 | 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
| Focus of a Parabola | A
simple proof of the fact that parabolas focus light (or sound, for that
matter) | | Focus of an Ellipse | A
simple proof of the fact that light from one focus of an ellipse is
reflected to the other | | Focus of a Hyperbola | A
simple proof that light heading for the "back" focus of a hyperbola is
reflected to the "front" focus | | Calculus | In which we provide visual motivation for a number of facts from elementary calculus. [This
page is actually in the "Basic Stuff section; I'm linking to it from
here because it also fits well with the "Proof by Picture" theme.] | | Why
partials commute | A simple fact from
basic calculus which, none
the less, is typically only proved symbolically | Pythagoras's
Theorem:
The "Aha!" Proof | (this one's
certainly not original!) | The "Aha!" Proof,
Version 2 | This
is a slight variation on the classic "Aha!" proof, in which the
triangles have been placed on the outside of the inner square, rather
than the inside of the outer square. A small difference but none
the less I find it a bit clearer than the original. |
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 |
