You are here

Concise Calculus

Sheng Gong
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
, on

The back cover asserts this text offers a

method of teaching Calculus to help students from physics, engineering and other sciences disciplines understand Calculus faster and deeper in order to meet the needs of applications in their own fields. […] The practical examples provided in the book bring motivation to students to learn Calculus.

Essential elements promised are: illustrative, practical applications and effective economy. Both promises are partly met in this edition, resulting in a good single volume supplemental to several undergraduate university courses.

The first seven of eleven chapters take students from the fundamental theorems of calculus through vector calculus topics such as gradient, divergence, and curl. This all in four hundred pages, roughly a fifth of the equivalent pages in a sampling of the textbooks available for comparison on my shelf. This brevity need not be an issue in a lecture presentation over multiple semesters, but it does feel too thin to support independent study. Expected textbook features such as section exercises and solution back matter are present and comprehensive. Effective and explanatory applications and examples (generally from physics and mechanics) speak directly to “students from physics, engineering and other sciences disciplines” from the onset, but trail off in Chapter 4, covering ODEs.

The bulk of the material is bivariate analysis with only one chapter focused on calculus of several variables. The text takes students through a thorough discussion of the ε-δ definition of limits as well as infinite series and improper (the author says “infinite”) integrals and topics in Fourier analysis. The book also includes introductions to partial derivatives, surface integrals, Stokes’ and Green’s Theorems, and much more in less than seven hundred pages. This material is presented in a form that is more cursory than concise and will require significant amplification in lecture for most students.

Tom Schulte says goodbye to Michigan and hello to Louisiana, where he hopes to be able to continue lecturing in mathematics to undergraduates. 

  • Basic Concepts
  • Calculations of Derivatives and Integrals
  • Some Applications of Differentiation and Integration
  • Ordinary Differential Equations
  • Vector Algebra and Analytic Geometry in Three-Dimensional Space
  • Multiple Integrals and Partial Derivatives
  • Line Integrals, Surface Integrals and Exterior Differential Forms
  • Some Applications of Calculus in Several Variables
  • The ε-δ Definitions of Limits
  • Infinite Series and Infinite Integrals
  • Fourier Series and Fourier Integrals