Detailed
description:
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- curves in the plane (regular curves, examples, curvature,
evolutes,
involutes, pedal curves, the fundamental theorem of plane curves)
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- curves in space (curvature and torsion, the Frenet
formulas, examples, the fundamental theorem of space curves, space
curves that lie on a sphere)
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- brief review of vector calculus
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- surfaces in the space (regular patches, regular
surfaces, examples, tangent spaces, surface mappings and tangent maps,
level surfaces, examples)
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- orientability of regular surfaces (definition, the Gauss
map, examples of non-orientable surfaces)
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- metrics on surfaces (the first fundamental form of a
surface, distances and areas on surfaces, isometries and
conformal maps, examples)
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- the shape operator (definition, normal curvature, principal
curvature and principal curves, the second fundamental form, the
Weingarten equations, Gaussian and mean curvature)
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- ruled surfaces (definition, examples, curvature
calculations)
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- surfaces of revolution (definition, examples,
curvature calculations)
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- Gauss's Theorema Egregium
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- geodesics on surfaces (definition, examples)
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- the Gauss-Bonnet Theorem (local and global versions)
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- minimal surfaces (definition, normal variation, examples,
the Gauss map of a minimal surface, isothermal coordinates, the
Weierstrass representation, Costa's minimal surface)
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