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

Read more about this topic:  Divergence Theorem

Famous quotes containing the word examples:

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)