Page 445 - Introduction to Continuum Mechanics

Green's Theorem 429

Since

Thus

Let s be the arc length measured along the boundary curve C in the counterclockwise direction
and let x = x(s) and y = y(s) be the parametric equations for the boundary curve. Then,
dy/ds = +n x , Thus,

which is Eq. (7.1.1).
Equation (7.1.2) can be proven in a similar manner.

Example 7.1.1

2
For P(xy) = xy , evaluate / P(xy}n^s along the closed path OABC (Fig. 7.2). Also,
c
evaluate the area integral J(dP/dx)dA. Compare the results.
R
Solution. We have

On the other hand,

Thus,