## Morse theory

Suppose *f* is a continuously differentiable function of 2 real
variables *x,y* satisfying

*f(x+1,y)=f(x,y+1)=f(x,y).*

Then there are at least 3 distinct pairs of numbers *x,y* in
the interval *[0,1)* so that both partial derivatives of *f*
are 0 at these points (i.e. *(x,y)* are critical points).
The displayed equation guarantees that *f* is determined by
its values
on the unit square *[0,1]x[0,1]*
, and further the
values on the opposite sides of the square coincide. So it is natural
to identify (or glue together) the opposite sides to obtain a * torus*
(popular science literature refers to it as a ``donut''). The
point is that *f* is really defined on the torus. There is an
extremely useful method, called Morse theory, that relates the
structure of a space (e.g. the torus) and the structure of the set of
critical points. Try constructing a function *f* as above that
has
exactly 3 critical points.

This is the picture of the gradient vector field of the function
f(x,y)=sin(2Àx)+sin(2Ày). Observe sources, sinks, and
saddle points.