Solutions to Exercises for Chapter 4
Solutions to Exercises for Chapter 5
Solutions to Exercises for Chapter 6
Solutions to Exercises for Chapter 7
Solutions to Exercises for Chapter 8
Back Matter
Index
Back Cover
