Robert's Mistakes

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)

1/…

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?

2/…

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!

3/…

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.

4/…

A jet-bundle is just another, fancier name for a jet-space. That's it for now, folks, but as there's still lots of fun stuff to talk about I feel compelled get back at them soon. Corrections, questions, or simply showing interest can help make that happen ;^)

https://en.wikipedia.org/wiki/Jet_(mathematics) https://en.wikipedia.org/wiki/Jet_space https://en.wikipedia.org/wiki/Taylor_expansions

jet #bundle, #Taylor expansions, #smooth #maths on a #manifold

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