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\newtheorem{df}{Definicija}\newtheorem{tm}{Teorem}\newtheorem{lm}{Lema}\newtheorem{kor}{Posljedica}\newtheorem{pr}{Primjer}[chapter]\newtheorem{prpr}{Primjer}[pr]\newtheorem{zd}{Zadaci}[chapter]\newtheorem{ak}{Aksiom}\newtheorem{alg}{Algoritam}\newtheorem{prg}{\textit{Mathematica} program}
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\title{\bf MATEMATIKA III}
\author{Salih Suljagić}
\date{\today}

\makeindex


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% latex2html id marker 30611
$\displaystyle \left[\begin{array}[center]{ccc}
2 & 3 & -1 \\  1 & 0 & 2
\end{array}\right]$%
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% latex2html id marker 30617
$\displaystyle \boldsymbol{x}_1 = \left[\begin{array}[center]{c}
1 \\  -2 \\  2
\end{array}\right], \quad 
\boldsymbol{x}_2 = \left[\begin{array}[center]{c}
-1 \\  3 \\  2
\end{array}\right], \quad 
\boldsymbol{x}_3 = \left[\begin{array}[center]{c}
-3 \\  1 \\  2
\end{array}\right]$%
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% latex2html id marker 30684
$\displaystyle \boldsymbol{e}_1=\left[\begin{array}{c} 1 \\  0
\\  \vdots \\  0 \end{array} \right],\hspace{1cm} \boldsymbol{e}_2=\left[\begin{array}{c} 
0 \\  1 \\  \vdots \\  0 \end{array} \right],\hspace{1cm}
\cdots,\hspace{1cm} \boldsymbol{e}_n=\left[\begin{array}{c} 0 \\  0 \\  \vdots \\  1
\end{array} \right].$%
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% latex2html id marker 30690
$\displaystyle \left[\begin{array}{c} 
a_{1} \\  a_{2} \\  \vdots \\  a_{n}
\end{array} \right]=a_{1}\left[\begin{array}{c} 
1 \\  0 \\  \vdots \\  0
\end{array} \right]+a_{2}\left[\begin{array}{c} 
0 \\  1 \\  \vdots \\  0
\end{array} \right] +\cdots+a_{n}\left[\begin{array}{c} 
0 \\  0 \\  \vdots \\  1
\end{array} \right]=a_{1}\,\boldsymbol{e}_1+a_{2}\,\boldsymbol{e}_2+\cdots+a_{n}\,\boldsymbol{e}_n,$%
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% latex2html id marker 30692
$\displaystyle \lambda_1\left[\begin{array}{c} 
1 \\  0 \\  \vdots \\  0
\end{array} \right]+\lambda_2\left[\begin{array}{c} 
0 \\  1 \\  \vdots \\  0
\end{array} \right] +\cdots+\lambda_n\left[\begin{array}{c} 
0 \\  0 \\  \vdots \\  1
\end{array} \right]=
\left[\begin{array}{c} 
\lambda_1 \\  \lambda_2 \\  \vdots \\  \lambda_n
\end{array} \right]=\left[\begin{array}{c} 
0 \\  0 \\  \vdots \\  0
\end{array} \right]$%
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% latex2html id marker 30701
$\displaystyle E_{11}=\left[\begin{array}{cccc}
1  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array}\right],\; E_{12}=\left[\begin{array}{cccc}
0  &  1  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array}\right],\;
\cdots,\; E_{mn}=\left[\begin{array}{cccc}
0  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  1
\end{array} \right].$%
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% latex2html id marker 30707
$\displaystyle \left[\begin{array}{cccc}
a_{11}  &  a_{12}  &  \cdots  &  a_{1n}  \\ 
a_{21}  &  a_{22}  &  \cdots  &  a_{2n}  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
a_{m1}  &  a_{m2}  &  \cdots  &  a_{mn}
\end{array} \right]=a_{11}\left[\begin{array}{cccc}
1  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array} \right]+a_{12}\left[\begin{array}{cccc}
0  &  1  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array} \right] +$%
