Gravity Anomaly of Polyhedral Bodies Having a Polynomial Density Contrast

Abstract

We analytically evaluate the gravity anomaly associated with a polyhedral body having an arbitrary geometrical shape and a polynomial density contrast in both the horizontal and vertical directions. The gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Density contrast is assumed to be a third-order polynomial as a maximum but the general approach exploited in the paper can be easily extended to higher-order polynomial functions. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and is expressed as a sum of algebraic quantities that only depend upon the 3D coordinates of the polyhedron vertices and upon the polynomial density function. The accuracy, robustness and effectiveness of the proposed approach are illustrated by numerical comparisons with examples derived from the existing literature.

Appendices

Appendix 1: Algebraic Expression of Integrals

We are going to show that the 2D integrals

$$\begin{aligned} \int \limits _{F_i} \frac{[\otimes {\varvec{\rho }}_i,m]}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\rm d}A_i \quad m\in [0,4] \end{aligned}$$

(167)

can be evaluated analytically. As a matter of fact, we only need to evaluate the integrals for $m=3$ and $m=4$

$$\begin{aligned} {\mathfrak {C}}_{F_i}= \int \limits _{F_i} \frac{ {\varvec{\rho }}_i \otimes {\varvec{\rho }}_i \otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}} {\rm d}A_i\qquad {\mathfrak {D}}_{F_i}= \int \limits _{F_i} \frac{{\varvec{\rho }}_i \otimes {\varvec{\rho }}_i \otimes {\varvec{\rho }}_i \otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}} {\rm d}A_i\,, \end{aligned}$$

(168)

since the additional ones in (167) have been already computed in D’Urso (2013a, 2014a, b). For completeness, these last ones are reported in Appendix 2.

A further integral, namely

$$\begin{aligned} {\varvec{\varPsi }}_{F_i}= \int \limits _{F_i}\frac{{\varvec{\rho }}_i \otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} {\rm d}A_i\,, \end{aligned}$$

(169)

required for the computation of the integrals (168), will be dealt with at the end of this Appendix.

The rationale for evaluating the integrals (168) is to first apply the generalized Gauss’ theorem D’Urso (2013a, 2014a) to transform them into 1D integrals and, subsequently, to compute such integrals by means of algebraic expressions depending upon the 2D coordinates of the vertices that define the face $F_i$.

In order to apply Gauss’ theorem to the integrals in (168), let us first prove the identity

$$\begin{aligned} {\text {grad}}\left[ \varphi \,( \mathbf {a}\otimes \mathbf {b})\right] = (\mathbf {a}\otimes \mathbf {b})\otimes {\text {grad}}\varphi + \varphi \, {\text {grad}}\mathbf {a}\otimes \mathbf {b}+ \varphi \, \mathbf {a}\otimes {\text {grad}}\mathbf {b}, \end{aligned}$$

(170)

holding for scalar $\varphi$ and vector $( \mathbf {a}, \mathbf {b})$ differentiable fields.

It can be easily verified by applying the chain rule to the ijk component of the third-order tensor on the left-hand side

$$\begin{aligned} \left\{ {\text {grad}}\left[ \varphi \,( \mathbf {a}\otimes \mathbf {b})\right] \right\} _{jkq}= \left( \varphi \, a_j b_k\right) _{/q} =\varphi _{/q}\,a_j\,b_k + \varphi \,a_{j/q} \, b_{k} + \varphi \,a_{j} \,b_{k/q}\,. \end{aligned}$$

(171)

In a similar fashion, one can prove the further differential identity involving fourth-order tensors

$$\begin{aligned} \begin{array}{cc} {\text {grad}}\left[ \varphi \,( \mathbf {a}\otimes \mathbf {b}\otimes {\mathbf {c}})\right] = (\mathbf {a}\otimes \mathbf {b}\otimes {\mathbf {c}}) {\text {grad}}\varphi + \varphi \, {\text {grad}}\mathbf {a}\otimes \mathbf {b}\otimes {\mathbf {c}}+ \varphi \, \mathbf {a}\otimes {\text {grad}}\mathbf {b}\otimes {\mathbf {c}}+\varphi \, \mathbf {a}\otimes \mathbf {b}\otimes {\text {grad}}{\mathbf {c}}\,. \end{array} \end{aligned}$$

(172)

Let us now apply the identity (171) as follows

$$\begin{aligned} \left[ {\text {grad}}\left( \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\right) \right] _{jkq} = -\left[ \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}\right] _{jkq} + \frac{({\varvec{\rho }}_i)_{j/q}({\varvec{\rho }}_i)_k}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}+ \frac{({\varvec{\rho }}_i)_{j}({\varvec{\rho }}_i)_{k/q}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} \end{aligned}$$

