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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J^k_x0 f (x)

To be clear: we intend to vary x0. And instead of evaluating the polynomial at some x we want to keep it so we can look at the numbers it's made of.

Here's a funny homomorphism involving function composition:

J^k_x0(f o g) = (J^k_x0 f) o (J^k_x0 g)

You could replace the composition symbol o by a multiplication sign and get another homomorphism. But the one about function composition is the surprising (nice, important, …) one.

Can you guess what a jet space is?

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