Divergence Theorem - Examples

Examples

To verify the planar variant of the divergence theorem for a region R, where

and R is the region bounded by the circle

The boundary of R is the unit circle, C, that can be represented parametrically by:

such that where s units is the length arc from the point s = 0 to the point P on C. Then a vector equation of C is

At a point P on C:

Therefore,

\begin{align}\oint_C \mathbf{F} \cdot \mathbf{N}\, ds &= \,\int_{0}^{2 \pi} ( 2 \sin s \mathbf{i} + 5 \cos s \mathbf{j}) \cdot (\cos s \mathbf{i} + \sin s \mathbf{j})\, ds\\ &= \,\int_{0}^{2 \pi} (2 \sin s \cos s + 5 \sin s \cos s)\, ds\\ &= \,7\int_{0}^{2 \pi} \sin s \cos s\, ds\\ &= \,0.\end{align}\

Because, and because, . Thus

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