Editing Earnshaw's theorem (section)

===Magnetic dipole aligned with external field lines===
The case of a paramagnetic or diamagnetic dipole is considered first. The energy is given by
<math display="block">U = -k|\mathbf{B}|^2 = -k \left (B_x^2 + B_y^2 + B_z^2 \right ).</math>

Expanding and rearranging terms,
<math display="block">\begin{align}
\nabla^2 |\mathbf{B}|^2 &= \nabla^2 \left (B_x^2 + B_y^2 + B_z^2 \right ) \\
                        &= 2\left( |\nabla B_x|^2 + |\nabla B_y|^2 + |\nabla B_z|^2 +B_x\nabla^2 B_x + B_y\nabla^2 B_y + B_z\nabla^2 B_z \right)
\end{align}</math>

but since the Laplacian of each individual component of the magnetic field is zero,
<math display="block">\nabla^2 |\mathbf{B}|^2 = 2\left( | \nabla B_x |^2 + | \nabla B_y |^2 + | \nabla B_z |^2 \right);</math>

and since the square of a magnitude is always positive,
<math display="block">\nabla^2 |\mathbf{B}|^2 \geq 0.</math>

As discussed above, this means that the Laplacian of the energy of a paramagnetic material can never be positive (no stable levitation) and the Laplacian of the energy of a diamagnetic material can never be negative (no instability in all directions).

Further, because the energy for a dipole of fixed magnitude aligned with the external field will be the square root of the energy above, the same analysis applies.