Jacobian and derivatives on manifolds

Question

I'm studying a paper about the optimal transport formulation of Einstein's equations, but I'm not understanding this fact: introduce on a Lorentzian manifold the Lagrangian $$\mathcal{L}_p(v):=-\frac{1}{p}\big(-g(v,v)\big)^{\frac{p}{2}}$$ if $v\in\mathcal{C}:=\mathrm{Cl}(\{v\in TM:g(v,v)<0\mbox{ and }g(v,X)<0\})\subset TM$, where $X$ is the vector field that gives the time orientation on $M$ (basically, $\mathcal{C}$ is the set of future-pointing causal vectors w.r.t. $X$). Then a calculation in coordinates gives that $$\frac{\partial\mathcal{L}_p}{\partial v^i}=\big(-g(v,v)\big)^{\frac{p-2}{2}}g_{ik}v^k,\quad i=1,\dots,n.$$ Then later in the paper there are some functions, say $\phi: M\rightarrow \mathbb{R}$, for which the authors write $d\phi(x)$ for the differential at $x\in M$. But also they write $D\mathcal{L}_p$ for some sort of Jacobian of $\mathcal{L}_p$. So my question is what is the difference between $d$ and $D$ for differentials? Is $D$ just a symbol for a particular derivative in coordinates or do I have to interpret it as the matrix of the linear map given by the differential?