I'm quite lost what $B$ and $H$ is. It seams to me that most of the texts I read do quite poor job in explaining them properly. They are explained only in cases when magnetic susceptibility is constant in the space.
First law says that $M = \chi H$, $M$ is magnetization and to my understanding $H$ is something like external magnetic field (for example Earth magnetic field when I do experiment in some lab).
Next law says that magnetic field is $B = \mu_0(M+H) = \mu H$. This has to be wrong when $\chi$ is not constant because than we are not guarantied that divergence of $B$ is zero. $$ \nabla \cdot B = \nabla \cdot (\mu H ) \neq \mu \nabla \cdot H = 0 $$
Can you please direct me to some book when this is explained properly?
My aim: I want to calculate magnetic field around paramagnetic material. I cannot use formula $B = \mu H$ because outside paramagnet is $\mu = \mu_0$ so magnetic field would be unchanged.