Page 444 - Introduction to Continuum Mechanics

428 Integral Formulation of General Principles

where the subscript C denotes the line integral around the closed curve C in the counter-
clockwise direction. For the proof, let us assume for simplicity that the region R is such that
every straight line through an interior point and parallel to either axis cuts the boundary in
exactly two points. Figure 7.1 shows one such region. Let a and b be the least and the greatest
values ofy on C (points G and H in the figure). Let x - xi(y) and x = ^(y) be equations for
the boundaries HAG and GBH respectively. Then

Fig. 7.1

Now

Thus,