What is the proper name for H?

[sniffer]

in electrodynamics, we have magnetization M.
the other quantity associated with free current is H.

i haven't seen a book that gives the name to H. what is the proper name assign to H?

 

[Bystander]

CRC: magnetic field strength, or magnetic field intensity.

 

[Galileo]

Unfortunately, many authors call H and not B the magnetic field and invent a new word for B like flux density or or induction (very poor choice, since that gives the word yet another meaning). H doesn't really have an appropriate name, but Griffith's names the chapter on it the 'Auxiliary field 'H'', but agrees to just call it "H". It seems A. Sommerfeld has written in his book on electrodynamics:

The unhappy term "magnetic field" for H should be avoided as far as possible. It seems to us that this term has led into error none less than Maxwell himself..."

 

[Antiphon]

Unfortunately, many authors call H and not B the magnetic field and invent a new word for B like flux density or or induction (very poor choice, since that gives the word yet another meaning). H doesn't really have an appropriate name, but Griffith's names the chapter on it the 'Auxiliary field 'H'', but agrees to just call it "H". It seems A. Sommerfeld has written in his book on electrodynamics:

In MKS units E is the electric field, H is the magnetic field, D is the
electric flux and B is the magnetic flux.

This usage is self-consistent even if traditionally the magnetic flux is what
has been referred to as the magnetic field.

To make this plain, by Gauss' law the electric flux D always equals the
charge enclosed (to within a multiplicative constant) regardless of the
material around the charge. The electric field would vary.

If there were magnetic monopoles, the magnetic field H would vary in the
presence of magnetic materials but the magnetic flux B would only
represent the presence of the enclosed magnetic charge.

The historical physics terminology is inconsistent in this regard.

 

[Galileo]

In MKS units E is the electric field, H is the magnetic field, D is the
electric flux and B is the magnetic flux.

This usage is self-consistent even if traditionally the magnetic flux is what
has been referred to as the magnetic field.

To make this plain, by Gauss' law the electric flux D always equals the
charge enclosed (to within a multiplicative constant) regardless of the
material around the charge. The electric field would vary.

It is not a matter of inconsistency, but of inconsequent terminology.
The FLUX of some vector field (through some surface S) is:
$$\Phi_F = \int_S \vec F \cdot \vec dr$$
So $\Phi_E$ is the electric flux and $\Phi_B$ the magnetic flux. This makes it seemingly reasonable to name E and B electric- and magnetic flux density.

This can already lead to confusion in your post. You said the electric flux D equals the charge enclosed within. D is a vector field, it does not contain charge. You mean $\Phi_D$ through a closed surface equals the enclosed charge (to within some constant ofcourse).