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A1 and A2 are the bounding surface areas on the negative z direction and positive z direction, respectively. Thus, n • n3 is negative on A1 (because on A1, n is pointing toward the negative z while n3 is pointing toward the positive z). n • n3 is positive on A2 (by parallel reasoning). Therefore, f d dV = f { S3 ( x, y, z2 )}[ n • n3 ] dAx +J { S3 (x,y,z1 )}[(n • n3)] dAx

Considering the x and y directions, respectively, and following similar steps: f -S dV =f S3[n • n1 ] dA (64)

Si and S2 are the scalar components of S on the x and y axes, respectively, and fil and n2 are the unit vectors on the x and y axes, respectively.

Adding Eqs. (63), (64), and (67) produces the Gauss-Green divergence theorem. That is, dS3 r dSi (• dS2

f ^ dV + f S dV + f --r2 dV = f S3[n • n3] dA + f S1[n • n1 ] dA JV oz JV ox JV oy ¿a J a

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