Editing Earnshaw's theorem (section)

===Summary of proofs===
For a magnetic dipole of fixed orientation (and constant magnitude) the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -(M_x B_x + M_y B_y + M_z B_z),</math>
where ''M<sub>x</sub>'', ''M<sub>y</sub>'' and ''M<sub>z</sub>'' are constant. In this case the Laplacian of the energy is always zero,
<math display="block">\nabla^2 U = 0,</math>
so the dipole can have neither an energy minimum nor an energy maximum. That is, there is no point in free space where the dipole is either stable in all directions or unstable in all directions.

Magnetic dipoles aligned parallel or antiparallel to an external field with the magnitude of the dipole proportional to the external field will correspond to paramagnetic and diamagnetic materials respectively. In these cases the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -k\mathbf{B}\cdot\mathbf{B} = -k \left (B_x^2 + B_y^2 + B_z^2 \right ),</math>
where ''k'' is a constant greater than zero for paramagnetic materials and less than zero for diamagnetic materials.

In this case, it will be shown that
<math display="block">\nabla^2 \left (B_x^2 + B_y^2 + B_z^2 \right ) \geq 0,</math>
which, combined with the constant {{math|''k''}}, shows that paramagnetic materials can have energy maxima but not energy minima and diamagnetic materials can have energy minima but not energy maxima. That is, paramagnetic materials can be unstable in all directions but not stable in all directions and diamagnetic materials can be stable in all directions but not unstable in all directions. Of course, both materials can have saddle points.

Finally, the magnetic dipole of a ferromagnetic material (a permanent magnet) that is aligned parallel or antiparallel to a magnetic field will be given by
<math display="block">\mathbf{M} = k{\mathbf{B} \over |\mathbf{B}|},</math>

so the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -k\frac{\mathbf{B}\cdot\mathbf{B}}{ |\mathbf{B}|} = -k\frac{|\mathbf{B}|^2 }{ |\mathbf{B}|} = -k\left (B_x^2 + B_y^2 + B_z^2 \right )^{\frac{1}{2}};</math>

but this is just the square root of the energy for the paramagnetic and diamagnetic case discussed above and, since the square root function is monotonically increasing, any minimum or maximum in the paramagnetic and diamagnetic case will be a minimum or maximum here as well. There are, however, no known configurations of permanent magnets that stably levitate so there may be other reasons not discussed here why it is not possible to maintain permanent magnets in orientations antiparallel to magnetic fields (at least not without rotation—see [[spin-stabilized magnetic levitation]].