This announces Math 462 Differential Geometry for Fall 2017.
The course is accessible for interested students past the 200 level Math
classes, but it runs infrequently, so if you are interested, you should
take it NOW.
Math 462 can make a pairing with Math 467 (offered typically every spring)
for purposes of the sequence requirement in the Math major.
Neither of 462,467 has the other as a prerequisite, they can be taken
independently.
Differential geometry of curves and surfaces in 3-space (or curves in
the plane) basically uses calculus to describe geometric notions, mainly
revolving about notions of curvature. You would have seen the very first
small glimpses of it in Calc 3 when discussing parametrized curves and
surfaces.
One highlight is the `theorema egregium' which in applications implies
that it is not possible to have a to-scale map of a larger portion of
the globe on `flat' paper. (Even though it *is* possible to have a map
that preserves angles while distorting lengths -- many maps in your world
atlas are of such a kind, and this property has been very convenient for
the navigation of seafarers of old.)
Another highlight is the Gauss-Bonnet theorem, one version of which
explains in particular why, for triangles on a sphere, the area is related
to the sum of its angles. (In stark contrast to plane geometry, where the
sum of the angles is always 180 degrees, regardless of area.)
We may have a small overlap with Math 460 inasmuch as
hyperbolic geometry is included in our discussion.
Generally, the notion of what is `straight' on a curved surface,
i.e., the notion of geodesics, will be part of our discussions.
Students that plan to study Riemannian geometry (500-level) later do *not*
need 462 as a prerequisite, but 462 does provide a good experiential
background to make the more abstract notions in advanced courses (including
those as are used in Einstein's General Relativity) more intuitive. It will
give some insight in what `curved space' could mean (by analogy to curved
surfaces) to everybody in the audience, even if they do not take the more
advanced courses.
The textbook is an affordable Dover-book: DoCarmo: `Differential Geometry
of Curves and Surfaces'
If you have further questions, feel free to contact me.
If you'd like to have the course as an honors course with
individual extras added under `Honors by Contract' provisions,
let me know and if there are such requests, I'll try to accommodate
them (subject to approval -- I'll address this issue if/when there is
interest). This option (if ok'ed) may become relevant to students
desiring to use 462+467 as an *honors* sequence.
Jochen Denzler
Ayres 317
974-5325
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