User harecare - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:39:41Z http://mathoverflow.net/feeds/user/15454 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66377/why-is-differentiating-mechanics-and-integration-art/66391#66391 Answer by harecare for Why is differentiating mechanics and integration art? harecare 2011-05-29T19:43:43Z 2011-05-29T19:43:43Z <p>"Inverse" does not always mean "symmetric". It is pretty mechanical to map \$x\mapsto x^3+x^2-5\$, and it was an art of 16th century to find an \$x\$ which maps, say, to \$0\$.</p> <p>Or, less mathematical, it is easy to mix a glass of rice with a glass of oats, and it is pretty more a matter of art to get them back to proper glasses.</p> http://mathoverflow.net/questions/66377/why-is-differentiating-mechanics-and-integration-art/66391#66391 Comment by harecare harecare 2011-05-30T07:09:04Z 2011-05-30T07:09:04Z A map is a rule of preparing for each object from A one object from B. Let me illustrate. Suppose you prepare from each decimal number the sum of its digits: from 231 you get 6, and so on. This is absolutely routine. But now you ask, which numbers give you 6, the preimages of 6? Looks more challenging, isn't it? Finding preimages is not a map, because for one object you may get several. And what if you are asked to find minimal of those preimages? This weird task is in the spirit of integration problems of traditional calculus: you are asked to find a very special form of answer. http://mathoverflow.net/questions/66377/why-is-differentiating-mechanics-and-integration-art/66391#66391 Comment by harecare harecare 2011-05-29T20:30:03Z 2011-05-29T20:30:03Z Exactly for the same reason as for polynomials above: differentiating is a map from functions to functions and integration is trying to revert it - to find a preimage. Nobody ever promised that the inverse action should be as easy as the original one.