(173)

since

$$\begin{aligned} {\text {grad}}\left[ \frac{1}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\right] = -\frac{{\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}\,. \end{aligned}$$

(174)

Thus, being $({\varvec{\rho }}_i)_{j/q}=\delta _{jq}$ we infer from (173)

$$\begin{aligned} {\text {grad}}\left( \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\right) = -\frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}} + \frac{\mathbf {I}_{2{\rm D}} \otimes _{23}{\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}+ \frac{{\varvec{\rho }}_i\otimes \mathbf {I}_{2{\rm D}}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} \end{aligned}$$

(175)

where $\mathbf {I}_{2{\rm D}}$ is the 2D identity tensor and $\otimes _{23}$ denotes the tensor product obtained by interchanging the second and third index of the rank-three tensor $\mathbf {I}_{2{\rm D}}\otimes {\varvec{\rho }}_i$.

The integral over $F_i$ of the first addend in the formula above can be transformed into a boundary integral by exploiting the differential identity (Bowen and Wang 2006)

$$\begin{aligned} \int _\varOmega {\text {grad}}\,\mathbf {S}{\rm d}V = \int _{\partial \varOmega } \mathbf {S}\otimes \mathbf {n}{\rm d}A \end{aligned}$$

(176)

where $\mathbf {S}$ is a continuous tensor field.

Thus, integrating over $F_i$ the previous relation and recalling the definition (64) one has

$$\begin{aligned} \int \limits _{F_i} \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i }{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\rm d}A_i& = -\int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i + \mathbf {I}_{2{\rm D}}\otimes _{23}{\varvec{\psi }}_{F_i}+ {\varvec{\psi }}_{F_i}\otimes \mathbf {I}_{2{\rm D}} \end{aligned}$$

(177)

where ${\varvec{\nu }}$ is the unit normal pointing outwards the boundary $\partial F_i$ of the i-th face $F_i$ of the polyhedron.

Hence, the first integral on the right-hand side of (177) becomes

$$\begin{aligned} \int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i =\sum _{j=1}^{N_{E_i}}\int \limits _{0}^{l_j} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i){\rm d}s_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\otimes {\varvec{\nu }}_j \end{aligned}$$

(178)

since ${\varvec{\nu }}$ is constant on each of the $N_{E_i}$ edges belonging to $\partial F_i$.

Recalling (68) and (73), formula (178) becomes

$$\begin{aligned} \int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i =\sum _{j=1}^{N_{E_i}}\int \limits _{0}^{1} \frac{\hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}}\otimes \Delta {\varvec{\rho }}_{j}^\perp \end{aligned}$$

(179)

and the integral on the right-hand side can be further transformed by defining

$$\begin{aligned} \mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}={\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\quad \mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}={\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j+\Delta {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\quad \mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}=\Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j\,. \end{aligned}$$

(180)

Actually, recalling the parametrization (67) one has

$$\begin{aligned}&\hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)=\mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ \lambda _j \mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ \lambda _j^2\mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\,, \end{aligned}$$

(181)

$$\begin{aligned}& \int \limits _{0}^{1} \frac{\hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}} = I_{0j}\,\mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,\mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{2j}\,\mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j} \end{aligned}$$

(182)

where the explicit expression of the integrals

$$\begin{aligned} \begin{array}{rr} I_{0j}= \int \limits _{0}^{1} \frac{d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}} &{} \qquad I_{1j}=\int \limits _{0}^{1} \frac{\lambda _j d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}} \\ \\ I_{2j}=\int \limits _{0}^{1} \frac{\lambda _j^2 d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}} &{} \\ \end{array} \end{aligned}$$

(183)

is provided in Appendix 2.

In conclusion, it turns out to be

$$\begin{aligned} \int \limits _{\partial {F}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i= \sum _{j=1}^{N_{E_i}} \left[ I_{0j}\,\mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,\mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{2j}\,\mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right] \otimes \Delta {\varvec{\rho }}_{j}^\perp \,, \end{aligned}$$

(184)

so that the integral of interest can be computed as follows on account of (177)

$$\begin{aligned} {\mathfrak {C}}_{F_i}=\int \limits _{F_i} \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i }{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\rm d}A_i &= -\sum _{j=1}^{N_{E_i}} \left[ I_{0j}\,\mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,\mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{2j}\,\mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right] \otimes \Delta {\varvec{\rho }}_{j}^\perp \\ &\quad+ \mathbf {I}_{2{\rm D}}\otimes _{23}{\varvec{\psi }}_{F_i} + {\varvec{\psi }}_{F_i}\otimes \mathbf {I}_{2{\rm D}} \end{aligned}$$

(185)

where the expression of ${\varvec{\psi }}_{F_i}$ as an explicit function of the position vectors defining the boundary of $F_i$ is provided at the end of this Appendix.

