Understanding k- jets

Question

K-jets can be realized as space of (up to equivalence classes) of Taylor polynomials of order k. There is a statement in V. Arnold's lectures in PDE stating that 1-jets of functions in the plane can be viewed a 5-dimensional space. Why is that? 1) Can someone clarify the concept? 2)How are jets defined for Riemannian manifolds?

Answer 1

At each point in the plane, a $1$-jet is determined by three numbers, since a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the plane, so the total space has dimension $3+2=5$.

All this works just as well over a smooth manifold. Taking a coordinate patch one can define jets as Taylor expansions in the local coordinates there. Then one checks that these behave well when we transition between coordinate patches. It's the same sort of argument that's used to show that the tangent bundle of a manifold makes sense.