Zde mus�me za��t parametrizac� k�ivky; nav�c u~integr�lu druh�ho druhu
 mus�me db�t na orientaci, tedy na smysl, ve kter�m je k�ivka prob�h�na.
 Definice k�ivkov�ho integr�lu 2. druhu p�es k�ivku {$ C, $}
 parametrizovanou vektorovou funkc� {$z=(x,y)= \varphi (t)
 \ (\text{tj. }x=x(t)=\varphi
 _{1}(t),y=y(t)=\varphi _{2}(t),)\   t\in (\alpha ,\beta ) $}, z~{\bf
 vektorov�} funkce {$ f(z)=(f_{1}(z),f_{2}(z)) $}:
 {$ \int_{C}^{}f(z)\dsvec:=\int_{\alpha }^{\beta }(f(\varphi (t)),\varphi
 '(t))\dt =                \int_{\alpha }^{\beta }(f_{1}(\varphi
 (t))x'(t)+                                      (f_{2}(\varphi
 (t))y'(t))\dt$}.                   \par
(Z�pis pou�it� v~zad�n� se �asto pou��v� a je ospravedln�n form�ln�
rovnost� {$ \dx=x'\dt, \dy=y'\dt. $})    \par
V~na�em p��pad� parametrizujeme elipsu s~u�it�m eliptick�ch sou�adnic 
z~p��kladu 2, v~nich� polo��me {$ r=1 $} a {$ \varphi \in\left( 0,2\pi
\right)  $} vezmeme za parametr; takto zvolen� sou�adnice  zaru�uj�
spr�vnou orientaci k�ivky:
$$
  \alignat3
             x&=a\cos \varphi ,\quad &\dx&=&-&a\sin \varphi \dpf\cr
             y&=b\sin \varphi ,\quad &\dy&=& & b\cos \varphi \dpf\cr
  \endalignat
$$
 (Vyj�d�en� diferenci�l� ve druh�m sloupci jest ov�em ch�pati form�ln�.)
 \par
 Dosazen�m t�chto v�raz�  dostaneme
 $$\multline
  \int_{C}^{}(x+y)\dx+(x-y)\dy=\cr=\int_{0}^{2\pi } \left( (a\cos \varphi
  +b\sin\varphi )(-a\sin\varphi )+(a\cos \varphi -b\sin \varphi
  )(b\cos\varphi )\right)\dpf=\cr
  =\int_{0}^{2\pi }\left( ab(\cos^{2}\varphi -\sin^{2}\varphi )-
  (a^{2}+b^{2})\sin\varphi\cos\varphi
  \right)\dpf=\cr=\int_{0}^{2\pi }\left( ab\cos 2\varphi
  -\frac{a^{2}+b^{2}}{2}\sin 2\varphi \right)\dpf = 0.
\endmultline
$$
