Editing Earnshaw's theorem (section)

===Background===
The proofs considered here are based on the following principles.

The energy U of a [[magnetic dipole]] with a [[magnetic dipole moment]] '''M''' in an external magnetic field '''B''' is given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -(M_x B_x + M_y B_y + M_z B_z).</math>

The dipole will only be stably levitated at points where the energy has a minimum. The energy can only have a minimum at points where the Laplacian of the energy is greater than zero. That is, where
<math display="block">\nabla^2 U = \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2} > 0.</math>

Finally, because both the divergence and the curl of a magnetic field are zero (in the absence of current or a changing electric field), the Laplacians of the individual components of a magnetic field are zero. That is,
<math display="block">\nabla^2 B_x = \nabla^2 B_y = \nabla^2 B_z = 0.</math>

This is proven at the very end of this article as it is central to understanding the overall proof.