Differential Topology Hirsch Chapter 2 Section 4 Problem 11: Compute the Dimension of $J^r(M, N)$
$J^r(M, N)$ is the set of all $r$-jets from $M$ to $N$. This is an equivalence class $[x, f, U]_r$ of triples $(x, f, U)$, where $U \subset M$ is an open set, $x \in U$, and $f: U \rightarrow N$ is a $C^r$ map; the equivlence relation is: $[x, f, U]_r = [x' f', U']_r$ if $x = x'$ and in some (hence any) pair of charts adapted to $f$ at $x$, $f$ and $f'$ have the same derivatives up to order $r$.
I wanted to check to see if this was right:
$J^r(M, N)$ seems to me to only distinguish among different points and functions whose derivatives differ at order $r+1$ and up. Since $x \in M$ we know that the dimension of $J^r(M, N)\geq dim M$. Now we just have to figure out the dimension of all functions that differ at order $r+1$ and up and add it to dim$M$? I can only think that this is infinite...
One useful way of thinking of jets is as a generalization of Taylor approximations, or as a coordinate-independent way of encoding all derivatives of order $\le r$. In this respect,your last paragraph seems to have things backward: the $r$-jets at $p$ do not distinguish between functions whose derivatives at $p$ differ only at order $>r$.
Choose local coordinates $x^1,\dots,x^m$ on an open domain $V\subseteq M$ and $y^1\dots,y^n$ on $W\subseteq N$, and consider jets $[p,f,U]_r$ with $p\in V$ and $f(p)\in W$. The class $[p,f,U]_r$ is uniquely determined by $p$, $f(p)$, and the partial derivatives $$ \frac{\partial f^j}{\partial x^{i_1}\dots\partial x^{i_k}},\ \ \ 1\le k\le r,\ \ \ 1\le j\le n,\ \ \ 1\le i_1\le\dots\le i_k\le m $$ Furthermore, every such collection $(x^i,y^j,A^j_{i_1\dots i_k})$ with $A^{j}_{i_1\dots i_k}\in\mathbb{R}$ is the $r$-jet of some function $f$ (this can be seen by writing $f$ as a polynomial of order $r$ in coordinates). These are called the derivative coordinates on $J^r(M,N)$ induced by $x^i,y^i$. The number of coordinates is then the dimension of $J^r(M,N)$. $$ \dim J^r(M,N)=\dim M+\dim N+S(\dim M,r)\dim(N) $$ where $S(m,r)$ is the number of nondecreasing sequences $1\le i_1\le\dots\le i_k\le m$ with length $1\le k\le r$.
Showing that the derivative coordinates do indeed form a coordinate patch requires showing that they map smoothly into $J^r(M,N)$. This will depend on how you have chosen to define the smooth structure on $J^r(M,N)$.