Also of interest is the composition of the third-order tensor above with the vector ${\varvec{\kappa }}_i$ since it appears in the expressions (47), (50) and (49). For this end, let us first notice that

$$\begin{aligned} \left[ \left( \mathbf {I}_{2{\rm D}}\otimes _{23}{\varvec{\psi }}_{F_i}\right) \;{\varvec{\kappa }}_i\right] _{jk} &=\left( \mathbf {I}_{2{\rm D}}\otimes _{23}{\varvec{\psi }}_{F_i}\right) _{jkp} \left( {\varvec{\kappa }}_i\right) _p = I_{jp}\left( {\varvec{\psi }}_{F_i}\right) _k \left( {\varvec{\kappa }}_i\right) _p\\ \\&=\delta _{jp} \left( {\varvec{\kappa }}_i\right) _p \left( {\varvec{\psi }}_{F_i}\right) _k = \left( {\varvec{\kappa }}_i\right) _j \left( {\varvec{\psi }}_{F_i}\right) _k =\left( {\varvec{\kappa }}_i\otimes {\varvec{\psi }}_{F_i}\right) _{jk}. \end{aligned}$$

(186)

Hence

$$\begin{aligned} {\mathfrak {C}}_{F_i}{\varvec{\kappa }}_i =\int \limits _{F_i} \frac{({\varvec{\rho }}_i\cdot {\varvec{\kappa }}_i)({\varvec{\rho }}_i\otimes {\varvec{\rho }}_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\rm d}A_i &= -\sum _{j=1}^{N_{E_i}} \left( {\varvec{\kappa }}_i\cdot \Delta {\varvec{\rho }}_{j}^\perp \right) \left( I_{0j}\,\mathbf {E}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,\mathbf {E}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{2j}\,\mathbf {E}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right) \\&\quad + {\varvec{\kappa }}_i\otimes {\varvec{\psi }}_{F_i} + {\varvec{\psi }}_{F_i}\otimes {\varvec{\kappa }}_i \end{aligned}$$

(187)

so that the right-hand side fulfills the symmetry of the tensor on the left-hand side of the previous expression.

To evaluate analytically the second integral in (168), we exploit the identity (172) to get

$$\begin{aligned} \left[ {\text {grad}}\left( \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\right) \right] _{jkpq} &= -\left[ \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}\right] _{jkpq} + \frac{\delta _{jq}({\varvec{\rho }}_i \otimes {\varvec{\rho }}_i)_{kp}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} \\ &\quad+\frac{\delta _{kq}({\varvec{\rho }}_i \otimes {\varvec{\rho }}_i)_{jp}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} +\frac{\delta _{pq}({\varvec{\rho }}_i \otimes {\varvec{\rho }}_i)_{jk}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\,, \end{aligned}$$

(188)

or equivalently

$$\begin{aligned} {\text {grad}}\left( \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}\right) = &-\frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}} + \frac{\mathbf {I}_{2{\rm D}} \otimes _{24}({\varvec{\rho }}_i\otimes {\varvec{\rho }}_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} \\& +\frac{({\varvec{\rho }}_i\otimes {\varvec{\rho }}_i) \otimes _{23}\mathbf {I}_{2{\rm D}}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} +\frac{({\varvec{\rho }}_i\otimes {\varvec{\rho }}_i) \otimes \mathbf {I}_{2{\rm D}}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} \end{aligned}$$

(189)

where $\otimes _{24}$ denotes the tensor product obtained by interchanging the second and fourth index of the rank-four tensor $\mathbf {I}_{2{\rm D}}\otimes ({\varvec{\rho }}_i\otimes {\varvec{\rho }}_i)$.

Integrating the previous relation over $F_i$ and applying Gauss’ theorem yields

$$\begin{aligned} {\mathfrak {D}}_{F_i}=\int \limits _{F_i} \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i \otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\rm d}A_i &= -\int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i \\& + \mathbf {I}_{2{\rm D}}\otimes _{24}{\varvec{\varPsi }}_{F_i} + {\varvec{\varPsi }}_{F_i}\otimes _{23}\mathbf {I}_{2{\rm D}} +{\varvec{\varPsi }}_{F_i}\otimes \mathbf {I}_{2{\rm D}} \end{aligned}$$

(190)

where ${\varvec{\varPsi }}_{F_i}$ is analytically evaluated in formula (208) of Appendix 2.

