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My book defines one-to-one mapping as

A mappingTisone-to-oneonD*if for (u,v) and (u',v') ∈ D*,

T(u,v) = T(u', v') implies that u = u' and v = v'

I don't really understand what they are trying to say, because right now what I'm getting from this information is that only functions that are equal to their derivative can be mapped one-to-one, but doesn't that mean only lines can be one to one functions?

Another thing

I know the Jacobian ≠ 0 has something to do with a function having an inverse and being mapped one-to-one, but my book just skips over that, so could someone please explain that. Also Jacobian Determinant is the determinant of the derivative matrix what does that have anything to do with inverse functions, and moreso what does a determinant even mean.

Help, I really want to learn mathematics and physics, but I always get bogged in technicalities and poorly written text books :(

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# One-to-One Mapping

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