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Free Calculus (A Liberation from Concepts and Proofs)

Qun Lin
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Kenneth A. Ross
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The title of this book is ambiguous; I do not, in fact, understand it. It is an uneven 82-page book with 50 pages of introductory calculus, 22 pages on differential equations, and 10 pages on functional analysis.

The calculus, intended for beginners, is given "with plain English and without mathematical symbols." The result is a real mish-mash. The author offers classroom motivation and pictures, which he calls proofs. Later he reverts to pure calculation, so that pages 24–27 consist of line after line of intimidating formulas giving versions of Taylor's Theorem. They involve integrals and derivatives, all undefined notation. Only later, pages 33–39, does he give the basic rules of differentiation (for sums, products, composition, etc)!

The complete statement of the Fundamental Theorem of Calculus is

Total true height = integral of differentials.

There are no discernible hypotheses. Many terms are undefined, including "integral" and "free calculus." He does define "tangent": A straight line closest to the curve near a node. Any questions?

The author presents his approach as one upon which "all of the calculus experts frown," but it's never quite clear what he is trying to do. But then, it's not clear to me what a calculus liberated from "concepts" would be. The cultural divide and the language barrier make it harder to figure it all out. (The author is not a native English speaker, and the publisher has, as far as I can tell, done no copyediting.)

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author many books, including  Elementary Analysis: The Theory of Calculus.


  • Calculus in Terms of Images
  • Official Calculus
  • Differential Equations of First Order
  • Differential Equations of Second Order
  • Free Calculus
  • Appendix: 
  • Calculus of Functional Analysis Becomes Elementary Algebra