In view of the ensuing developments, we further set

$$\begin{aligned}&{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,{\varvec{\rho }}_j}={\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\qquad {\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}={\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j+{\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j+ \Delta {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j \end{aligned}$$

(191)

$$\begin{aligned}&{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}={\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j+\Delta {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j+ \Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j\otimes {\varvec{\rho }}_j \end{aligned}$$

(192)

$$\begin{aligned}&{\mathbf{\mathbb{E}}}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}=\Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_j \end{aligned}$$

(193)

yielding

$$\begin{aligned} \hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)={\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ \lambda _j{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ \lambda _j^2 {\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ \lambda _j^3{\mathbf{\mathbb{E}}}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\,. \end{aligned}$$

(194)

Accordingly, the integral on the right-hand side in (190) becomes

$$\begin{aligned} \int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i &= \sum _{j=1}^{N_{E_i}}\int \limits _{0}^{1}\left\{ \frac{\hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)\otimes \hat{{\varvec{\rho }}}_i(\lambda _j)d\lambda _j}{\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2}}\otimes \Delta {\varvec{\rho }}_{j}^\perp \right\} \\ & \begin{aligned}=&-\sum _{j=1}^{N_{E_i}}\left[ I_{0j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j} \right. \\&\left. +\,I_{2j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{3j}\,{\mathbf{\mathbb{E}}}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right] \otimes \Delta {\varvec{\rho }}_{j}^\perp \end{aligned}\end{aligned}$$

(195)

where the integrals $I_{0j}$, $I_{1j}$, $I_{2j}$ and $I_{3j}$ are explicitly evaluated in Appendix 2.

In conclusion one has

$$\begin{aligned} \int \limits _{\partial {F_i}} \frac{{\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i)}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{\rm d}s_i =& \sum _{j=1}^{N_{E_i}} \left[ I_{0j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j} \right. \\&\left. +\,I_{2j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+ I_{3j}\,{\mathbf{\mathbb{E}}}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right] \otimes \Delta {\varvec{\rho }}_{j}^\perp \\& + \mathbf {I}_{2{\rm D}}\otimes _{24}{\varvec{\varPsi }}_{F_i} + {\varvec{\varPsi }}_{F_i}\otimes _{23}\mathbf {I}_{2{\rm D}} +{\varvec{\varPsi }}_{F_i}\otimes \mathbf {I}_{2{\rm D}}\,. \end{aligned}$$

(196)

The composition of the previous integral with ${\varvec{\kappa }}_i$, a quantity that is needed in (175) and (to be displayed), yields a third-order tensor. The contribution to the jkp component of this tensor provided by the tensor product ${\varvec{\varPsi }}_{F_i}\otimes _{23}\mathbf {I}_{2{\rm D}}$ is given by

$$\begin{aligned} \left[ \left( {\varvec{\varPsi }}_{F_i}\otimes _{23}\mathbf {I}_{2{\rm D}}\right) \;{\varvec{\kappa }}_i\right] _{jkp} &=\left( {\varvec{\varPsi }}_{F_i}\otimes _{23}\mathbf {I}_{2{\rm D}}\right) _{jkpq} \left( {\varvec{\kappa }}_i\right) _q = \left( {\varvec{\varPsi }}_{F_i}\right) _{jp} \left( \delta _{kq}\right) \left( {\varvec{\kappa }}_i\right) _q\\ \\ & =\left( {\varvec{\varPsi }}_{F_i}\right) _{jp}\left( {\varvec{\kappa }}_i\right) _k =\left( {\varvec{\varPsi }}_{F_i}\otimes _{23}{\varvec{\kappa }}_i\right) _{jkp}. \end{aligned}$$

(197)

Analogously

$$\begin{aligned} \left[ \left( \mathbf {I}_{2{\rm D}}\otimes _{24}{\varvec{\varPsi }}_{F_i}\right) \;{\varvec{\kappa }}_i\right] _{jkp} &=\left( \mathbf {I}_{2{\rm D}}\otimes _{24}{\varvec{\varPsi }}_{F_i}\right) _{jkpq} \left( {\varvec{\kappa }}_i\right) _q = \left( \delta _{jq}\right) \left( {\varvec{\varPsi }}_{F_i}\right) _{pk} \left( {\varvec{\kappa }}_i\right) _q\\ \\ &{} =\left( {\varvec{\kappa }}_i\right) _j\left( {\varvec{\varPsi }}_{F_i}\right) _{pk}=\left( {\varvec{\kappa }}_i\right) _j\left( {\varvec{\varPsi }}_{F_i}\right) _{kp} =\left( {\varvec{\kappa }}_i\otimes {\varvec{\varPsi }}_{F_i} \right) _{jkp} \end{aligned}$$

