Calculus: Concepts and Methods

Citation

Binmore, K., & Davies, J. (2002). Calculus: Concepts and Methods. Cambridge University Press.

Chapter Summary

The book takes the reader gently from single-variable to multivariable calculus and on to simple differential and difference equations. Ideas from univariate calculus and linear algebra are introduced when needed, often from a fresh angle, and the two-dimensional case is studied in detail before the generalisation to higher dimensions. There are no theorems or formal proofs; the explanations rely on geometric devices, worked examples and diagrams.

Chapter 1: Matrices and Vectors

Calculus opens with linear algebra: matrices and vectors, with their operations and transformations, are set up first because multivariable calculus is written in their language. The economic illustrations include commodity bundles and price vectors.

Chapter 2: Functions of One Variable

A review of single-variable calculus from the book's own viewpoint, covering power, exponential and trigonometric functions, derivatives, their calculation and higher-order derivatives.

Chapters 3–4: Functions of Several Variables; Stationary Points

Functions of several variables are studied through partial derivatives and the gradient. The next chapter finds and classifies stationary points, for functions of one variable and several variables, with and without constraints.

Chapters 5–6: Vector Functions; Optimisation of Scalar Valued Functions

Vector-valued functions and their derivatives lead to the chain rule and to Taylor series for scalar-valued functions of several variables. The optimisation chapter treats convex and concave functions and constrained optimisation by Lagrange's method.

Chapters 7–9: Inverse Functions; Implicit Functions; Differentials

These chapters cover when a function can be inverted, what the implicit function theorem says about equations that define a variable implicitly, and how differentials give small-change approximations.

Chapters 10–11: Sums and Integrals; Multiple Integrals

Integration begins with sums, integration by parts and the fundamental theorem of calculus, and extends to multiple integrals, including integrals over unbounded regions.

Chapters 12–14: Differential Equations of Order One; Complex Numbers; Linear Differential and Difference Equations

First-order differential equations are followed by complex numbers, which are needed for the last chapter on linear differential and difference equations. The book ends with answers to the starred exercises, with some hints and solutions, running to about 80 pages.

Key Concepts

Chapters 1–2: Matrices, Vectors and One Variable

• Matrix and vector operations: the algebra in which the multivariable material is expressed.
• Derivative of a function of one variable: reconsidered as a base for the multivariable chapters.

Chapters 3–6: Several Variables and Optimisation

• Partial derivative and gradient: the rates of change of a function of several variables.
• Stationary point: a point where the first-order conditions hold, classified with the second-order information.
• Lagrange's method: constrained optimisation, applied to economic problems.

Chapters 7–11: Inverses, Implicit Functions and Integrals

• Implicit function theorem: the condition under which an equation defines one variable as a function of others.
• Differential: the linear approximation used for small changes.
• Multiple integral: integration over regions in the plane and in higher dimensions.

Chapters 12–14: Differential and Difference Equations

• First-order differential equation: a rate of change specified in terms of the current state.
• Linear difference equation: the discrete-time counterpart, used for models that evolve in steps.

Critical Analysis

Strengths

• Gentle but serious. Cambridge describes the book as introductory yet "serious", with conceptual issues explained carefully through geometric devices, worked examples, diagrams and exercises (345 exercises and 649 illustrations).
• A route for non-specialists. The publisher says the text appeals to students of mathematics, economics and science, and that the range of applications in business and economics is unusual for a calculus text.
• Linear algebra built in. Ideas from linear algebra are covered as needed, so a reader does not have to study a separate matrix course before reaching multivariable calculus.
• Visual material. Mathematica was used to produce many of the illustrations, which helps with surfaces and other objects that are hard to draw by hand.
• Support for self-study. Solutions are available in the book (the answers to starred exercises with hints and solutions fill about 80 pages) or on the web.

Limitations

• No proofs. The publisher states that there are no theorems or formal proofs. Students who need to see arguments and their logic will require a rigorous analysis text afterwards.
• Concentrated on two dimensions at first. The treatment of higher dimensions is by generalisation from the plane, so readers who want a full abstract treatment of vector calculus in several variables will need more.
• Not a general survey of vector analysis. The chapter list has no chapters on line and surface integrals, divergence and curl, so physics and engineering students who need vector field theory must look elsewhere.

Applications in the Book

The emphasis on business and economics is part of the book's character. The text illustrates matrices and vectors with commodity bundles and price vectors, uses optimisation with Lagrange's method for economic problems, and applies inverse and implicit functions and differentials to economic concepts such as contract curves, shadow prices and the Slutsky equation. Integration is linked to probability, and difference equations are applied to economic and social models. Scientific examples are included as well.

Where It Fits in the Course

Among the calculus books on the course list, Binmore and Davies is aimed at readers who need multivariable calculus, optimisation and simple differential and difference equations, particularly in economics and the social sciences. It is not a first course in single-variable calculus. Strang and Herman's free Calculus Volume 1 and the informal primer Calculus Made Easy are more suitable starting points for the single-variable ideas, and the very large Thomas' Calculus covers the standard full sequence including multivariable calculus.

Compared with Thomas' Calculus, Binmore and Davies puts matrices and optimisation up front, and treats Lagrange multipliers, implicit functions and difference equations as central topics. Wrede's Advanced Calculus in Schaum's series is a problem-driven outline and a good supplement for extra practice.

How to Use This Book

The book is meant for students of economics, business, science and mathematics who want to understand multivariable calculus and its main uses without a course on formal proofs. It suits readers who already know some single-variable calculus; Chapter 2 revisits it, so a short refresher is built in.

A sensible plan is Chapters 1 and 2 as groundwork, Chapters 3 to 6 as the multivariable and optimisation core, Chapters 7 to 9 for inverses, implicit functions and differentials, Chapters 10 and 11 for integration, and Chapters 12 to 14 for equations. Economics students will want the optimisation and differentials chapters most; scientists may prefer to move quickly to the integral and equation chapters.

Do the exercises: the starred ones have answers and hints at the back, which is the main route to checking your work. Use the illustrations to build a picture before working through the algebra. Readers who want proofs afterwards should move to a first course in analysis.