Proof
by Picture |

More material will
be added to this
page!
I find proofs that rely too heavily on algebra and abstract symbol manipulation are often unsatisfactory. I'm left with the feeling that I know that the fact to be proved is true, but I still don't understand why
it's true.Many facts in mathematics can be proved or explained in more than one way, and many things which are usually proven using abstract reasoning can also be shown to be true using pictures. Perhaps the best-known example is the "Aha!" proof of Pythagoras's theorem. Other facts which are easily shown via pictures include such simple stuff as (a/b)*(c/d) = (ac/bd) -- something which was taught by rote in the school I attended. On this page I've listed a few things which I had to work to understand, and for which I eventually found satisfactory explanations which can be displayed as pictures. A
few simple things
Some
more advanced items
And one item which isn't actually a proof-by-picture at all, just because I felt like it... The Glue Function A good
"picture proof" may take only a few seconds to
understand.
However, constructing the picture to start with can be time
consuming. My plans for this page are ambitious, but it's
likely
to be quite a while before most of these things are actually
done. Someday, maybe, the following sections will be included:*Lie derivative, equivalence to Lie bracket (I had a reasonably good picture for this one as long ago as 6/22/05 but unfortunately the supporting text -- which must include a definition of the Lie derivative, of course, as well as an explanation of what the picture is supposed to show! -- turned out to be long and tedious to write.)**Equivalence of parallel-transport and longest-path defs for geodesic**Exterior derivative -- maybe??**Stokes' theorem (the big one, not the little one) -- maybe??*
And we have...
*Variational path principle ... done!*
Page created on 8/11/04, and last
updated 2/16/2008 |