General-purpose integral formulae to compute the magnetic field of permanent magnets and electromagnets

What are the general-purpose integral formulae to compute, for any $(x,y,z) \in \mathbb{R}$ (in particular, not just near or far), the magnetic field $\vec B(x,y,z)$ of:

1. a permanent magnet with 1.1) known, arbitrary geometry (ie. it can be a sphere, cylinder, torus, box, horseshoe, pyramid, etc.) and 1.2) known, uniform/constant, arbitrarily oriented magnetization vector field $\vec M(x,y,z)$ (ie. the magnet can be magnetized axially or in any other magnetization direction, but it's constant inside the magnet and it vanishes outside of it),
2. a volumetric (ie. not infinitely thin) electromagnet with 2.1) known, steady-state DC (eg. 2 amps) and 2.2) known, arbitrary geometry (ie. it can be an $n$-layer solenoid, a complicated knot, or something else).

"Known geometry" means that, for instance, the surface of the magnet is given by a known parametric function $\vec f(u,v)$ taking values in $\mathbb{R}^3$, or as a triangle mesh (where each triangle is given by its vertices $\vec v_1, \vec v_2, \vec v_3 \in \mathbb{R}^3$), or in any other manner that is convenient and explicit.

I think the hypotheses rule out the need to consider time retardation, wave propagation delay, etc.

For 1), assuming that $\vec M(\vec x)$ is constant inside the magnet and vanishes outside, the following might work:

$\vec B(x,y,z) = \vec B(\vec x) = \nabla \times \vec A(\vec x) = {\mu_0 \over 4\pi} \nabla \times \iint _S {\vec M(\vec x') \times \hat n(\vec x') \over |\vec x - \vec x'|} dS(\vec x'),$

where the integral runs over the surface/boundary $S$ of the magnet, $\vec x' \in S$ are the points in the surface $S$, $\hat n(\vec x')$ is the unit normal at $\vec x'$, and $dS(\vec x')$ is the infinitesimal area of the differential area element at $\vec x'$ (ie. it's a scalar).

This is nice, but I think there's at least 2 ways to go about this: one is to use the magnetic monopole method (ie. assume there's a known distribution of magnetic monopoles inside the volume of the magnet), and the other is to use the Amperian current loop model (ie. assume there's a known distribution of current loops on the surface of the magnet). I wonder what method this integral corresponds to, and what integral the other method would yield. (Eg. I think there's a way to do this using the magnetic scalar potential, rather than the vector potential.)

For 2), I think Biot-Savart might work:

$\vec B(x,y,z) = \vec B(\vec x) = {\mu_0 \over 4\pi} \iiint_V {\vec J(\vec x') \times (\vec x - \vec x') \over |\vec x - \vec x'|^3} dV(\vec x)',$

where the integral runs over the 3D volume $V$ of the magnet, $\vec x' \in V$ are the points in the volume $V$, $\vec J(\vec x')$ is the current density at $\vec x'$, and $dV(\vec x')$ is the infinitesimal volume of the differential volume element at $\vec x'$ (ie. it's a scalar).
(But if you're running some known current, eg. 2 amps, over your electromagnet, then what is the current density $\vec J(\vec x)$? Is it just uniform?)