If $Q$ is a differentiable manifold and $L:TQ\to \Bbb R$ is a Lagrangian invariant under a $1$-parameter group $(\varphi_s)_{s\in \Bbb R}$ of diffeomorphisms of $Q$, then the Noether charge $\mathscr{J}:TQ\to \Bbb R$ defined by $$\mathscr{J}(x,v) = \mathbb{F}L(x,v)\left(\frac{\rm d}{{\rm d}s}\bigg|_{s=0}\varphi_s(x)\right),$$where $\mathbb{F}L:TQ\to T^*Q$ is the fiber derivative of $L$, is constant along curves $x:[a,b]\to Q$ which are critical points of the action functional of $L$. This is Noether's Theorem stated in the cleanest way I know.
I am interested in the version of the theorem for Lagrangians whose domain is $TQ^{\oplus m}$ for some $m\geq 1$, i.e., with more than one tangent vector as input. Coordinates $(q^1,\ldots, q^n)$ in $Q$ induce coordinates in each copy of $TQ$ inside $TQ^{\oplus m}$, and so we get coordinates $$(q^1,\ldots , q^n, v^1_{(1)},\ldots ,v^n_{(1)},\ldots, v^1_{(m)},\ldots, v^n_{(m)}).$$Fix $\Omega\subseteq \Bbb R^m$ a compact subset with non-empty interior and regular boundary (and coordinates $(u^1,\ldots u^m)$), so that the domain of the action functional of $L:TQ^{\oplus m}\to \Bbb R$ consists of "$m$-surfaces" $x:\Omega\to Q$. For a Lagrangian like this invariant over $(\varphi_s)_{s \in \Bbb R}$, I got that that $$\sum_{\ell=1}^m\frac{\partial}{\partial u^\ell}\left( \sum_{k=1}^n \frac{\partial L}{\partial v^k_{(\ell)}}(x(u),\nabla x(u))\frac{\partial q^k}{\partial s}(0,u)\right) = 0.$$ This is clearly the divergence of something. I don't know how to describe in an intrinsic way what is this something, in terms of (partial?) fiber derivatives or whatever. I would like to possibly describe this as some map $TQ^{\oplus m}\to ?$ that is constant along critical $m$-surfaces.
Physics texts are completely unintelligible for me, and the few mathematics texts that could possibly say something useful about this discuss a level of generality that goes way over my head. Help?
I got some help outside here, so I'll summarize it: we define $\mathscr{J}\colon TQ^{\oplus m} \to \Bbb R^m$ by $\mathscr{J} = (\mathscr{J}^1,\ldots, \mathscr{J}^m)$, where $$\mathscr{J}(x,v_1,\ldots, v_m) = (\mathbb{F}L)^\ell(x,v_1,\ldots, v_m)\left(\frac{{\rm d}}{{\rm d}s}\bigg|_{s=0}\varphi_s(x)\right),$$and $(\mathbb{F}L)^\ell$ is the $\ell$-th partial fiber derivative of $L$ (i.e., the restriction of $\mathbb{F}L$ to the $\ell$-th copy of $TQ$ inside $TQ^{\oplus m}$). What I had missed is that while $\mathscr{J}$ is not a vector field, whenever we have a $m$-surface $x\colon \Omega \to Q$, the composition $$\Omega \ni u \mapsto \mathscr{J}(x(u), \nabla x(u)) \in \Bbb R^m $$does define a vector field along $\Omega$, whose divergence is zero when $x$ is a critical point of the action functional associated to $L$ (this is the content of the theorem).