Visual Calculus

Visual Calculus by Mamikon Mnatsakanian is an approach to solving a variety of integral calculus problems. [ [http://www.cco.caltech.edu/~mamikon/calculus.html Visual Calculus] Mamikon Mnatsakanian] Many problems which would otherwise seem quite difficult yield to the method with hardly a line of calculation, what Martin Gardner calls "aha! solutions" or Roger Nelsen a proof without words. [Nelsen, Roger B. (1993). "Proofs without Words", Cambridge University Press. ISBN 9780883857007. ] [Martin Gardner (1978) "Aha! Insight", W.H. Freeman & Company; ISBN 0-7167-1017-X]

The method was devised by Mamikon in 1959 while a young undergraduate. It is based on the old puzzle about what is the area of a ring if the tangent to the inner circle is 6" long? With Mamikon's insight the solution becomes obvious - the area is the same as that swept by a tangent from the inner circle to the outer circle, and the tangents can all be translated parallel to themselves to make a smaller circle the points of tangency at the centre and with same radius as the tangent length. Thus it doesn't really matter that the inner and outer curves are circles, just that the tangent to one side of the inner curve should have a constant length.

Mamikon generalised the insight to Mamikon's theorem::"The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve."

Tom Apostol has produced a very readable introduction to the subject. [ [http://www.its.caltech.edu/~mamikon/VisualCalc.html A VISUAL Approach to CALCULUS problems] An introduction by Tom Apostol] In it he shows that the problems of finding the area of a cycloid and tractrix can be solved by very young students. To quote him "Moreover, the new method also solves some problems "unsolvable by calculus, and allows many incredible generalizations yet unknown in mathematics"." Solutions to many other problems appear on Mamikon's Visual Calculus site.

See also

* Hodograph This is a related construct which maps the velocity of a point using a polar diagram.

References

External

* [http://www.mamikon.com/ Mamikon] Main web site (The menu is on the right of the ... graphics)
* [http://www.its.caltech.edu/~mamikon/ ProjMath Mamikon]
* [http://mathworld.wolfram.com/ProofwithoutWords.html Proof without Words] from MathWorld