1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

y Interior

6.3 • THE CAUCHY-GOURSAT THEOREM 215

y

Exterior

-1-----1--4-+.x

Figure 6.15 The interior and exterior of simple closed contours.

y y

(a) A simply connected domain. (b) A simply c onnected domain.

y y

... .... ~.

I ,._,\.

(c) A multiply ronnected domain. (d) A multiply connected domain.

Figure 6.16 Simply connected and multiply connected domains.

p ositively (counterclockwise); otherwise, C is oriented negatively (clockwise).
If C is positively oriented, then - C is negatively oriented. Figure 6.17 illustrates
the concept of positive and negative orienta tion.
Green's theorem is an important result from the calculus of real variables.
It tells you how to evaluate the line integral of real-valued functions.

y

~

CJ::::J"V

-1-------•x

(a) A positively oriented contour.

y

~

e:o~

-1---------.x

(b) A negatively oriented contour.

Figure 6.17 Simple closed contours that are positively and negatively oriented.