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\begin_body

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
setlength{
\backslash
columnseprule}{0.2pt}
\backslash
begin{multicols}{2}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Podmnožice v evklidskih prostorih
\end_layout

\begin_layout Standard
\begin_inset Formula $A$
\end_inset

 zaprta, če 
\begin_inset Formula $\forall$
\end_inset

 zaporedje s členi v 
\begin_inset Formula $A:$
\end_inset

 vsa stekališča v 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $A$
\end_inset

 kompaktna, če 
\begin_inset Formula $\forall$
\end_inset

 zaporedje s členi v 
\begin_inset Formula $A$
\end_inset

:
\begin_inset Formula $\exists$
\end_inset

 stekališče v 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 je kompozitum zveznih 
\begin_inset Formula $\Rightarrow$
\end_inset

 
\begin_inset Formula $f$
\end_inset

 zvezna
\end_layout

\begin_layout Standard
za 
\begin_inset Formula $x\in\mathbb{R}^{k}$
\end_inset

: 
\begin_inset Formula $\text{\left|\left|x\right|\right|\ensuremath{\coloneqq}}\sqrt{x_{1}^{2}+\cdots+x_{k}^{2}}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
\end_inset

 omejena 
\begin_inset Formula $\Leftrightarrow\exists M\in\mathbb{R}\forall x\in A:\left|\left|x\right|\right|<M$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $K\left(s\in\mathbb{R}^{k},M\in\mathbb{R}\right)\coloneqq\left\{ s+x\in\mathbb{R}^{k};\text{\left|\left|x\right|\right|}<M\right\} $
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $x\in A$
\end_inset

 notranja 
\begin_inset Formula $\Leftrightarrow\exists\varepsilon\ni:K\left(x,\varepsilon\right)\subset A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
\end_inset

 odprta 
\begin_inset Formula $\Leftrightarrow\forall x\in A:x$
\end_inset

 notranja
\end_layout

\begin_layout Standard
\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
\end_inset

 zaprta 
\begin_inset Formula $\Leftrightarrow\mathbb{R}^{k}\setminus A$
\end_inset

 odprta
\end_layout

\begin_layout Standard
\begin_inset Formula $x\in\mathbb{R}^{k}$
\end_inset

 stekališče 
\begin_inset Formula $A\Leftrightarrow\forall\varepsilon>0:K\left(x,\varepsilon\right)\cup\left(A\setminus\left\{ x\right\} \right)\not=\emptyset$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $x\in\mathbb{R}^{k}$
\end_inset

 izolirana točka 
\begin_inset Formula $A\Leftrightarrow x$
\end_inset

 ni stekališče 
\begin_inset Formula $A$
\end_inset


\end_layout

\begin_layout Standard
Zaporedje 
\begin_inset Formula $a_{n}:\mathbb{N}\to\mathbb{R}^{k}$
\end_inset

 konvergira proti 
\begin_inset Formula $a\in\mathbb{R}^{k}$
\end_inset

 kadar 
\begin_inset Formula $\forall\varepsilon>0\exists N\in\mathbb{N}\forall n>N:\left|\left|a_{n}-a\right|\right|<\varepsilon$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $s\in\mathbb{R}^{k}$
\end_inset

 stekališče zap.
 
\begin_inset Formula $\Leftrightarrow$
\end_inset

 v 
\begin_inset Formula $\text{\ensuremath{\varepsilon}}-$
\end_inset

okolici 
\begin_inset Formula $s$
\end_inset

 je 
\begin_inset Formula $\infty$
\end_inset

mnogo členov
\end_layout

\begin_layout Standard
Vsako omejeno zaporedje ima stekališče.
\end_layout

\begin_layout Section
Funkcije več spremenljivk
\end_layout

\begin_layout Standard
fja 
\begin_inset Formula $k$
\end_inset

 spremenljivk je preslikava 
\begin_inset Formula $f:D\subseteq\mathbb{R}^{k}\to\mathbb{R}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $L\in\mathbb{R}$
\end_inset

 je limita 
\begin_inset Formula $f:D\subseteq\text{\ensuremath{\mathbb{R}^{k}}}\to\mathbb{R}$
\end_inset

 v stekališču 
\begin_inset Formula $a\in D$
\end_inset

, če 
\begin_inset Formula $\forall\varepsilon>0\exists\delta>0\forall x\in D\setminus\left\{ a\right\} :\left|\left|x-a\right|\right|<\delta\Rightarrow\left|f\left(x\right)-L\right|<\varepsilon$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\lim_{x\to a}\left(f\oslash g\right)x=\lim_{x\to a}fx\oslash\lim_{x\to a}gx$
\end_inset

, če obstajata.
 
