## 7 Computation of Magnetization in Body Fragments from Vases 1-6
### Formulation of Remament Magnetization in Vases with Arbitrary Rotational Symmetry
sin\(\varphi\), M\({}_{\rm z}\) = M.cos\(\theta\). The magnetization, **M**, and the magnetic field, **B**, diverge at the same angle, \(\varphi=\varphi_{\rm B}\), from the horizontal x-axis. The angles \(\tan\varphi=\frac{\mathrm{M_{z}}}{\mathrm{M_{x}}}=\frac{\mathrm{B_{z}}}{\mathrm{B_{ s}}}\), \(\cos\theta=\frac{\mathrm{M_{y}}}{\mathrm{M}}\) are calculated from the magnetic field measurements of the body fragments, while the angle \(\gamma\) is calculated from the magnetic field measurements in co-belonging fragments of the vase base. * The height-position of the fragments in each longitudinal section of the vase is determined by the inclination, \(\kappa\), which is computed by the equation: \[\kappa=\tan^{-1}\frac{\mathrm{M_{z}.\sin\gamma.\sin\omega-M_{y}.\cos\gamma}}{ \mathrm{M_{y}.\sin\gamma.\sin\omega+M_{z}.\cos\gamma}}\] (12)  
Determining the change in directionality of remanent magnetization, M, in the vase body in a manner dictated by its rotational symmetry, is performed from the calculated angles \(\omega\), \(\kappa\) at each measurement position, using the inverse rotational transform of the calculated magnetization components at an angle -\(\kappa\) around the x-axis (Figure 19a), from the reference system (xyz), where the magnetic field measurements are taken, to the (xyz)* reference system (Figure 19b) of the cylindrical vase:  
\[\mathrm{M_{x}*}=\mathrm{M_{x}}=\mathrm{M_{x}}=\mathrm{M_{y}.\cos\omega,\,M_{ y}*}=\mathrm{M_{y}.\cos\kappa}+\mathrm{M_{z}}\,\sin\kappa=\mathrm{M_{.}\sin \gamma.\sin\omega,} \tag{13}\]  
In the reverse transformation (Figure 19b), the components \(\mathrm{M_{x}*}\),\(\mathrm{M_{y}*}\) and the angle \(\omega\) should display similar values in each longitudinal section of the body, based on the above equations, while \(\mathrm{M_{z}*}\), which depends only on angle \(\gamma\), should display similar values in each measurement position. Indicative experimental results with the reverse transformation are given from measurements of body fragments (Table 3) from vase 6. Figure 19: (**a**). Illustration of the inverse transformation at an angle -\(\kappa\) around the x-axes, from the reference systems (xyz), where the magnetic field measurements are taken, to the reference systems (xyz)*of the cylindrical vase. (**b**). Visualization of reference systems, (xyz)*, of the cylindrical vase, in a vase with arbitrary rotational symmetry. The components \(\mathrm{M_{x}}^{\star}\),\(\mathrm{M_{y}}^{\star}\) and the angle \(\omega\) display similar values (Table 3a) in each longitudinal section of the body. Inclination, \(\kappa\), displays similar values in each transverse groove of the vase. \(\mathrm{M_{z}}^{\star}\) displays similar values (Table 3b) in each measurement position and with that which computed for the base fragments of the vase. Experimental Results: Screening of Co-Belonging Fragments and Determination of Their Position in the Body of a Vase from Their Magnetization  
The computed values for the inclination, \(\kappa\), and the angle \(\omega\) are displayed (Figures 20-31) on the bodies of the six vases. The magnetization magnitude M (mA/m) in body fragments displays similar values (Tables 4-9), and with those calculated from the base fragments of the vases. The slope, \(\kappa\) (Figures 20, 22, 24, 26, 28 and 30) which displays similar values in each cross-section and determines the height-position of the fragments, is compared with the measured inclination of the walls with respect to the axis of symmetry in each vase. The computed values for the inclination, \(\kappa\), from the sensor readings are similar to the measured ones.