# What are the physical interpretations of $\nabla \times \textbf{P}$ and $\nabla \cdot \textbf{M}$?

In the text I am reading, it is made clear that it is incorrect to assume that the electric displacement $\textbf{D}$ is identical to the electric field with the exception that it is raised from the free charge instead of total charge. A similar case is made with $\textbf{H}$. For the electric displacement, we know that $\nabla \times \textbf{D}=\nabla \times \textbf{P}$. We also know that $\nabla \cdot \textbf{H}=\nabla \cdot \textbf{M}$. So what would the physical interpretation of $\nabla \times \textbf{P}$ and $\nabla \cdot \textbf{M}$ be? Furthermore, are there any materials in which $\nabla \times \textbf{P}$ and $\nabla \cdot \textbf{M}$ are zero?