Advanced Calculus

Citation

Wrede, R. C., & Spiegel, M. R. (2010). Schaum's Outline of Advanced Calculus (3rd ed.). McGraw-Hill.

Chapter Summary

Context and Argument

This is a Schaum's Outline, a format in which each chapter states the essential definitions, theorems and methods in compact form and then works through solved problems and sets supplementary exercises. The publisher describes the third edition as containing 1,370 fully solved problems, a complete review of the course fundamentals and concise explanations of the concepts of advanced calculus. It is designed to be compatible with a classroom text, not to replace one.

Its sixteen chapters run from the real numbers and sequences, through the differential and integral calculus of one and several variables and vector analysis, to series, Fourier methods, special functions and complex variables. The first half supplies the foundations and the calculus of several variables; the second half covers series and transforms and the functions of a complex variable.

Chapters 1–3: Numbers; Sequences; Functions, Limits and Continuity

The book begins with the real number system: sets, decimal and geometric representations, exponents, roots and logarithms, and the axioms on which the real numbers rest. Chapter 2 treats sequences, limits of sequences, boundedness and Cauchy's convergence criterion, with a first look at infinite series. Chapter 3 turns to functions, limits and continuity, including piecewise and uniform continuity.

Chapters 4–5: Derivatives; Integrals

Chapter 4 gives the derivative, the rules of differentiation and its applications. Chapter 5 gives the definite integral and its uses in calculating areas and volumes, with attention to numerical evaluation.

Chapters 6–8: Partial Derivatives; Vectors; Applications of Partial Derivatives

Chapter 6 extends limits, continuity and derivatives to functions of several variables. Chapter 7 covers vector algebra and geometry, vector functions, and the differential operators gradient, divergence and curl. Chapter 8 applies partial derivatives to geometry and to optimisation, including Lagrange multipliers and differentiation under the integral sign.

Chapters 9–10: Multiple Integrals; Line Integrals, Surface Integrals and Integral Theorems

Chapter 9 treats double and triple integrals and changes of variables. Chapter 10 introduces line and surface integrals and the integral theorems of vector calculus, namely Green's theorem, Stokes' theorem and the divergence theorem.

Chapters 11–12: Infinite Series; Improper Integrals

Chapter 11 covers tests of convergence for series, power series and Taylor series. Chapter 12 treats improper integrals and their convergence.

Chapters 13–15: Fourier Series; Fourier Integrals; Gamma and Beta Functions

Chapter 13 introduces Fourier series and their properties, with applications to boundary-value problems. Chapter 14 extends these ideas to Fourier integrals and transforms. Chapter 15 covers the gamma and beta functions and the relations between them.

Chapter 16: Functions of a Complex Variable

The last chapter gives an introduction to complex analysis: complex functions, integration, series and residues.

Key Concepts

Chapters 1–5: Foundations

• Completeness of the real numbers: the axiomatic property that underlies limits.
• Cauchy's convergence criterion: convergence of a sequence tested without knowing its limit.
• Uniform continuity: a stronger form of continuity that behaves well over a whole interval.
• Fundamental theorem of calculus: the link between differentiation and integration.

Chapters 6–10: Several Variables and Vector Calculus

• Partial derivatives and the Jacobian: derivatives of functions of several variables and the factor in a change of variables.
• Gradient, divergence and curl: the differential operators of vector analysis.
• Lagrange multipliers: optimisation under constraints.
• Green's, Stokes' and divergence theorems: relations between integrals over regions and over their boundaries.

Chapters 11–16: Series, Transforms and Complex Variables

• Convergence tests, power series and Taylor series: ways to decide whether a series converges and to represent a function by one.
• Fourier series and integrals: representation of functions by trigonometric terms.
• Gamma and beta functions: special functions defined by integrals.
• Residues: a tool of complex analysis for evaluating integrals.

Critical Analysis

Strengths

• Problem-solving focus. With 1,370 solved problems, the book gives a reader worked models for most techniques in the course, and supplementary problems for practice.
• Compact coverage. The sixteen chapters collect the topics of a typical advanced calculus course, from sequences and continuity to vector integral theorems, Fourier methods and complex variables, in a single volume.
• Compatible with other texts. The outline is designed to be used beside a classroom textbook, which makes it a good source of extra practice and quick review.
• Clear organisation. Each chapter follows the same pattern of statement and then solved problems, so a reader can find the rule needed for an exercise quickly.

Limitations

• An outline, not a full text. Theory is stated briefly, so a reader who wants motivation or complete proofs will need a textbook of analysis beside it.
• Limited narrative. The format favours method over explanation, and a first reader may find that the connections between chapters are not developed at length.
• Level and scope. The book covers the calculus of one and several variables and does not extend to measure theory or functional analysis, so it is not a substitute for a rigorous real analysis course.

Where It Fits in the Course

Advanced Calculus is the supplement to a multivariable or advanced calculus course. It assumes the single-variable calculus covered by the other books on the list: Calculus Made Easy is an informal primer for the core ideas, Calculus Volume 1 is a free first-course text on single-variable calculus, and Binmore's Calculus: Concepts and Methods and Thomas' Calculus carry the subject through multivariable calculus with explanations and exercises.

Read those first if the ideas of limit and derivative are new. This book then works well as a reference and practice source for partial derivatives, vector calculus, series, Fourier methods and complex variables, where a student wants more worked examples than the main text offers.

How to Use This Book

The book is meant for undergraduate students of mathematics, physics and engineering who have met introductory calculus and want worked practice in the topics of an advanced course.

Use it alongside lectures or a main text. For each topic, read the summary of definitions and theorems first, then try the solved problems yourself before reading the solutions, and finish with the supplementary problems. Chapters 6–10 on several variables and vector calculus are the core for most science and engineering courses, and Chapters 11–16 are the next block for courses on series, Fourier analysis and complex variables.

Because the theory is condensed, pair it with a textbook whenever a theorem needs a full proof or motivation. Use Chapters 1–5 as a review of the foundations, and skip what your course has already covered.