From 178253be0ada30420f39720e7e0fa4ee4058bce0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= <anton@sijanec.eu>
Date: Sat, 1 Jun 2024 23:00:49 +0200
Subject: razne stvari iz b

---
 "\305\241ola/ana2/kolokvij1.lyx" | 976 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 976 insertions(+)
 create mode 100644 "\305\241ola/ana2/kolokvij1.lyx"

(limited to 'šola/ana2/kolokvij1.lyx')

diff --git "a/\305\241ola/ana2/kolokvij1.lyx" "b/\305\241ola/ana2/kolokvij1.lyx"
new file mode 100644
index 0000000..a4f9569
--- /dev/null
+++ "b/\305\241ola/ana2/kolokvij1.lyx"
@@ -0,0 +1,976 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\usepackage{amsmath}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\DeclareMathOperator{\g}{g}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cir}{Cir}
+\DeclareMathOperator{\ecc}{ecc}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\diam}{diam}
+\newcommand\euler{e}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics xetex
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 1cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\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
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