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% latex2html id marker 30709
$\displaystyle \cdots+a_{mn}\left[\begin{array}{cccc}
0  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  1
\end{array} \right]=a_{11}\,E_{11}+a_{12}\,E_{12}+\cdots+a_{mn}\,E_{mn},$%
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% latex2html id marker 30711
$\displaystyle \lambda_{11}\left[\begin{array}{cccc}
1  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array} \right]+\lambda_{12}\left[\begin{array}{cccc}
0  &  1  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array} \right] +\cdots +\lambda_{mn}\left[\begin{array}{cccc}
0  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  1
\end{array} \right]$%
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% latex2html id marker 30713
$\displaystyle =\left[\begin{array}{cccc}
\lambda_{11}  &  \lambda_{12}  &  \cdots  &  \lambda_{1n}  \\ 
\lambda_{21}  &  \lambda_{22}  &  \cdots  &  \lambda_{2n}  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
\lambda_{m1}  &  \lambda_{m2}  &  \cdots  &  \lambda_{mn}
\end{array} \right]=\left[\begin{array}{cccc}
0  &  0  &  \cdots  &  0  \\ 
0  &  0  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  0
\end{array} \right]$%
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\begin{displaymath}
% latex2html id marker 30925
A=
\left[
\begin{array}{cccc}
  a_{11}  &  a_{12}  &  \cdots  &  a_{1n}  \\ 
  a_{21}  &  a_{22}  &  \cdots  &  a_{2n}  \\ 
  \vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
  a_{m1}  &  a_{m2}  &  \cdots  &  a_{mn}
\end{array}
\right],\hspace{1cm}
A_b=
\left[
\begin{array}{ccccc}
  a_{11}  &  a_{12}  &  \cdots  &  a_{1n} & b_1 \\ 
  a_{21}  &  a_{22}  &  \cdots  &  a_{2n} & b_2 \\ 
  \vdots     &  \vdots     &  \ddots  &  \vdots    & \vdots \\ 
  a_{m1}  &  a_{m2}  &  \cdots  &  a_{mn} & b_m
\end{array}
\right].\end{displaymath}%
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% latex2html id marker 30965
$\displaystyle \begin{array}{rrrrrr}
a_{11}\,x_1 & +a_{12}\,x_2  & +\cdots & +a_{1n}\,x_n   & = & b_1 \\ 
&  a'_{22}\,x_2 & +\cdots & +a'_{2n}\,x_n  & = & b'_2 \\ 
&             & \cdots  &              &   & \\  
&  a'_{m2}\,x_2 & +\cdots & +a'_{mn}\,x_n  & = & b'_m, 
\end{array} $%
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% latex2html id marker 30981
$\displaystyle \begin{array}{rrrrrr}
x_1 &      & -\,x_3 & +2\,x_4  & = & -3 \\    
& 2\,x_2 & +\,x_3 &  +\,x_4  & = & 1 \\ 
x_1 & +\,x_2 & -\,x_3 &        & = & -1 \\ 
2\,x_1 & +\,x_2 & +\,x_3 &  +\,x_4  & = & 3.
\end{array} $%
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% latex2html id marker 30987
$\displaystyle \begin{array}{rrrrrr}
x_1 &      & -\,x_3 & +2\,x_4  & = & -3 \\    
& 2\,x_2 & +\,x_3 &  +\,x_4  & = & 1 \\ 
&  x_2 &      & -2\,x_4  & = & 2 \\ 
&  x_2 &+3\,x_3 & -3\,x_4  & = & 9.
\end{array} $%
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% latex2html id marker 30993
$\displaystyle \begin{array}{rrrrrr}
x_1 &      & -\,x_3 & +2\,x_4  & = & -3 \\    
&  x_2 &      & -2\,x_4  & = & 2 \\ 
&      &  x_3 & +5\,x_4  & = & -3 \\ 
&      & 3\,x_3 &  -\,x_4  & = & 7.
\end{array} $%
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% latex2html id marker 30997
$\displaystyle \begin{array}{rrrrrr}
x_1 &      & -x_3 & +2\,x_4  & = & -3 \\    
&  x_2 &      & -2\,x_4  & = & 2 \\ 
&      &  x_3 & +5\,x_4  & = & -3 \\ 
&      &      &-16\,x_4  & = & 16.
