(See Section 2 for the definition and properties of the Neumann Green's function.) Instead, [PERSON] ignored the integral over \(\partial D\) in Equation (2)--called the bilinear concomitant--but by adding a harmonic function and assuming the no-slip boundary condition that \(\partial_{n}v(\mathbf{x})=0\ \forall\mathbf{x}\in\partial D\) gave a potential \(u\) that satisfied the boundary condition that \(\partial_{n}u(\mathbf{x})=0\ \forall\mathbf{x}\in\partial D\), though with unstated reference potential. By taking the difference of such potentials, [PERSON] avoided the requirement to state a reference potential for \(u\). Here, we utilize the rigorous and comprehensive theory of [PERSON]'s functions as particular fundamental solutions, formally defined via [PERSON]' distribution theory, though these methods were not widely integrated into the applied sciences at the time. Substituting \(\
abla^{2}u=B\partial v/\partial y\), as noted above, and integrating by parts in the \(y\) direction, [PERSON] [11]derived the potential difference between electrodes at \((x,y)=(0,1)\) and \((x,y)=(0,-1)\) in a pipe of radius 1 as  
\[\Delta u=\frac{2B}{\pi}\iint_{D}v(x,y)W(x,y)\;\mathrm{d}\;x\;\mathrm{d}\;y \tag{4}\]  
where \(W\) is [PERSON]'s weight function. Contours of \(W\) are plotted in Figure 1, which, as noted by [PERSON] [11], are Cassini ovals. Importantly, for our purposes, and as is evident in Figure 1, the measured potential difference is much more sensitive to fluid flow in the pipe near the electrodes, than further away. That is, a given magnitude of flow in a region near the electrode will cause a greater electrical response than when the same flow occurs in a region further away from the electrode. Because the weighting function \(W\) approaches \(\infty\) at the electrodes, reverse flow near one electrode could cause negative readings for positive bulk flow, and, indeed, the measurement \(\Delta u\) could be anywhere between \(+\infty\) and \(-\infty\) for a given average, bulk flow rate. [PERSON] [11] also noted this, and that the common statement at that time that flowmeters are insensitive to the velocity profile is clearly erroneous, except in the special case of radially symmetric axial flow. [PERSON]'s weighting function was subsequently extended by [PERSON] [15] to accommodate general flow fields, and also gave the conditions for an EM flowmeter to be insensitive to flow profile [16]. [PERSON]'s formulation defines the weight function \(\mathbf{W}\) by  
\[\Delta u=\int\mathbf{v}\cdot\mathbf{W} \tag{5}\]  
with  
\[\mathbf{W}=\mathbf{B}\times\mathbf{j}, \tag{6}\]  
where \(\mathbf{j}\) is the current density when unit current is passed between the electrodes. We generalize both [PERSON]'s and [PERSON]'s formulation in Section 2. Further notes on the historical development of electromagnetic flowmeters are presented in [17]. Many papers have been published since, further exploring this weighting function applied to different magnetic field arrangements, and extending to more measurements than the original two electrodes; see, e.g., [18; 19; 20; 21]. This paper further explores the sensitivity of a flowmeter to spatial variation in flow and imposed magnetic field. As noted by [PERSON] [11], even in the restricted case of axial flow in a pipe, the spatial distribution of flow can dominate the measurement made, with a measurement anywhere between \(+\infty\) and \(-\infty\) theoretically possible for a given  
Figure 1: Cross section of pipe showing contours of [PERSON]’s weight function, \(W\), for constant \(\mathbf{B}\) field in the \(x\)-direction and for small sensing electrodes at \((x,y)=(0,1)\) and \((x,y)=(0,-1)\), redrawn from [11]. bulk flow rate. It is therefore critical to understand the response of flowmeters to spatial flow distribution when designing a flowmeter intended for detecting bulk flow rate or for imaging flow distribution, especially when calibration within a restricted flow regime is not feasible or when simplifying assumptions about the flow distribution are not valid.