$$\begin{array}{ccc}\hfill detA& =& {u}_{x}(a,b){v}_{y}(a,b){u}_{y}(a,b){v}_{x}(a,b)\hfill \\ & =& {\left({u}_{x}(a,b)\right)}^{2}+{\left({v}_{x}(a,b)\right)}^{2}\hfill \\ & =& ({u}_{x}(a,b)+i{v}_{x}(a,b))({u}_{x}(a,b)i{v}_{x}(a,b))\hfill \\ & =& {f}^{\text{'}}\left(c\right)\overline{{f}^{\text{'}}\left(c\right)}\hfill \\ & =& {f}^{\text{'}}{\left(c\right)}^{2},\hfill \end{array}$$
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.
Let
$c$ be a point in
$C,$ and let
$S$ be the geometric set that is a closed disk
${\overline{B}}_{r}\left(c\right).$ Let
$\phi $ be the parameterization of the boundary
${C}_{r}$ of
$S$ given by
$\phi \left(t\right)=c+r{e}^{it}$ for
$t\in [0,2\pi ].$ For each integer
$n\in Z,$ define
${f}_{n}\left(z\right)={(zc)}^{n}.$
 Show that
${\int}_{{C}_{r}}{f}_{n}(\zeta \phantom{\rule{0.166667em}{0ex}}d\zeta =0$ for all
$n\ne 1.$
 Show that
$${\int}_{{C}_{r}}{f}_{1}\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta ={\int}_{{C}_{r}}\frac{1}{\zeta c}\phantom{\rule{0.166667em}{0ex}}d\zeta =2\pi i.$$
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
$S\cap \tilde{{\widehat{S}}^{0}},$ 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)\phantom{\rule{0.166667em}{0ex}}d\zeta $ and
${\int}_{{C}_{\widehat{S}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}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:

${S}_{1}$ is determined by the interval
$[a,\widehat{a}]$ and the two bounding functions
$u$ and
$l$ restricted to that interval.

${S}_{2}$ is determined by the interval
$[\widehat{a},\widehat{b}]$ and the two bounding functions
$u$ and
$\widehat{u}$ restricted to that interval.

${S}_{3}$ is determined by the interval
$[\widehat{a},\widehat{b}]$ and the two bounding functions
$\widehat{l}$ and
$l$ restricted to that interval.

${S}_{4}$ 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] ,
$$\begin{array}{ccc}\hfill {\int}_{{C}_{S}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta & =& {\int}_{{C}_{\widehat{S}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta +\sum _{k=1}^{4}{\int}_{{C}_{{S}_{k}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta \hfill \\ & =& {\int}_{{C}_{\widehat{S}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta ,\hfill \end{array}$$
as desired.
 Draw a picture of the five geometric sets in the proof above and
justify the claim that the sum of the four contour integrals around the geometric sets
${S}_{1},...,{S}_{4}$ is the integral around
${C}_{S}$ minus the integral around
${C}_{\widehat{S}}.$
 Let
${S}_{1},...,{S}_{n}$ be pairwise disjoint, piecewise smooth geometric sets, each
having a boundary of finite length, and each containedin a piecewise smooth geometric set
$S$ whose boundary also has finite length.
Prove that the
${S}_{k}$ 's are some of the elements of a partition
$\left\{{\tilde{S}}_{l}\right\}$ of
$S,$ each of which is piecewise smooth and has a boundary of finite length.
Show that, by reindexing,
${S}_{1},...,{S}_{n}$ can be chosen to be the first
$n$ elements of the partition
$\left\{{\widehat{S}}_{l}\right\}.$ HINT: Just carefully adjust the proof of
[link] .
 Suppose
$S$ is a piecewise smooth geometric set whose boundary has finite length,
and let
${S}_{1},...,{S}_{n}$ be a partition of
$S$ for which each
${S}_{k}$ is piecewise smooth
and has a boundary
${C}_{{S}_{k}}$ of finite length.
Suppose
$f$ is continuous on each of the boundaries
${C}_{{S}_{k}}$ of the
${S}_{k}$ 's as well as the boundary
${C}_{S}$ of
$S,$ and assume that
$f$ is continuous on each of the
${S}_{k}$ 's, for
$1\le k\le m,$ and differentiable at each point of their interiors.
Prove that
$${\int}_{{C}_{S}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta =\sum _{k=m+1}^{n}{\int}_{{C}_{{S}_{k}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta .$$
 Prove the following generalization of the Cauchy Theorem:
Let
${S}_{1},...,{S}_{n}$ be pairwise disjoint, piecewise smooth geometric sets whose boundaries have finite length, all contained
in the interior of a piecewise smooth geometric set
$S$ whose boundary also has finite length.
Suppose
$f$ is continuous at each point of
$S$ that is not in the interior of any of the
${S}_{k}$ 's,
and that
$f$ is differentiable at each point of
${S}^{0}$ that is not an element of any of the
${S}_{k}$ 's.
Prove that
$${\int}_{{C}_{S}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta =\sum _{k=1}^{n}{\int}_{{C}_{{S}_{k}}}f\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta .$$