(198)

where the identity $\left( {\varvec{\varPsi }}_{F_i}\right) _{pk}=\left( {\varvec{\varPsi }}_{F_i}\right) _{kp}$ stems from the symmetry of ${\varvec{\varPsi }}_{F_i}$. Accordingly, we infer from (190) and (196)

$$\begin{aligned} {\mathfrak {D}}_{F_i}{\varvec{\kappa }}_i= \int \limits _{F_i} \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i\otimes {\varvec{\rho }}_i{\rm d}A_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}}{\varvec{\kappa }}_i & = -\sum _{j=1}^{N_{E_i}} \left( {\varvec{\kappa }}_i\cdot \Delta {\varvec{\rho }}_{j}^\perp \right) \left( I_{0j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,{\varvec{\rho }}_j}+ I_{1j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j} \right. \\&\left. \quad +\,I_{2j}\,{\mathbf{\mathbb{E}}}_{{\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}+I_{3j}\,{\mathbf{\mathbb{E}}}_{\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j\,\Delta {\varvec{\rho }}_j}\right) \\&\quad +{\varvec{\varPsi }}_{F_i}\otimes {\varvec{\kappa }}_i + {\varvec{\varPsi }}_{F_i}\otimes _{23}{\varvec{\kappa }}_i +{\varvec{\kappa }}_i\otimes {\varvec{\varPsi }}_{F_i}\,. \end{aligned}$$

(199)

The expression (185) for ${\mathfrak {C}}_{F_i}$ and (190) for ${\mathfrak {D}}_{F_i}$ requires the computation of the integral ${\varvec{\varPsi }}_{F_i}$ defined in formula (169); it is evaluated analytically by invoking the differential identity

$$\begin{aligned} {\text {grad}}\left[ \varphi \mathbf {a}\right] =\mathbf {a}\otimes {\text {grad}}\varphi + \varphi \,{\text {grad}}\mathbf {a} \end{aligned}$$

(200)

holding for differentiable scalar $(\varphi )$ and vector $(\mathbf {a})$ fields. Actually, applying the previous identity as follows

$$\begin{aligned} {\text {grad}}\left[ ({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}{\varvec{\rho }}_i\right] =\frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}} + ({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}\mathbf {I}_{2{\rm D}}\,, \end{aligned}$$

(201)

integrating over $F_i$ and setting

$$\begin{aligned} \iota _{F_i} = \int \limits _{F_i}\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}{\rm d}A_i \end{aligned}$$

(202)

one has

$$\begin{aligned} {\varvec{\varPsi }}_{F_i}= \int \limits _{\partial F_i}\left[ {\varvec{\rho }}_i(s_i)\cdot {\varvec{\rho }}_i(s_i)+d_i^2\right] ^{1/2}{\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}_i(s_i){\rm d}s_i- \iota _{F_i} \mathbf {I}_{2{\rm D}}. \end{aligned}$$

(203)

To compute the domain integral (202), we apply the differential identity

$$\begin{aligned} {\text {div}}\left[ \varphi \mathbf {a}\right] ={\text {grad}}\varphi \cdot \mathbf {a}+ \varphi \,{\text {div}}\mathbf {a} \end{aligned}$$

(204)

to the vector field $\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}{\varvec{\rho }}_i$ to get

$$\begin{aligned} {\text {div}}\left[ \left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}{\varvec{\rho }}_i\right] =\frac{{\varvec{\rho }}_i\cdot {\varvec{\rho }}_i}{\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}} + 2\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}\,. \end{aligned}$$

(205)

Adding and subtracting $d_i^2$ to the numerator yields

$$\begin{aligned} {\text {div}}\left[ \left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}{\varvec{\rho }}_i\right] = 3\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2} - \frac{d_i^2}{\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2}} \,, \end{aligned}$$

(206)

so that, upon integrating over $F_i$ and applying Gauss’ theorem, one has

$$\begin{aligned} \iota _{F_i} = \frac{1}{3}\int \limits _{\partial F_i}\left[ {\varvec{\rho }}_i(s_i)\cdot {\varvec{\rho }}_i(s_i)+d_i^2\right] ^{1/2}{\varvec{\rho }}_i(s_i)\cdot {\varvec{\nu }}(s_i){\rm d}s_i-\frac{d_i^2}{3}\psi _{F_i}\,, \end{aligned}$$

