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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