## 2 Materials and Methods
### Mathematical Apparatus for Magnetic Hysteresis Curve Reconstruction  
This sub-chapter is devoted to the derivation of a mathematical equation, based on which it is possible to quantify the magnetic polarization, in this case as a quantity defining the magnetic state of the open-ended sample, from the measured value of the stray magnetic field. The entire analytical process of derivation is purposefully processed into individual extensive and systematically consecutive steps, which, with the use of theorems of classical magnetostatics, lead to a final definition of a sought equation. The reason for this complex approach is that current authors that deal with similar issues use only the resulting mathematical relationships, which alone are insufficient for understanding the physical essence of the measurement method and the meaning of the theorems used to define the final mathematical relation. The construction of a typical magnetic hysteresis curve, like dependence of a magnetization \(M\) or polarization \(J\) on the excitation field \(H_{ex}\), is given by linear proportionality between measured stray magnetic field \(B_{stray}\) of the ferromagnetic open-ended sample and its actual state of magnetization. This interdependence stems from the physical nature of the magnetization process, during which the infinitesimal magnetic moments of the ferromagnetic sample gradually align in the preferred direction corresponding to the excitation (applied) magnetic field \(H_{ex}\), leading to the formation of a dominant magnetic moment. As a consequence of magnetization, surface and volume magnetic poles, collectively called free or isolated magnetic poles, are created. Since, according to magnetostatic theory, free magnetic poles of the magnetized material represent locally uncompensated dipole magnetic moments, their surface and volume distribution on and inside the open-end sample, respectively, must naturally give rise to an internal and external magnetic field \(H\) which can be in an arbitrary position inside or around the sample generally expressed as a divergence of a magnetic scalar potential \(\varphi_{m}\)[44]:  
\[\mathbf{H}(\mathbf{r})=-\
abla\varphi_{m} \tag{1}\]  
This statement is valid only in situations where the magnetic field \(\mathbf{H}\) is not produced by the free currents. The field \(\mathbf{H}\) is then conservative, whereas the magnetic field \(\mathbf{B}\) act solenoidal; therefore, [PERSON]'s law becomes:  
\[\
abla\times\mathbf{H}=0 \tag{2}\]  
And [PERSON]'s law for magnetism states:  
\[\
abla\cdot\mathbf{B}=0 \tag{3}\]  
Consequently, from the definition of the equation for divergence-free field \(\mathbf{B}\), it follows that:  
\[\
abla\cdot\mathbf{B}=\mu_{0}\
abla\cdot(\mathbf{H}+\mathbf{M})=0\]  
\[\
abla\cdot\mathbf{H}=-\
abla\cdot\mathbf{M} \tag{4}\]  
Considering Equations (1) and (4), the magnetic scalar potential must satisfy [PERSON]'s equation:  
\[-\
abla^{2}\varphi_{m}=-\Delta\varphi_{m}=\
abla\cdot\mathbf{M} \tag{5}\]  
Discontinuities of the magnetic field \(\mathbf{H}\), described in terms of divergence of magnetization \(\mathbf{M}\), implicitly suggest the existence of the above-mentioned surface and volume magnetic poles. These free magnetic poles, consisting of positive and negative \"monopoles\", thus act as sources of magnetization, the discontinuity of which can occur either at the sample boundaries or in the interior.