(207)

by recalling definition (62). In conclusion, we infer from (203) and the previous expression

$$\begin{aligned} {\varvec{\varPsi }}_{F_i} &= \int \limits _{\partial F_i}\left[ {\varvec{\rho }}_i(s_i)\cdot {\varvec{\rho }}_i(s_i)+d_i^2\right] ^{1/2} {\varvec{\rho }}_i(s_i)\otimes {\varvec{\nu }}(s_i){\rm d}s_i \\&\quad -\frac{\mathbf {I}_{2{\rm D}}}{3}\left\{ \int \limits _{\partial F_i}\left[ {\varvec{\rho }}_i(s_i)\cdot {\varvec{\rho }}_i(s_i)+d_i^2\right] ^{1/2} {\varvec{\rho }}_i(s_i)\cdot {\varvec{\nu }}(s_i){\rm d}s_i - d_i^2\psi _{F_i} \right\} \\ &= \ \sum _{j=1}^{N_{E_i}}\left\{ \left[ \int \limits _0^{l_j}\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2} {\varvec{\rho }}_i{\rm d}s_j\right] \otimes {\varvec{\nu }}_j\right. \\& \left. \quad - \frac{\mathbf {I}_{2{\rm D}}}{3} \left[ \left( {\varvec{\rho }}_j\cdot {\varvec{\nu }}_j\right) \int \limits _0^{l_j}\left( {\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2\right) ^{1/2} {\rm d}s_j\right] \right\} +\frac{ d_i^2}{3} \psi _{F_i} \\ &= \ \sum _{j=1}^{N_{E_i}}\left\{ \left[ \int \limits _0^1\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2} \left( {\varvec{\rho }}_j+ \lambda _j\Delta {\varvec{\rho }}_j\right) d\lambda _j\right] \otimes \Delta {\varvec{\rho }}_{j}^\perp \right. \\&\qquad \left. - \frac{\mathbf {I}_{2{\rm D}}}{3} \left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right) \int \limits _0^1\left[ \hat{{\varvec{\rho }}}_i(\lambda _j)\cdot \hat{{\varvec{\rho }}}_i(\lambda _j)+d_i^2\right] ^{1/2} d\lambda _j\right\} +\frac{ d_i^2}{3} \left( \psi _i - |d_i|\alpha _i\right) \\ &= \ \sum _{j=1}^{N_{E_i}}\left[ \left( I_{4j}{\varvec{\rho }}_j+ I_{5j}\Delta {\varvec{\rho }}_j\right) \otimes \Delta {\varvec{\rho }}_{j}^\perp - \frac{\mathbf {I}_{2{\rm D}}}{3} ({\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp )I_{4j}\right] +\frac{ d_i^2}{3} \left( \psi _i - |d_i|\alpha _i\right) \end{aligned}$$

(208)

where $\psi _i$ is defined in (219).

We have numerically verified that the sum over the $N_{E_i}$ edges of the first addend on the right-hand side returns a symmetric rank-two tensor as the one on the left-hand side.

Appendix 2: Available Expressions of Integrals

We hereby collect some known formulas in order to allow the reader to implement the expression of the gravity anomaly contributed in the main body of the paper.

We first report the algebraic expression of some definite integrals that will be repeatedly referred to in the sequel; they have been computed elsewhere D’Urso (2013a, 2014a, b) though with a different denomination. Making reference to the quantities $p_j$, $q_j$, $u_j$, $v_j$ introduced in formula (71), we set

$$\begin{aligned}&{\rm ATN}1_j = { \arctan \frac{|d_i|(p_j+q_j)}{\sqrt{p_j u_j-q_j^2} \sqrt{p_j +2 q_j +v_j}}}\,, \end{aligned}$$

(209)

$$\begin{aligned}&{\rm ATN}2_j = { \arctan \frac{|d_i|q_j}{\sqrt{p_j u_j-q_j^2} \sqrt{v_j}} } \end{aligned}$$

(210)

where the suffix $(\cdot )_j$ has been added to remind that they all refer to the j-th edge of the generic face $F_i$.

Also of interest are the following integrals

$$\begin{aligned}&I_{0j}= \int \limits _0^1\frac{d\lambda _j}{\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}} = \ln k_j = \ln \frac{p_j+q_j + \sqrt{p_j}\sqrt{p_j +2 q_j +v_j}}{q_j + \sqrt{p_j v_j}}={\rm LN}_j\,, \end{aligned}$$

(211)

$$\begin{aligned}& I_{1j}= \int \limits _0^1\frac{\lambda _j d\lambda _j}{\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}} = \frac{1}{p_j}\left\{ \sqrt{p_j+2q_j +v_j} - \sqrt{v_j}-\frac{q_j}{\sqrt{p_j}}I_{0j}\right\}, \end{aligned}$$

