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## Book Description

The pebbles used in ancient abacuses gave their name to the calculus, which today is a fundamental tool in business, economics, engineering and the sciences. This introductory book takes readers gently from single to multivariate calculus and simple differential and difference equations. Unusually the book offers a wide range of applications in business and economics, as well as more conventional scientific examples. Ideas from univariate calculus and linear algebra are covered as needed, often from a new perspective. They are reinforced in the two-dimensional case, which is studied in detail before generalisation to higher dimensions. Although there are no theorems or formal proofs, this is a serious book in which conceptual issues are explained carefully using numerous geometric devices and a wealth of worked examples, diagrams and exercises. Mathematica has been used to generate many beautiful and accurate, full-colour illustrations to help students visualise complex mathematical objects. This adds to the accessibility of the text, which will appeal to a wide audience among students of mathematics, economics and science.

1. Cover
2. Half Title
3. Dedications
4. Title Page
6. Contents
7. Preface
8. Acknowledgements
9. 1. Matrices and vectors
1. 1.1 Matrices◊
2. 1.2 Exercises
3. 1.3 Vectors in R2
4. 1.4 Exercises
5. 1.5 Vectors in R3
6. 1.6 Lines
7. 1.7 Planes
8. 1.8 Exercises
9. 1.9 Vectors in Rn
10. 1.10 Flats
11. 1.11 Exercises
12. 1.12 Applications (optional)
10. 2. Functions of one variable
1. 2.1 Intervals◊
2. 2.2 Real valued functions of one real variable◊
3. 2.3 Some elementary functions◊
4. 2.4 Combinations of functions◊
5. 2.5 Inverse functions◊
6. 2.6 Inverses of the elementary functions◊
7. 2.7 Derivatives◊
8. 2.8 Existence of derivatives◊
9. 2.9 Derivatives of inverse functions◊
10. 2.10 Calculation of derivatives
11. 2.11 Exercises
12. 2.12 Higher order derivatives
13. 2.13 Taylor series for functions of one variable
14. 2.14 Conic sections
15. 2.15 Exercises
11. 3. Functions of several variables
1. 3.1 Real valued functions of two variables
2. 3.2 Partial derivatives
3. 3.3 Tangent plane
5. 3.5 Derivative
6. 3.6 Directional derivatives
7. 3.7 Exercises
8. 3.8 Functions of more than two variables
9. 3.9 Exercises
10. 3.10 Applications (optional)
12. 4. Stationary points
13. 5. Vector functions
14. 6. Optimisation of scalar valued functions
1. 6.1 Change of basis in quadratic forms◊
2. 6.2 Positive and negative definite
3. 6.3 Maxima and minima
4. 6.4 Convex and concave functions
5. 6.5 Exercises
6. 6.6 Constrained optimisation
8. 6.8 Lagrange’s method – optimisation with one constraint
9. 6.9 Lagrange’s method – general case♣
10. 6.10 Constrained optimisation – analytic criteria♣
11. 6.11 Exercises
12. 6.12 Applications (optional)
15. 7. Inverse functions
1. 7.1 Local inverses of scalar valued functions
2. 7.2 Local inverses of vector valued functions
3. 7.3 Coordinate systems
4. 7.4 Polar coordinates
5. 7.5 Differential operators♣
6. 7.6 Exercises
7. 7.7 Application (optional): contract curve
16. 8. Implicit functions
17. 9. Differentials
18. 10. Sums and integrals
1. 10.1 Sums◊
2. 10.2 Integrals◊
3. 10.3 Fundamental theorem of calculus◊
4. 10.4 Notation◊
5. 10.5 Standard integrals◊
6. 10.6 Partial fractions◊
7. 10.7 Completing the square◊
8. 10.8 Change of variable◊
9. 10.9 Integration by parts◊
10. 10.10 Exercises
11. 10.11 Infinite sums and integrals♣
12. 10.12 Dominated convergence♣
13. 10.13 Differentiating integrals♣
14. 10.14 Power series♣
15. 10.15 Exercises
16. 10.16 Applications (optional)
19. 11. Multiple integrals
1. 11.1 Introduction
2. 11.2 Repeated integrals
3. 11.3 Change of variable in multiple integrals♣
4. 11.4 Unbounded regions of integration♣
5. 11.5 Multiple sums and series♣
6. 11.6 Exercises
7. 11.7 Applications (optional)
20. 12. Differential equations of order one
21. 13. Complex numbers
2. 13.2 Complex numbers
3. 13.3 Modulus and argument
4. 13.4 Exercises
5. 13.5 Complex roots
6. 13.6 Polynomials
7. 13.7 Elementary functions♣
8. 13.8 Exercises
9. 13.9 Applications (optional)
22. 14. Linear differential and difference equations
1. 14.1 The operator P(D)
2. 14.2 Difference equations and the shift operator E
3. 14.3 Linear operators♣
4. 14.4 Homogeneous, linear, differential equations♣
5. 14.5 Complex roots of the auxiliary equation
6. 14.6 Homogeneous, linear, difference equations
7. 14.7 Nonhomogeneous equations♣
8. 14.8 Convergence and divergence♣
9. 14.9 Systems of linear equations♣
10. 14.10 Change of variable♣
11. 14.11 Exercises
12. 14.12 The difference operator (optional)♣
13. 14.13 Exercises
14. 14.14 Applications (optional)
23. Answers to starred exercises with some hints and solutions
24. Appendix
25. Index