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> In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the oth
by wolfgke 10y ago
> In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the other set,
This is something math and computer science students typically learn in the first two weeks in their mandatory math lectures at least at German universities
- Bahamut 10y agoIn the US, I don't think I saw it until a course in set theory, although it could get rolled up in a topology course.
- wolfgke 10y agoI mean: It really makes sense to put it at the beginning since otherwise it's ugly to understand why a diffeomorphism is defined as it is: You surely know that an isomorphism of sets (bijective function) has an inverse that is also an isomorphism of sets. For differentiable functions a similar statement does not hold in general (just consider [-1,1] -> [-1,1]; x \mapsto x^3; its inverse is not differentiable everywhere on [-1,1]; so its inverse exists as an isomorphism of sets, but not as a differentiable function). Since diffeomorphisms for R^d are introduced in the 2nd semester for math students (typically in the context of the inverse function theorem), one better has already understood the basics before.