(212)

$$\begin{aligned} I_{2j}= \int \limits _0^1\frac{\lambda _j^2 d\lambda _j}{\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}} &= \frac{1}{2p_j^2}\left[ (p_j-3q_j) \sqrt{p_j+2q_j +v_j} + 3q_j \sqrt{v_j}\right] \\&\quad +\frac{3q_j^2 -p_j v_j}{2p_j^{5/2}} I_{0j}, \end{aligned}$$

(213)

$$\begin{aligned} I_{3j}= \int \limits _0^1\frac{\lambda _j^3 d\lambda _j}{\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}} &= \frac{1}{6p_j^3}\left[ (2p_j^2-5p_jq_j-4p_jv_j +15q_j^2) \sqrt{p_j+2q_j +v_j} \right. \\&\left. \quad +(4p_jv_j -15q_j^2) \sqrt{v_j}\right] +\frac{3p_jq_jv_j -5q_j^3}{2p_j^{7/2}} I_{0j}\,, \end{aligned}$$

(214)

$$\begin{aligned}& I_{4j}= \int \limits _0^1\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}d\lambda _j = \frac{(p_j+q_j)\sqrt{p_j+2q_j +v_j} - q_j\sqrt{v_j}}{2p_j}+\frac{p_jv_j -q_j^2}{2p_j^{3/2}} I_{0j}\,, \end{aligned}$$

(215)

$$\begin{aligned} I_{5j}= \int \limits _0^1\lambda _j\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}d\lambda _j &= \frac{1}{6p_j^2}\left[ (2p_j^2+p_jq_j+2p_jv_j -3q_j^2) \sqrt{p_j+2q_j +v_j} \right. \\&\quad \left. -(2p_jv_j -3q_j^2) \sqrt{v_j}\right] +\frac{q_j^3 -p_jq_jv_j}{2p_j^{5/2}} I_{0j}\,, \end{aligned}$$

(216)

$$\begin{aligned}& I_{6j}= \int \limits _0^1\frac{\left[ p_j \lambda ^2 + 2q_j \lambda _j + v_j\right] ^{1/2}}{p_j \lambda ^2 + 2q_j \lambda _j + u_j}d\lambda _j = \frac{|d_i|}{\sqrt{p_j u_j -q_j^2}}\left[ {\rm ATN}1_j -{\rm ATN}2_j\right] + \frac{1}{\sqrt{p_j}} {\rm LN}_j\,. \end{aligned}$$

(217)

Let us now consider the evaluation of 2D integrals having either $({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}$ or $({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{3/2}$ in the denominator. The first domain integral to consider is

$$\begin{aligned} \psi _{F_i}= \int \limits _{F_i}\frac{{\rm d}A_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} = \psi _i-|d_i|\alpha _i \end{aligned}$$

(218)

where

$$\begin{aligned} \psi _i&= \sum\limits _{j=1}^{N_{E_i}}\left( {\varvec{\rho }}_j\cdot {\varvec{\nu }}_j \right) \int \limits _0^{l_j}\frac{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}}{{\varvec{\rho }}_i\cdot {\varvec{\rho }}_i} {\rm d}s_j = \sum\limits _{j=1}^{N_{E_i}}\left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right) \int \limits _0^1\frac{(p_j\lambda _j^2 +2q_j\lambda _j +v_j)^{1/2}}{p_j\lambda _j^2 +2q_j\lambda _j +u_j} d\lambda _j\\ \\ &= { \sum\limits _{j=1}^{N_{E_i}} } \left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right) \left\{ { \frac{|d_i|}{\sqrt{p_j u_j-q_j^2}} \left[ {\rm ATN}1_j-{\rm ATN}2_j \right] + \frac{1}{\sqrt{p_j}} {\rm LN}_j } \right\} = \sum\limits _{j=1}^{N_{E_i}} \psi _j^i\, \left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right). \end{aligned}$$

(219)

The derivation of the previous expression can be found, e.g., in formula (19) of D’Urso (2013a) and (23) of D’Urso (2014a).

The scalar $\alpha _i$ in (218) is the two-dimensional counterpart of the quantity $\alpha _V$ in (26) and accounts for the singularity of $\psi _{F_i}$ when $d_i=0$ and ${\varvec{\rho }}={\varvec{ o }}$ where ${\varvec{ o }}=(0,\,0)$. Thus $\alpha _i$ represents the angular measure, expressed in radians, of the intersection between $F_i$ and a circular neighborhood of the singularity point ${\varvec{\rho }}={\varvec{ o }}$, see D’Urso (2013a, 2014a, b) for additional details. Although its computation is not required in the ensuing developments, we specify for completeness that $\alpha _i$ can be computed by means of the general algorithm detailed in D’Urso and Russo (2002).

