From a textbook on classical mechanics:
Here $H$ denotes the Hamiltonian, and $F$ denotes a function whose codomain is unclear to me (see below). I assume we are in $6N$-dimensional phase phase (with some $3N$-dimensional collection of location coordinates $q$, plus some $3N$-dimensional collection of momentum coordinates $p$).
Questions: This book is very good, but sometimes the author is loose on notation, leading me to the following questions:
Is $F$ necessarily a real valued function? That is, is $F(q, p) : \mathbb{R}^{6N} \rightarrow \mathbb{R}$, or could $F$ be any vector-valued function (where the codomain is in $\mathbb{R}^m$ for some $m \ge 1$)?
Are the partial derivative terms in this equation -- like $\frac{\partial F}{\partial q}$ -- only meant to be viewed component-wise (for example as $\frac{\partial F_i}{\partial q_i} \in \mathbb{R}$ for some $i \in \{1, \ldots, 3N\}$), or can they also be viewed as $3N$-dimensional gradient vectors?
Are the terms $\frac{\partial F}{\partial q} \frac{\partial H}{\partial p}$ and $\frac{\partial F}{\partial p} \frac{\partial H}{\partial q}$ meant to be viewed as the dot-products of $3N$-dimensional vectors? If so, doesn't that force the left-hand side $\frac{d}{dt} F$ to be a real-number?
The book (which book?) seems to be discussing a $2$-dimensional phase space with canonical coordinates $q,p$, unless there is some abuse of notation. The Hamiltonian $H$ is a smooth real-valued function on phase space. Hamilton's equations $$\frac{dq}{dt} = +\frac{\partial H}{\partial p}, \qquad \frac{dp}{dt}=-\frac{\partial H}{\partial q}$$ define the time evolution of the points $(q,p)$ in phase space under the Hamiltonian $H$: each point $(q_0,p_0)$ provides an initial condition for the ODE, and hence yields a flow $(q(t),p(t))$ starting at that point $q(0)=q_0, p(0)=p_0$. The time evolution of a smooth real-valued function $F$ on phase space (under the same flow) is then given by $$\frac{dF}{dt} = \frac{d}{dt}\left(F(q(t),p(t))\right)= \frac{\partial F}{\partial q}\frac{dq}{dt}+\frac{\partial F}{\partial p}\frac{dp}{dt}=\frac{\partial F}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial H}{\partial q}\mathrel{=:}\{F,H\}.$$
More generally in $2n$-dimensional phase space with canonical coordinates $q_1,\ldots,q_n,p_1,\ldots,p_n$, Hamilton's equations are $$\frac{dq_k}{dt}=+\frac{\partial H}{\partial p_k},\qquad \frac{dp_k}{dt}=-\frac{\partial H}{\partial q_k};$$ the time evolution of a smooth real-valued function $F$ is then given by $$\frac{dF}{dt}=\frac{d}{dt}\left(\ldots\right)=\ldots = \sum_{k=1}^n\left(\frac{\partial F}{\partial q_k}\frac{\partial H}{\partial p_k}-\frac{\partial F}{\partial p_k}\frac{\partial H}{\partial q_k}\right)\mathrel{=:}\{F,H\}.$$
It is just the same calculation as above, repeated $n$ times (i.e. once for each pair of canonically conjugate coordinates).
To answer your questions: $F$ is a real-valued function. If the book uses an abuse of notation, then the above provides the correct interpretation of it. As you write, $n=3N$ (hence $6N$ coordinates in total) is common for direct descriptions of mechanical systems with $N$ point particles.