V6.
Multiplyconnected
Regions; Topology
In Section V5, we called a region D of the plane simplyconnected if it had no holes in it.
This is a typical example of what would be called in mathematics a topological property, that
is, a property that can be described without using measurement. For a curve, such properties
as having length
3,
or being a circle, or a line, or a triangle

these are not topological
properties, since they involve measurement, whereas the property of being closed, or of
intersecting itself once, would be topological.
The important thing about topological properties is that they are preserved when the
geometric figure is deformed continuously without adding or subtracting points, whereas
nontopological properties change under such a deformation. For example, if you deform a
circle, it will not stay a circle, but it will still remain a closed curve that does not intersect
itself.
Topology is that branch of mathematics which studies topological properties of geomet
rical figures; it's a kind of geometry, but at the opposite pole from Euclidean geometry,
which emphasizes measurement ("congruent triangles" "right angles", "circles"). Topology
is a large and active branch of mathematics today, one which is attracting attention from
other disciplines, like theoretical physics and molecular biology. Most students have never
heard of it, because topological properties don't enter very often into the first few years
of mathematics. However, they do right here, and in fact it was just in the study of the
possible values of a line integral around a closed curve that the central ideas of modern
topology first entered into mathematics, in the middle of the 1800's.
So let
F be a continuously differentiable vector field in a multiplyconnected

i.e., not
simplyconnected

region D of the xyplane, and suppose curl
F
=
0. What values can
jc
F
F.
dr have?
We begin by considering an earlier example (Section V2, Example 2) in greater detail,