7.1 Cauchy's theorem  (Page 2/3)

proving part (1).

The vectors ${\overrightarrow{V}}_1$ and ${\overrightarrow{V}}_2$ are the columns of the matrix $A,$ and so, from elementary linear algebra, we see that they are linearly independent if and only if the determinant of $A$ is nonzero. Hence, part (2) follows from part (1).Similarly, part (3) is a consequence of part (1).

It may come as no surprise that the contour integral of a function $f$ around the boundary of a geometric set $S$ is not necessarily 0 if the function $f$ is not differentiable at each point in the interior of $S.$ However, it is exactly these kinds of contour integrals that will occupy our attention in the rest of this chapter, and we shouldn't jump to any conclusions.

There is a remarkable result about contour integrals of certain functions that aren't differentiable everywhere within a geometric set, and it is what has been called theFundamental Theorem of Analysis, or Cauchy's Theorem. This theorem has many general statements, but we present one here that is quite broad and certainlyadequate for our purposes.

Cauchy's theorem, fundamental theorem of analysis

Let $S$ be a piecewise smooth geometric set whose boundary $C_S$ has finite length, and let $\widehat{S}\subseteq S^0$ be a piecewise smooth geometric set, whose boundary $C_{\widehat{S}}$ also is of finite length. Suppose $f$ is continuous on $S\cap \widetilde{{\widehat{S}}^0},$ i.e., at every point $z$ that is in $S$ but not in ${\widehat{S}}^0,$ and assume that $f$ is differentiable on $S^0\cap \tilde{\widehat{S}},$ i.e., at every point $z$ in $S^0$ but not in $\widehat{S}.$ (We think of these sets as being the points “between” the boundary curves of these geometric sets.) Then the two contour integrals ${\int }_{C_S}f\left(\zeta \right)d\zeta$ and ${\int }_{C_{\widehat{S}}}f\left(\zeta \right)d\zeta$ are equal.

Let the geometric set $S$ be determined by the interval $[a,b]$ and the two bounding functions $u$ and $l,$ and let the geometric set $\widehat{S}$ be determined by the subinterval $[\widehat{a},\widehat{b}]$ of $[a,b]$ and the two bounding functions $\widehat{u}$ and $\widehat{l}.$ Because $\widehat{S}\subseteq S^0,$ we know that $\widehat{u}\left(t\right)<u\left(t\right)$ and $l\left(t\right)<\widehat{l}\left(t\right)$ for all $t\in [\widehat{a},\widehat{b}].$ We define four geometric sets $S_1,...,S_4$ as follows:

1.   $S_1$ is determined by the interval $[a,\widehat{a}]$ and the two bounding functions $u$ and $l$ restricted to that interval.
2.   $S_2$ is determined by the interval $[\widehat{a},\widehat{b}]$ and the two bounding functions $u$ and $\widehat{u}$ restricted to that interval.
3.   $S_3$ is determined by the interval $[\widehat{a},\widehat{b}]$ and the two bounding functions $\widehat{l}$ and $l$ restricted to that interval.
4.   $S_4$ is determined by the interval $[\widehat{b},b]$ and the two bounding functions $u$ and $l$ restricted to that interval.

Observe that the five sets $\widehat{S},S_1,...,S_4$ constitute a partition of the geometric set $S.$ The corollary to [link] applies to each of the four geometric sets $S_1,...,S_4.$ Hence, the contour integral of $f$ around each of the four boundaries of these geometric sets is 0. So, by [link] ,

as desired.