## 2 Neumann Green's Function and Dipole Kernel for EM Flowmeters
Formally, one can take the radius of the disk \(D\) to be very large so the inverse point \(\boldsymbol{\xi}^{\prime}\) in (9) tends to \(\infty\) and the Neumann Green's function reduces to the free-space fundamental solution with reference potential at 'the point at' \(\infty\), as desired._In Figure 1, the measurement is the difference in the potentials measured at the two electrode locations, \(\mathbf{x}_{1}=(0,1)\) and \(\mathbf{x}_{2}=(0,-1)\). Using (9), a theoretical measurement, i.e., without any measurement errors, for this geometry is given by  
\[\Delta u=u(\mathbf{x}_{1})-u(\mathbf{x}_{2}) =\iint_{D}B\frac{\partial v}{\partial y}(\xi)\Big{(}g_{\mathrm{N} }(\mathbf{x}_{1}|\xi)-g_{\mathrm{N}}(\mathbf{x}_{2}|\xi)\Big{)}\,\,\mathrm{d}\,\xi\] \[=B\iint_{D}v(\xi)\frac{\partial}{\partial y}\Big{(}g_{\mathrm{N} }(\mathbf{x}_{2}|\xi)-g_{\mathrm{N}}(\mathbf{x}_{1}|\xi)\Big{)}\,\,\mathrm{d}\,\xi \tag{10}\]  
where the second equality uses integration by parts and [PERSON]'s assumption that \(v=0\) on \(\partial D\). Comparing this with (4), we see that [PERSON]'s weight function \(W\) is the (scaled) \(y\)-gradient of the difference of [PERSON]'s functions. [PERSON]'s formulation [15; 16] allowed more general flow profiles and placement of electrodes, with the theoretical measurement that is the difference of potentials at electrodes at \(\mathbf{x}1\) and \(\mathbf{x}2\). Then, Equation (9) gives  
\[\Delta u=u(\mathbf{x}_{1})-u(\mathbf{x}_{2}) =\iint_{D}\
abla\cdot\big{(}\mathbf{v}\times\mathbf{B}\big{)}( \xi)\Big{(}g_{\mathrm{N}}(\mathbf{x}_{1}|\xi)-g_{\mathrm{N}}(\mathbf{x}_{2}| \xi)\Big{)}\,\,\mathrm{d}\,\xi\] \[=\iint_{D}\big{(}\mathbf{v}\times\mathbf{B}\big{)}(\xi)\cdot \
abla\Big{(}g_{\mathrm{N}}(\mathbf{x}_{2}|\xi)-g_{\mathrm{N}}(\mathbf{x}_{1 }|\xi)\Big{)}\,\,\mathrm{d}\,\xi\] \[=\iint_{D}\mathbf{v}\cdot\Big{(}\mathbf{B}\times\
abla\big{(}g_{ \mathrm{N}}(\mathbf{x}_{2}|\xi)-g_{\mathrm{N}}(\mathbf{x}_{1}|\xi)\big{)}\Big{)} \,\,\mathrm{d}\,\xi \tag{11}\]  
where the second equality uses the divergence theorem and the assumption that \(\mathbf{v}=0\) on \(\partial D\), and the third equality follows from application of the scalar triple product. Comparing this with Equations (5) and (6), we see that \(\mathbf{j}=\
abla\big{(}g_{\mathrm{N}}(\mathbf{x}_{2}|\xi)-g_{\mathrm{N}}( \mathbf{x}_{1}|\xi)\big{)}\), i.e., the gradient of the difference of Neumann Green's functions. This result is clear when one considers that the Neumann Green's function is the potential in the region due to unit current source at \(\xi\) with insulating boundary conditions, current is the gradient of potential, and reciprocity equates the form to current injection at electrodes. That is, Bevir's weight function \(\mathbf{W}\) is the cross product of the magnetic field with the gradient of the difference of Neumann Green's functions. The calculations leading to Equations (10) and (11) suggest a generalization of both [PERSON]'s and [PERSON]'s analysis that also does not bias the forward map towards any specific reference point, while acknowledging that only potential differences are measurable. In particular we evaluate measurement at a single boundary electrode given by Equation (9) with \(\psi=0\) as in Remark 1. Using vector identities to shift the divergence off the source term, \(f=\
abla\cdot\big{(}\mathbf{v}\times\mathbf{B}\big{)}\), (9) with \(\psi=\partial_{n}u=0\) can be written as  
\[u(\mathbf{x})=\iint_{D}-\
abla g_{\mathrm{N}}(\mathbf{x}|\xi)\cdot(\mathbf{v }\times\mathbf{B})\,\,\mathrm{d}\,\xi+\int_{\partial D}g_{\mathrm{N}}( \mathbf{x}|\xi)(\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\,\mathbf{S}. \tag{12}\]  
Since \(-\
abla\delta(\mathbf{x}-\xi)\cdot\mathbf{e}\) is the dipole distribution with moment \(\mathbf{e}\)[13], the first term describes the potential generated from dipolar source distribution \((\mathbf{v}\times\mathbf{B})\) in the domain \(D\), and the second describes the potential due to current source distribution on the boundary.