Definition of torque for a continuous body I am working on basic physics definitions. Given a particle at position $r$ (in some coorinate reference system) upon which acts a force $F$, the $torque$ $\tau$ is defined by
\begin{equation}
\tau:=r\times F
\end{equation}
Now let's suppose that we have a continuous body bounded by a region $D$; let also $\rho(r)$ be the mass distribution and let $F(r)$ be a force field. I'd like to derive a definition of torque suitable to this situation. Any help?
 A: I'll write my comments here as a full answer, as suggested by Floris. I won't use the moment of inertia tensor: it's simpler from pure angular momentum of each point particle.
We know that
$$\vec{L} = (\vec{r} \times \dot{\vec{r}})\,m .$$
So, for a point particle,
$$d\vec{L} = (\vec{r} \times \dot{\vec{r}})\, dm .$$
Noting that $\rho = \frac{dm}{dV}$,
$$d\vec{L} = (\vec{r} \times \dot{\vec{r}})\, \rho(\vec{r}) \, dV ,$$
which brings us to
$$\vec{L} = \int_\Sigma (\vec{r} \times \dot{\vec{r}})\, \rho(\vec{r}) \, dV ,$$
where $\Sigma$ is the region where the body's volume is defined on. Since the torque $\tau$ is the time derivative of $\vec{L}$, by assuming mass density doesn't vary in time we get
$$\vec{\tau} = \int_\Sigma \left(\dot{\vec{r}} \times \dot{\vec{r}} + \vec{r} \times \ddot{\vec{r}} \right)\, \rho(\vec{r}) \, dV \\
= \int_\Sigma (\vec{r} \times \ddot{\vec{r}} )\, \rho(\vec{r}) \, dV $$
A: $
\newcommand{\r}{\mathbf{r}}
\newcommand{\F}{\mathbf{F}}
\newcommand{\g}{\mathbf{g}}
\newcommand{\t}{\boldsymbol\tau}
$Let me start by defining $\g(\r)$ to be a position dependent force per unit mass. Then the force per unit volume $d\F$ is given by $d\F(\r) = \g(\r) \rho (\r) dV$. The torque per unit volume $d\t$ is given by $d\t = \r \times d\F$. The total torque is then given by $\t=\int d\t = \int \r \times d\F = \int \r \times \rho \g dV$.
Given this definition for torque, and the independent definition for angular momentum, we can then make a meaningful state about relating the torque and angular momentum (given only central forces are at play).
