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author | Anton Luka Šijanec <anton@sijanec.eu> | 2024-06-01 23:00:49 +0200 |
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committer | Anton Luka Šijanec <anton@sijanec.eu> | 2024-06-01 23:00:49 +0200 |
commit | 178253be0ada30420f39720e7e0fa4ee4058bce0 (patch) | |
tree | db6d574d5945c468fccb0d39250762c41b53c7f0 /šola/ana2 | |
parent | earhorn_m3u (diff) | |
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Diffstat (limited to '')
-rw-r--r-- | šola/ana2/fja.scad | 13 | ||||
-rw-r--r-- | šola/ana2/kolokvij1.lyx | 976 |
2 files changed, 989 insertions, 0 deletions
diff --git a/šola/ana2/fja.scad b/šola/ana2/fja.scad new file mode 100644 index 0000000..4d18a17 --- /dev/null +++ b/šola/ana2/fja.scad @@ -0,0 +1,13 @@ +echo(version=version()); +function sinh(x) = (exp(1)-exp(-x))/2; +function f(x, y, z) = sinh(z)*sinh(y)-sin(x); +epsilon = 0.01; +rob = 1; +korak = 0.1; +particle = 0.1; +for (x = [-rob : korak : rob]) + for (y = [-rob : korak : rob]) + for (z = [-rob : korak : rob]) + if (f(x, y, z) < epsilon) + translate([x, y, z]) + cube(particle);
\ No newline at end of file diff --git a/šola/ana2/kolokvij1.lyx b/šola/ana2/kolokvij1.lyx new file mode 100644 index 0000000..a4f9569 --- /dev/null +++ b/šola/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 |