\begin_inset Formula $\oslash\in\left\{ +,-,\cdot\right\} $
\end_inset

.
 
\begin_inset Formula $\oslash$
\end_inset

 je lahko deljenje, kadar 
\begin_inset Formula $\lim_{x\to a}gx\not=0$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 zvezna v 
\begin_inset Formula $a\Leftrightarrow\forall\varepsilon\exists\delta\forall x\in D:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-fa\right|<\varepsilon$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 zvezna 
\begin_inset Formula $\Leftrightarrow\forall a\in D:f$
\end_inset

 zvezna v 
\begin_inset Formula $a$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 zvezna v stekališču 
\begin_inset Formula $a\Leftrightarrow\lim_{x\to a}fx=fa$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
\end_inset

 kompaktna, 
\begin_inset Formula $f:A\to\mathbb{R}$
\end_inset

 zvezna
\begin_inset Formula $\Rightarrow f$
\end_inset

 omejena in doseže maksimum in minimum (obstoj globalnega ekstrema).
\end_layout

\begin_layout Standard
\begin_inset Formula $f,g$
\end_inset

 zv.
 
\begin_inset Formula $\Rightarrow f\oslash g$
\end_inset

 zv.
 
\begin_inset Formula $\oslash\in\left\{ +,-,\cdot,\circ\right\} ,\oslash=/\Leftrightarrow\forall x:gx\not=0$
\end_inset


\end_layout

\begin_layout Section
Odvodi funkcij več spremenljivk
\end_layout

\begin_layout Standard
\begin_inset Formula $f_{x_{i}}a$
\end_inset

, 
\begin_inset Formula $i\in\left\{ 1..k\right\} $
\end_inset

 je odvod fje 
\begin_inset Formula $x_{i}\to f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)$
\end_inset

 v točki 
\begin_inset Formula $a_{i}$
\end_inset

.
 
\begin_inset Formula $f_{x_{i}}a=\lim_{x_{i}\to a_{i}}\frac{f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)-fa}{x_{i}-a_{i}}$
\end_inset


\end_layout

\begin_layout Standard
Tang.
 ravn.
 v 
\begin_inset Formula $a=\left(b,c\right)$
\end_inset


\begin_inset Formula $a=\left(b,c\right)$
\end_inset

: 
\begin_inset Formula $z=fa+f_{x}a\cdot\left(x-b\right)+f_{y}a\cdot\left(y-c\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 je odvedljiva v 
\begin_inset Formula $a\Leftrightarrow\lim_{h\to\left(0,0\right)}\frac{R_{a}\left(a+h\right)}{\left|\left|h\right|\right|}=0$
\end_inset

, kjer je 
\begin_inset Formula $R_{a}\left(a+h\right)\coloneqq f\left(a+h\right)-fa-f_{x}\left(a\right)\cdot u+f_{y}\left(a\right)\cdot v$
\end_inset

 za 
\begin_inset Formula $h=\left(u,v\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $dfa\coloneqq\left[f_{x_{1}}a\cdots f_{x_{k}}a\right]=\nabla fa,dfa\cdot h\coloneqq f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 v 
\begin_inset Formula $a$
\end_inset

 odvedljiva 
\begin_inset Formula $\Rightarrow f$
\end_inset

 v 
\begin_inset Formula $a$
\end_inset

 zvezna
\end_layout

\begin_layout Standard
\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :\exists f_{x_{i}}\wedge f_{x_{i}}$
\end_inset

 zvezna 
\begin_inset Formula $\Rightarrow f$
\end_inset

 odvedljiva
\end_layout

\begin_layout Standard
Lagrange: 
\begin_inset Formula $fx_{1}-fx_{2}=f'\xi\left(x_{2}-x_{1}\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f\in C^{r}U\sim$
\end_inset

 
\begin_inset Formula $f$
\end_inset

 
\begin_inset Formula $r-$
\end_inset

krat zvezno odvedljiva v vsaki točki 
\begin_inset Formula $U$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f:U\subseteq\mathbb{R}^{k}\to\mathbb{R},f\in C^{2}U\Rightarrow\forall i,j\in\left\{ 1..k\right\} :f_{x_{i}}=f_{x_{j}}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\frac{df}{d\vec{s}}\left(x,y\right)=\lim_{t\to0}\frac{f((x,y)+t\vec{s})-f(x,y)}{t}=s_{1}f(x,y)+s_{2}f_{x}(x,y)$
\end_inset


