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)


Of course, you can also construct jets on functions between manifolds f:M->N.

If you have been following me lately, you may know a little about differential forms, and the desire to write down things in coordinate free form, without needing to explicitly set up base systems to relate to its particulars everywhere.

Structures for which this is possible are called equivariate tensors, but sadly jets don't fall in this class. There are, however, shortcuts for working with jets.


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