Monday, April 12, 2010

Monday Mornings With Math


This week's episode of Steven Strogatz's series -- Math: From Basic To Baffling -- is on calculus: Change We Can Believe In:

Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuan — and so are its textbooks. Many exceed 1,000 pages and work nicely as doorstops.

But within that bulk you’ll find two ideas shining through. All the rest, as Rabbi Hillel said of the Golden Rule, is just commentary. Those two ideas are the “derivative” and the “integral.” Each dominates its own half of the subject, named in their honor as differential and integral calculus.

Roughly speaking, the derivative tells you how fast something is changing; the integral tells you how much it’s accumulating. They were born in separate times and places: integrals, in Greece around 250 B.C.; derivatives, in England and Germany in the mid-1600s. Yet in a twist straight out of a Dickens novel, they’ve turned out to be blood relatives — though it took almost two millennia to see the family resemblance.

4 Comments:

  1. jbeck said...

    TAB,

    It wouldn't be out of place to highlight the pioneering contribution of the "Kerala school of mathematics" Here's JK's (the guy who blogs at Varnam) http://bit.ly/bowu8v on Nilakantha Somayaji. Here's a review by David Mumford (one time collaborator of the late C.P. Ramanujam) of the most comprehensive book on the subject to date by Dr. Kim Plofker. Dr. Mumford's review http://bit.ly/awz2wB and Dr. Plofker's home page at Union College http://bit.ly/bjmPhv and Dr.Plofker's publications http://bit.ly/9SmXaU

  2. gaddeswarup said...

    My impression which may be wrong. Nobody before Newton and Leibnitz had the Fundamental Theorem of Calculus which relates differentiation and integration for many functions. Despite the great achievements of others before, I think that this theorem is what made Calculus a machine like tool which can be used without really understanding the the fundamentals. Presumaly, this will be discussed next by Strogatz.

  3. gaddeswarup said...

    Did David Mumford and C.P. Ramanujam colloborate? I think that there was mutual admiration and famous paper of Ramanujam was inspired by Mumford's work (one of the few papers with home address, poonamalle high road, madras, possibly written after one of his breakdowns). I think that Ramanujam wrote some lecture notes for Mumfoed (Tata lectures on Theta?)and there was a rumour that Mumford wantd to make it a joint book but Ramanujam declined.

  4. gaddeswarup said...

    Oops; wrong again. Wikipedia
    http://en.wikipedia.org/wiki/Calculus says:
    "This realization, made by both Newton and Leibniz, who based their results on earlier work by Isaac Barrow, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences."
    Apparently Gregory had a version too.