Page 340 - Intro to Tensor Calculus

334

Figure 2.6-4. Sphere S containing sphere Σ.

Applying the Gauss divergence theorem to the top half of figure 2.6-4 gives

ZZ ZZ ZZ ZZZ
i T
i
F n dσ + F n i dσ + F n i dσ = F dτ (2.6.30)
i b T
i Σ T
i
,i
S T S b1 Σ T V T
where the n i are the unit outward normals to the respective surfaces S T , S b1 and Σ T . Applying the Gauss
divergence theorem to the bottom half of the sphere in figure 2.6-4 gives
ZZ ZZ ZZ ZZZ
i B
i
i Σ B
i b B
F n dσ + F n i dσ + F n i dσ = F dτ (2.6.31)
,i
i
S B S b2 Σ B V B
Observe that the unit normals to the surfaces S b1 and S b2 are equal and opposite in sign so that adding the
equations (2.6.30) and (2.6.31) we obtain
ZZ ZZ ZZZ
i (1)
i
i
F n i dσ + F n i dσ = F dτ (2.6.32)
,i
S Σ V T +V B