What's jet-space? A jet is a fancy way to write down and track a function together with a number of its derivatives in a fancy way. Say, there's a sufficiently smooth function f: R^n -> R^m, then we can approximate f using a Taylor expansion:

f(x) = f(x0)/0! + f'(x0)·x/1! + f"(x0)·x^2/2! ... f^(k) (x0)·x^k/k! + E(x0,k)

The E is called error term, and it's there so I'm allowed to use an equal sign. Or rather, let's introduce a jet symbol that doesn't include the error term:

J^k_x0 f (x)

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Let F be a class of smooth functions, that is, a subset of {f|f:R^m->R^n}. Now, the k-jet-space J^k_x0(F) at a point x0 is the space of the k-jets at x0 of all the functions in F!

I haven't yet tried to find out why that's a useful thing to have, beyond it coming in handy to shorten notation when handling jets. But I can tell you a bit more about jets, for example:

A 1-jet is the tangent space of a function. So k-jets are a generalization of tangent spaces!

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