Analogously, formulas (19), (77) and (79) of D’Urso (2014b) yield

$$\begin{aligned} {\varvec{\psi }}_{F_i}&= \int \limits _{F_i}\frac{{\varvec{\rho }}_i{\rm d}A_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i + d_i^2)^{1/2}} =\sum _{j=1}^{N_{E_i}}{\varvec{\nu }}_j\int \limits _0^{l_j}({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d_i^2)^{1/2} {\rm d}s_i\\ \\ &= \sum _{j=1}^{N_{E_i}}l_j{\varvec{\nu }}_j \int \limits _0^1 \left[ p_j\lambda _j^2 +2q_j\lambda _j +v_j\right] ^{1/2}d\lambda _j=\sum _{j=1}^{N_{E_i}}I_{4j}\,\Delta {\varvec{\rho }}_{j}^\perp \\ \end{aligned}$$

(220)

while formulas (37) and (81) of D’Urso (2014b)

$$\begin{aligned} \varphi _{F_i} &={ \int \limits _{F_i}\frac{{\rm d}A_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{3/2}} = \frac{\alpha _i}{|d_i|} - \sum _{j=1}^{N_{E_i}}\left[ \left( {\varvec{\rho }}_j\cdot {\varvec{\nu }}_j\right) \int \limits _{0}^{l_j}\frac{{\rm d}s_{j}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i)({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}} \right] }\\ &=\frac{\alpha _i}{|d_i|} - \sum _{j=1}^{N_{E_i}}\left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right) \int \limits _0^1\frac{\lambda _j}{(p_j\lambda _j^2 +2q_j\lambda _j +u_j)(p_j\lambda _j^2 +2q_j\lambda _j +v_j)^{1/2}}\\&=\frac{\alpha _i}{|d_i|} -\sum _{j=1}^{N_{E_i}} \left[ \frac{{\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp }{|d_i|\sqrt{p_j u_j - q_j^2}}({\rm ATN}1_j-{\rm ATN}2_j)\right] =\frac{\alpha _i}{|d_i|} -\sum _{j=1}^{N_{E_i}} \varphi _j \,\left( {\varvec{\rho }}_j\cdot {\varvec{\rho }}_{j+1}^\perp \right). \end{aligned}$$

(221)

Furthermore, on account of formulas (38) and (82) of D’Urso (2014b) it turns out to be

$$\begin{aligned} {\varvec{\varphi }}_{F_i}&= { \int \limits _{F_i}\frac{{\varvec{\rho }}_i {\rm d}A_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{3/2}} = } {\, -\sum _{j=1}^{N_{E_i}} \left( {\varvec{\nu }}_j\int \limits _{0}^{l_j} \frac{ {\rm d}s_{j}}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}} \right) }\\ &= \ - \sum _{j=1}^{N_{E_i}}\Delta {\varvec{\rho }}_{j}^\perp \int \limits _0^1\frac{d\lambda _j}{(p_j\lambda _j^2 +2q_j\lambda _j +v_j)^{1/2}} = -\sum _{j=1}^{N_{E_i}} I_{0j}\;\Delta {\varvec{\rho }}_{j}^\perp \end{aligned}$$

(222)

while one infers from formulas (40) and (83) of D’Urso (2014b)

$$\begin{aligned} {\mathbf \Phi }_{F_i} &= { \int \limits _{F_i} \frac{{\varvec{\rho }}_i\otimes {\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{3/2}} {\rm d}A_i }\\ &= -\sum _{j=1}^{N_{E_i}}\int \limits _0^{l_j} \frac{{\varvec{\rho }}_i}{({\varvec{\rho }}_i\cdot {\varvec{\rho }}_i+d^2_i)^{1/2}} {\rm d}s_i \otimes {\varvec{\nu }}_j\, + \psi _{F_i} \mathbf {I}_{2{\rm D}}\\ &= \ -\sum _{j=1}^{N_{E_i}}\int \limits _0^1\frac{{\varvec{\rho }}_j+\lambda _j\Delta {\varvec{\rho }}_j}{(p_j\lambda _j^2 +2q_j\lambda _j +v_j)^{1/2}}d\lambda _j \otimes \Delta {\varvec{\rho }}_{j}^\perp + \psi _{F_i}\mathbf {I}_{2{\rm D}} \\ &= \ -\sum _{j=1}^{N_{E_i}}\left[ {\rm LN}_j\;{\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_{j}^\perp +I_{1j}\;\Delta {\varvec{\rho }}_j\otimes \Delta {\varvec{\rho }}_{j}^\perp \right] + \psi _{F_i}\mathbf {I}_{2{\rm D}} \end{aligned}$$

(223)

where $\mathbf {I}_{2{\rm D}}$ is the rank-two two-dimensional identity tensor.