If a function is not one-to-one, then it has no inverse.Antineutrino said:The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?
Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
Diffeomorphisms are mathematical functions that are smooth and invertible, meaning they have a well-defined derivative and can be reversed.
Yes, diffeomorphisms must be one-to-one functions in order to be invertible. This means that each input has a unique output, and each output has a unique input.
One-to-one functions are important in diffeomorphisms because they ensure that the function can be reversed, which is necessary for certain mathematical operations and applications.
In some cases, diffeomorphisms may not be strictly one-to-one functions, but they must still be "locally" one-to-one. This means that in a small neighborhood around each point, the function is one-to-one.
Yes, diffeomorphisms can be applied to any smooth, differentiable mathematical space, such as curves, surfaces, or higher-dimensional manifolds.