Motivating differentiable manifolds

I'm reading lectures for the first time starting next week ^_^ The subject is calculus on manifolds I was told that the students I'll be lecturing are not motivated (at all), so I need to kick off the series with an impressive demonstration of what an cool subject it will be that I'll be able to periodically return to later on when I'll need to motivate specific concepts.

So far I came up with:

1) Projective plane, I'll demonstrate how conics morph into each other on different models, should be cool enough and motivate general-topological manifolds.

2) Plücker coordinates and their applications to line geometry and computer graphics (not sure if it will be easy to demonstrate), should motivate forms.

3) Expanding Universe, motion in relativity and perception of time, should motivate tangent vectors.

I'm not sure they will work, and I could always use more ideas. I would greatly appreciate examples and general advice too.

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I think Plucker coordinates are pretty cool, but that sounds like a kind of intense thing for a neat introductory motivation. Maybe you have an easy intuitive explanation? – Matt Feb 6 '11 at 0:56
@Matt I like how Slawomir Bialy approached them in the discussion on Wikipedia: en.wikipedia.org/wiki/… I think I can use intuition for wedge product as a parallelogram, but going from $\mathbb{RP}^3$ to $Gr(2,3)$ may be problematic, I haven't looked into it yet. – Alexei Averchenko Feb 6 '11 at 1:23

The universe is a manifold! And we don't know which. I think that is already plenty of motivation, but you might be interested in reading Weeks' The Shape of Space for some more motivation along these lines, as well as some interesting visual exercises and good exposition overall (although some of the physical possibilities he suggests later are, I think, out of date). I would also suggest that you draw a lot of pictures, e.g. of 2-manifolds.

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"I would also suggest that you draw a lot of pictures" I'll prepare animations, maybe even cut them together to form a few short films, I got a friend who I can do great voice overs 8) Thanks for the book! – Alexei Averchenko Feb 6 '11 at 0:17
While I'm not a big fan of string theory myself, you may also consider making the topic more sexy (in a physics sense) with some of the things discussed in Yau's book The Shape of Inner Space. At the very least, Calabi-Yau manifolds are rather pretty to look at. – Willie Wong Feb 6 '11 at 0:53

See this book by W.D Curtis and F.W Miller. Essentially they say to view the "evolution of a physical system as a curve in the space of states." Vector fields on state spaces describe the behavior an evolving physical system.

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I'm not sure configuration and phase spaces are very easy to explain. – Alexei Averchenko Feb 6 '11 at 0:04

Personally I always loved to link maths and physics.

For instance, why not introduce hysteresis? Some hysteresis phenomenons can be modelled using manifolds I believe.

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