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    • Thread Starter

    This is a retarded question. Brace yourself

    I want to be able to say

    f : S^1 \rightarrow \mathbb{R}^3

    f : x \mapsto \text{blah }

    Can I say f:\theta \rightarrow f(\theta) for \theta \in [0, 2\pi)? Because that's not really a mapping from the circle, it's a mapping from the parameter of a parameterization of the circle, but each theta basically IS a point on the circle - there's a one-to-one correspondence?
    • PS Helper

    PS Helper
    Well S_1 = \{ (\cos \theta, \sin \theta)\, :\, \theta \in [0, 2\pi) \}, and so instead of saying f : \theta \mapsto f(\theta) what you're really saying is (\cos \theta, \sin \theta) \mapsto \theta \mapsto f(\theta), but that's fine since \theta is uniquely determined by (\cos \theta, \sin \theta).
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Updated: April 5, 2011
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