\end_layout

\begin_layout Standard
Tangentna ravnina v 
\begin_inset Formula $\left(x_{0},y_{0},f\left(x_{0},y_{0}\right)\right)$
\end_inset

 je 
\begin_inset Formula $f_{x}\left(x_{0},y_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0},y_{0}\right)\left(y-y_{0}\right)-z+f\left(x_{0},y_{0}\right)$
\end_inset

 in razpenjata jo vektorja 
\begin_inset Formula $\left(1,0,f_{x}\right)$
\end_inset

 in 
\begin_inset Formula $\left(0,1,f_{y}\right)$
\end_inset

.
\end_layout

\begin_layout Section
Taylorjeva formula in verižno pravilo
\end_layout

\begin_layout Standard
\begin_inset Formula $f$
\end_inset

 diferenciabilna v 
\begin_inset Formula $a\in\mathbb{R}^{k}\Rightarrow f\left(a+h\right)\cong fa+dfa\cdot h=fa+f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $U\subseteq R^{k}$
\end_inset

 odprta in 
\begin_inset Formula $f:U\to\mathbb{R},f\in C^{n+1}U$
\end_inset

.
 Naj bo 
\begin_inset Formula $D_{f,r,a}$
\end_inset

 vektor vseh parcialnih odvodov reda 
\begin_inset Formula $r$
\end_inset

 v točki 
\begin_inset Formula $a$
\end_inset

.
 Primer: 
\begin_inset Formula $D_{f,2,a}=\left(f_{xx}a,2f_{xy}a+f_{yy}a\right)$
\end_inset

.
 
\begin_inset Formula $D_{f,0,a}\coloneqq f\left(a\right)$
\end_inset

.
 Naj bo 
\begin_inset Formula $H_{r}$
\end_inset

 vektor z vsemi kombinacijami dolžine 
\begin_inset Formula $r$
\end_inset

 komponent 
\begin_inset Formula $h$
\end_inset

.
 Primer: 
\begin_inset Formula $H_{2}=\left(uu,2uv,vv\right)$
\end_inset

.
 
\begin_inset Formula $H_{0}=1$
\end_inset

.
 
\begin_inset Formula $D_{f,r,a}\cdot H_{r}$
\end_inset

 je njun skalarni produkt.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
T_{f,a,n}\left(h_{1}=x-a,h_{2}=y-b\right)=\sum_{i=0}^{n}\frac{1}{i!}\left(D_{f,i,a}\cdot H_{i}\right)
\]

\end_inset


\end_layout

\begin_layout Section
Ekstremalni problemi
\end_layout

\begin_layout Standard
Kandidati so 
\begin_inset Formula $a$
\end_inset

, da 
\begin_inset Formula $\nabla fa=0$
\end_inset

 ali 
\begin_inset Formula $f$
\end_inset

 ni odv.
 v 
\begin_inset Formula $a$
\end_inset

 ali 
\begin_inset Formula $a$
\end_inset

 robna točka.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
H\left(a,b\right)=\left[\begin{array}{cc}
f_{xx}\left(a,b\right) & f_{xy}\left(a,b\right)\\
f_{yx}\left(a,b\right) & f_{yy}\left(a,b\right)
\end{array}\right]
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\det H\left(a,b\right)>0$
\end_inset

: 
\begin_inset Formula $f_{xx}\left(a,b\right)>0$
\end_inset

 l.
 min., 
\begin_inset Formula $f_{xx}\left(a,b\right)<0$
\end_inset

 l.
 max.
\end_layout

\begin_layout Standard
\begin_inset Formula $\det H\left(a,b\right)<0$
\end_inset

 sedlo
\end_layout

\begin_layout Standard
Izrek o implicitni funkciji: 
\begin_inset Formula $D\subseteq\mathbb{R}^{2}$
\end_inset

 odprta, 
\begin_inset Formula $f:D\to\mathbb{R}$
\end_inset

 zvezno parcialno odvedljiva.
 
