Consider the simple case of electromagnetic irradiation of a homogeneous isotropic dielectric, neglecting the dispersion of the refractive index. Assuming a transparent medium, the spatial density of forces acting on the dielectric in a static external electromagnetic field can be given as
$$\mathbf{f} = - \nabla p - \nabla \epsilon \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} - \nabla \mu \dfrac{\langle \mathbf{H}^2 \rangle}{8 \pi} + \nabla \left[ \left( \rho \dfrac{\partial{\epsilon}}{\partial{p}} \right)_T \dfrac{\langle \mathbf{E}^2 \rangle}{8 \pi} + \left( \rho \dfrac{\partial{\mu}}{\partial{\rho}} \right)_T \dfrac{\langle \mathbf{H}^2 \rangle}{8 \pi} \right] + \dfrac{\epsilon \mu - 1}{4 \pi c} \dfrac{\partial}{\partial{t}}\langle [ \mathbf{E} \times \mathbf{H}] \rangle.$$
$p$ is the pressure in the medium (for a given density $\rho$ and temperature $T$ in zero field.
$\epsilon$ and $\mu$ are the permittivity and magnetic permeability.
$c$ is the speed of light.
The angular brackets denote averaging over a time period far greater than the characteristic alternation period of light.
And my understanding is that $\mathbf{E} \times \mathbf{H}$ is the Poynting vector.
What I don't understand is the squared field terms $\mathbf{E}^2$ and $\mathbf{H}^2$. These field terms are vector fields, and so my understanding is that it is not mathematically valid to take a vector field (or any other vector) to an exponent. So what is meant by $\mathbf{E}^2$ and $\mathbf{H}^2$ in this context?
I would greatly appreciate it if people would please take the time to explain this.
