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18.02 Multivariable Calculus
Fall 2007
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Some Topological Questions
We consider once again the criterion for a gradient field. We know that
and inquire about the converse. It is natural to try to prove that
by using Stokes' theorem: if curl
F
=
0,
then for any closed curve C in space,
The difficulty is that we are given C, but not S. So we have to ask:
Question. Let D be a region of space in which
F is continuously differentiable. Given
a closed curve lying in D; is it the boundary of some twosided surface
S lying inside D?
We explain the "twosided" condition.
Some surfaces are only onesided: if you start
painting them, you can use only one color, if you don't allow abrupt color changes. An
example is
S below, formed giving three halftwists to a long strip of paper before joining
the ends together.
Surface S
Boundary C (trefoil knot)
S has only one side. This means that it cannot be oriented: there is no continuous choice
for the normal vector
n over this surface. (If you start with a given n and make it vary
continuously, when you return to the same spot after having gone all the way around, you
will end up with n, the oppositely pointing vector.) For such surfaces, it makes no sense to
speak of "the flux through S", because there is no consistent way of deciding on the positive
direction for flow through the surface. Stokes' theorem does not apply to such surfaces.
To see what practical difficulties this causes, even when the domain is all of 3space,
consider the curve C in the above picture. It's called the trefoil knot. We know it is the
boundary of the onesided surface
S, but this is no good for equation (3), which requires
that we find a twosided surface which has C for boundary.
There are such surfaces; try to find one. It should be smooth and not cross
itself. If successful, consider yourself a brownbelt topologist.
The preceding gives some ideas about the difficulties involved in finding a twosided
surface whose boundary is a closed curve C when the curve is knotted, i.e., cannot be con
tinuously deformed into a circle without crossing itself at some point during the deformation.
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 Spring '08
 Auroux
 Logic, Multivariable Calculus

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