    <p>
                    The notions of curves in the complex plane that are smooth, piecewise smooth, simple, closed, and
                    simple
                    closed are easily formulated in terms of the vector function (\ref{parcurve}).
                    Suppose the derivative of (\ref{parcurve}) is
                    $z'(t) = x'(t) + iy'(t).$ We say a curve $C$ in the complex plane is <em>smooth</em> if $z'(t)$
                    is continuous and never zero in the
                    in