Tensor Calculus for Engineers and Physicists by Emil de Souza Sánchez Filho

Author:Emil de Souza Sánchez Filho
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(3.2.3)

Adding expressions (3.2.2) and (3.2.3) results in

To prove the validity of this theorem for the more general cases being the region , in which the integrals are calculated for each subregion (Fig. 3.7). In the segment ST the line integrals are calculated twice, but as they are of different direction they cancel each other when they are added, hence

Fig. 3.7Division of the simply connected region into two simply connected regions

Therefore, the expression of Green’s theorem is valid for the subdivision of region R (Fig. 3.7). This ascertaining is generalized for a finite region comprising N simple regions, with the outline curves , then

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