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.
P.S.: Please, wiki-hammer this question.