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\title[Matematika 2]
{Matematika 2}

\subtitle{4. vaja, parcialni odvodi}

\author[Borut] % 
{B.~Jur\v{c}i\v{c}~Zlobec\inst{1}}

\institute[Univerza v Ljubljani Slovenia] % 
{
  \inst{1}%
    Univerza v Ljubljani,\\Fakulteta za Elektrotehniko\\ 
    1000 Ljubljana, Tr\v{z}a\v{s}ka 25, Slovenija
}


\date[\today] % 
{Matematika FE, Ljubljana, Slovenija \today}

\subject{Matematika, parcialni odvodi}

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\section{Funkcije ve"c spremenljivk}
\subsection{Definicijsko obmo\v{c}je}
\begin{frame}{Dolo"ci definicijsko obmo"cje funkcije.}
\textcolor{blue}{$$
	f(x)=\sqrt{1-|x|-|y|}
$$}
\begin{itemize}
	\item 
\end{itemize}
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\subsection{Parcialni odvodi}
\begin{frame}{Osnovna pravila}
\begin{itemize}
\item {\bf Totalni diferencial:}
\begin{eqnarray*}
z = f(x,y)\quad\text{je}\quad dz & = & \frac{\partial z}{\partial x} dx + 
\frac{\partial z}{\partial y} dy \\
u = f(x,y,z)\quad\text{je}\quad du & = & \frac{\partial u}{\partial x} dx + 
\frac{\partial u}{\partial y} dy + \frac{\partial u}{\partial z} dz
\end{eqnarray*}
\item {\bf Sestavljene funkcije:} $z = f(x,y)$, $x = \varphi (u,v)$, $y = \psi (u,v)$.
\begin{eqnarray*}
\frac{\partial z}{\partial u} & = & \frac{\partial z}{\partial x} \: 
\frac{\partial \varphi}{\partial u} + \frac{\partial z}{\partial y} \: 
\frac{\partial \psi}{\partial u} \\
\frac{\partial z}{\partial v} & = & \frac{\partial z}{\partial x} \: 
\frac{\partial \varphi}{\partial v} + \frac{\partial z}{\partial y} \: 
\frac{\partial \psi}{\partial v} \\
\end{eqnarray*}
\item {\bf Implicitna funkcija:} $F(x,y,z) = 0$.
$$ \frac{\partial z}{\partial x} = -\frac{\frac{\partial F}{\partial x}}
{\frac{\partial F}{\partial z}} \mbox{  in  }
 \frac{\partial z}{\partial y} = -\frac{\frac{\partial F}{\partial y}}
{\frac{\partial F}{\partial z}} $$
\end{itemize}
\end{frame}

\subsection{Ekstremi in vezani ekstremi}
\begin{frame}{Osnovne formule}
\begin{itemize}
\item {\bf Stacionarna to"cka funkcije} $z = f(x,y)$:
$$ \frac{\partial z}{\partial x} = 0 \mbox{ in } \frac{\partial z}{\partial y} = 0.$$
\item {\bf Zadosten pogoj za nastop ekstrema.}
V stacionarni to"cki  $(x_0,y_0)$  nastopi ekstrem, "ce je $\Delta(x_0,y_0) > 0$,  kjer je   
\begin{equation*}
  \Delta = AC-B^2,\quad  
  A = \frac{\partial^2 z}{\partial x^2},\quad 
  B = \frac{\partial^2 z}{\partial x \partial y},\quad 
  C = \frac{\partial^2 z}{\partial y^2}.
\end{equation*}
\begin{itemize}
  \item "Ce je  $A > 0$,  nastopi ekstrem {\em minimum},
  \item "ce je  $A < 0$,  nastopi ekstrem {\em maksimum},
  \item "ce je  $\Delta  < 0$,  ekstrema ni in 
  \item "ce je  $\Delta = 0$, je odvisno od vi"sjih odvodov. 
\end{itemize}
\item {\bf Vezani ekstrem} 
funkcije  $z = f(x,y)$  pri pogoju  $\varphi(x,y) = 0$ 
lahko nastopi v stacionarnih to"ckah Lagrangeove funkcije
$$F(x,y,\lambda) = f(x,y) + \lambda \varphi (x,y).$$  	
\end{itemize}
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\end{document}