\begin_inset Formula $K=\left\{ \left(x,y\right)\in D;f\left(x,y\right)=0\right\} $
\end_inset

.
 Za 
\begin_inset Formula $\left(a,b\right)\in D$
\end_inset

, 
\begin_inset Formula $f\left(a,b\right)=0\wedge\nabla f\left(a,b\right)\not=0\exists h\left(a\right)=b,f\left(x,h\left(x\right)\right)=0\forall x\in U$
\end_inset

.
\end_layout

\begin_layout Standard
Vezani ekstrem: 
\begin_inset Formula $D^{\text{odp.}}\subseteq\mathbb{R}^{n}$
\end_inset

, 
\begin_inset Formula $f:D\to\mathbb{R}$
\end_inset

 diferenciabilna na 
\begin_inset Formula $D$
\end_inset

.
 let 
\begin_inset Formula $g:D\to\mathbb{R}$
\end_inset

 zvezno parcialno odvedljiva, 
\begin_inset Formula $A\coloneqq\left\{ x\in D;gx=0\right\} $
\end_inset

.
 
\begin_inset Formula $\exists$
\end_inset

 vezani ekstrem 
\begin_inset Formula $f$
\end_inset

 pri pogoju 
\begin_inset Formula $g\Leftrightarrow\nabla fa=\lambda\nabla ga$
\end_inset

.
 Kandidati za vezane ekstreme so stac.
 točke fje 
\begin_inset Formula $F\left(x,\lambda\right)=fx-\lambda gx$
\end_inset

.
\end_layout

\begin_layout Section
Krivulje in ploskve
\end_layout

\begin_layout Standard
Pot v 
\begin_inset Formula $\mathbb{R}^{3}\sim\vec{r}:I\to\mathbb{R}^{3},I\in\mathbb{R}$
\end_inset

 interval.
 
\begin_inset Formula $\forall t\in I:\vec{r}t=\left(xt,yt,zt\right)$
\end_inset

.
\end_layout

\begin_layout Standard
Odvod poti: 
\begin_inset Formula $\dot{\vec{r}}\left(t\right)=\left(\dot{x}\left(t\right),\dot{y}\left(t\right),\dot{z}\left(t\right)\right)$
\end_inset

.
 
\begin_inset Formula $\dot{\vec{r}}t$
\end_inset

 je tangentni vektor na krivuljo v točki 
\begin_inset Formula $\vec{r}t$
\end_inset

.
\end_layout

\begin_layout Standard
Dolžina poti 
\begin_inset Formula $\vec{r}:\left[a,b\right]\to\mathbb{R}^{3}$
\end_inset

 je 
\begin_inset Formula 
\[
L=\int_{a}^{b}\left|\dot{\vec{r}}t\right|dt=\int_{a}^{b}\sqrt{\dot{x}^{2}t+\dot{y}^{2}t}dt
\]

\end_inset

.
\end_layout

\begin_layout Standard
Ploščina območja, ki ga omejuje krivulja, če je parametrizacija taka, da
 je krivulja levo od 
\begin_inset Formula $\vec{r}t$
\end_inset

: 
\begin_inset Formula $\text{Pl\left(D\right)=\ensuremath{\frac{1}{2}\int_{a}^{b}\left(xt\dot{y}t-\dot{x}tyt\right)dt}}$
\end_inset

.
\end_layout

\begin_layout Standard
Ploskev eksplicitno kot graf 
\begin_inset Formula $f:D^{\text{odp.}}\subseteq\mathbb{R}^{2}\to\mathbb{R}$
\end_inset

.
 
\begin_inset Formula $f$
\end_inset

 difer.
 v 
\begin_inset Formula $\left(x,y\right)\Rightarrow$
\end_inset

 v 
\begin_inset Formula $\left(x,y,f\left(x,y\right)\right)$
\end_inset

 definiramo tangentno ravnino z normalo 
\begin_inset Formula $\left(-f_{x}\left(x,y\right),-f_{y}\left(x,y\right),1\right)$
\end_inset


\end_layout

\begin_layout Standard
Ploskev implicitno: 
\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$
\end_inset

 zvezno parcialno odvedljiva.
\begin_inset Formula 
\[
P=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3};f\left(x,y,z\right)=0\right\} 
\]

\end_inset

Če 
\begin_inset Formula $\forall\left(x,y,z\right)\in P:\nabla f\left(x,y,z\right)\not=0\Rightarrow P$
\end_inset

 ploskev v 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

, saj je po izreku o implicitni fji 
\begin_inset Formula $P$
\end_inset

 lokalno graf fje dveh spremenljivk.
 Normala tangentne ravnine v 
\begin_inset Formula $\left(x,y,z\right)\in P$
\end_inset

 ima normalo 
\begin_inset Formula $\nabla f\left(x,y,z\right)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{multicols}
\end_layout

\end_inset


\end_layout

\end_body
\end_document