magnetic fields
Hematite has certain magnetic qualities, such as being paramagnetic (meaning it is attracted to magnetic fields), but I have read that if you heat it up it becomes strongly magnetic.
But I have a string of magnetic hematite beads. If they are magnetic, does that mean someone has heated them? Or is hematite only magnetic while it is hot? How do you get hematite to stay magnetic?
First doubt: Why do magnetic field lines form closed curves?
The premise is false!
Take the following image I generated as an example. The black circles here are two current loops arranged haphazardly. The blue line is a single magnetic field line, plotted for a really long length. It's still going, and it isn't ending any time soon.
The only statement of importance is that $\nabla \cdot \vec{B}=0$. This can be interpreted differently: the divergence of a vector field at a point can be approximated by the flux into a very small sphere of volume $V$ at that point:$$\nabla \cdot \vec{B}=\lim_{V\to 0}\frac{\oint_S\vec{B}\cdot d\vec{a}}{V} $$ ($S$ denotes the surface of the sphere volume $V$ centered at the point in question, and $d\vec{a}$ denotes a vector area element). Therefore, if a magnetic field line penetrates the tiny sphere and ends, and has some magnitude, then $\nabla \cdot \vec{B}\neq 0$ and you've violated a Maxwell law!
But a magnetic field line can actually end. For example, imagine two single loop solenoids on top of each other, pointing in opposite directions. As derived on this page, we might have:
$$B_z=-\frac{\mu_0 R^2 I}{2((z-a)^2+R^2)^{3/2}}+\frac{\mu_0 R^2 I}{2((z+a)^2+R^2)^{3/2}}$$
At $z=0$, the field is zero. At $z<0$, the field is positive and along the z axis. At $z>0$, the field is negative and along the z axis. So clearly the field line heads towards zero, but never reaches it.
Second doubt: Why do we say that the strength of the magnetic field is more where the lines are closer together?
The following page defines $B=\sqrt{\vec{B}\cdot \vec{B}}$ and $\hat{b}=\vec{B}/B$, and proves that as you walk along a field line:
$$\frac{dB}{d\ell}=\hat{b}\cdot \nabla B=-B \nabla \cdot \hat{b}$$
If the field lines are converging then $\nabla \cdot \hat{b}<0$ and so $B$ is increasing in magnitude, and if the field lines are diverging then $\nabla \cdot \hat{b}>0$ and so $B$ is decreasing in magnitude. So there's your vector calculus proof.
J.D Callen, Fundamentals of Plasma Physics, chapter 3
Third: Why do iron fillings acquire exactly the design of the magnetic field?
This is more complicated. Each iron filing forms a little magnet that attracts its neighbors, so the iron filings can't fill up space and instead join end to end in directions induced by the magnetic field. So they form lines. Which field lines are chosen depends on the whole, ugly dynamics of the situation.
Last doubt: The diagram of the magnetic field lines that we see (the 2D diagram with many curves), is that diagram 3D in reality?
Yep, Maxwell's equations in their vector calculus form work only in 3D, so the lines you get, in general, are three dimensional lines.
It is energetically favourable for diamagnetic moments to anti-align with an applied $B$ field.
In a liquid all these microscopic moments are free to rotate (subject to thermal fluctuations) and as such all anti-align with the applied field.
An example of magnetic moments rotating in a fluid is in the "super paramagnetic" ferrofluid (except here the ferromagnetic nanoparticles align with the $B$ field).
For further reading I recommend "Magnetism and Magnetic Materials" by Coey.
Best Answer
The magnetic haematite beads commonly sold as being good for you in some ill defined way are actually a mixture of haematite and magnetite. The magnetic field is generated by the magnetite.
At room temperature haematite is weakly ferromagnetic but not enough to be easily noticeable. It is only paramagnetic above 948K.