\end{array} $%
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% latex2html id marker 31008
$\displaystyle \begin{array}{rrrrrrr}
x_1 &      & -\,x_3 & +2\,x_4  & +\,x_5 & = & -3 \\    
& 2\,x_2 & +\,x_3 &  +\,x_4  & +\,x_5 & = & 1 \\ 
x_1 & +\,x_2 & -\,x_3 &        & +\,x_5 & = & -1 \\ 
2\,x_1 & +\,x_2 & +\,x_3 &  +\,x_4  & +\,x_5 & = & 3
\end{array}.$%
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% latex2html id marker 31010
$\displaystyle \begin{array}{rrrrrrr}
x_1 &      & -\,x_3 & +2\,x_4  & +\,x_5   & = & -3 \\    
&  x_2 &      & -2\,x_4  &        & = & 2 \\ 
&      &  x_3 & +5\,x_4  & +\,x_5   & = & -3 \\ 
&      &      &   x_4  &+\frac{1}{4}\,x_5 & = & -1
\end{array}.$%
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% latex2html id marker 31014
$\displaystyle \begin{array}{rrrrrrr}
x_1 &      & -\,x_3 &        &+\frac{2}{4}\,x_5 & = & -1 \\    
&  x_2 &      &        &+\frac{2}{4}\,x_5 & = & 0 \\ 
&      &  x_3 &        &-\frac{1}{4}\,x_5 & = & 2 \\ 
&      &      &   x_4  &+\frac{1}{4}\,x_5 & = & -1
\end{array}.$%
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{\newpage\clearpage
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% latex2html id marker 31016
$\displaystyle \begin{array}{rrrrrrr}
x_1 &      &      &        &+\frac{1}{4}\,x_5 & = & 1 \\    
&  x_2 &      &        &+\frac{2}{4}\,x_5 & = & 0 \\ 
&      &  x_3 &        &-\frac{1}{4}\,x_5 & = & 2 \\ 
&      &      &   x_4  &+\frac{1}{4}\,x_5 & = & -1
\end{array}.$%
\lthtmlindisplaymathZ
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\stepcounter{subsubsection}
\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 31134
$\displaystyle \left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 0 & 1 
\end{array}\right]\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]=\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
1 & 1 & -1  \\ 
0 & 2 & 1  \\ 
2 & 1 & 1
\end{array} \right],$%
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{\newpage\clearpage
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% latex2html id marker 31136
$\displaystyle \left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]\left[ \begin{array}{rrr}
1 & 0 & 0  \\ 
0 & 0 & 1  \\ 
0 & 1 & 0
\end{array} \right]=\left[ \begin{array}{rrr}
1 & -1 & 0  \\ 
0 & 1 & 2  \\ 
1 & -1 & 1  \\ 
2 & 1 & 1
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31139
$\displaystyle \left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 3 & 0 \\ 
0 & 0 & 0 & 1 
\end{array}\right]\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]=\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
3 & 3 & -3  \\ 
2 & 1 & 1
\end{array} \right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31141
$\displaystyle \left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]\left[ \begin{array}{rrr}
1 & 0 & 0  \\ 
0 & 3 & 0  \\ 
0 & 0 & 1
\end{array} \right]=\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 6 & 1  \\ 
1 & 3 & -1  \\ 
2 & 3 & 1
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31152
$\displaystyle \left[\begin{array}{rrrr}
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
-2 & 0 & 0 & 1 
\end{array}\right]\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]=\left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
0 & 1 & 3
\end{array} \right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31154
$\displaystyle \left[ \begin{array}{rrr}
1 & 0 & -1  \\ 
0 & 2 & 1  \\ 
1 & 1 & -1  \\ 
2 & 1 & 1
\end{array}\right]\left[ \begin{array}{rrr}
\par 1 & 0 & 0  \\ 
0 & 1 & 0  \\ 
-2 & 0 & 1
\end{array} \right]=\left[ \begin{array}{rrr}
3 & 0 & -1  \\ 
-2 & 2 & 1  \\ 
3 & 1 & -1  \\ 
0 & 1 & 1
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 31185
$\displaystyle A=\left[\begin{array}{cccc} 
\boldsymbol{a}_{\cdot\, 1} & \boldsymbol{a}_{\cdot\, 2} & \cdots & \boldsymbol{a}_{\cdot\, n}
\end{array} \right],\hspace{1cm}
A=\left[\begin{array}{c} 
\boldsymbol{a}_{1\,\cdot} \\  \boldsymbol{a}_{2\,\cdot} \\  \vdots \\  \boldsymbol{a}_{m\,\cdot}
\end{array} \right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31210}%
% latex2html id marker 31210
$\displaystyle T\,A=T\,\left[\begin{array}{cccc}
\boldsymbol{a}_{\cdot\, 1} & \boldsymbol{a}_{\cdot\, 2} & \cdots & \boldsymbol{a}_{\cdot\, n}
\end{array} \right] = \left[\begin{array}{cccc} T\,\boldsymbol{a}_{\cdot\, 1} &
T\,\boldsymbol{a}_{\cdot\, 2} & \cdots & T\,\boldsymbol{a}_{\cdot\, n} \end{array} \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31275
$\displaystyle \left[\begin{array}{cccccc} 
1      & \cdots &    0        &   0    & \cdots &  0    \\ 
\vdots & \ddots & \vdots      & \vdots & \ddots & \vdots \\ 
0      & \cdots &   1         &   0    & \cdots &  0  \\ 
0      & \cdots &   0         &   0    & \cdots &  0 \\ 
\vdots & \ddots & \vdots      & \vdots & \ddots & \vdots \\ 
0      & \cdots &   0         &   0    & \cdots &  0
\end{array}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31288}%
% latex2html id marker 31288
$\displaystyle A=\left[ \begin{array}{rrrrrr}
1 & 0 & -1 & 2  & 1 &  -3 \\    
0 & 2 & 1 &  1  & 1 &  1 \\ 
1 & 1 & -1 & 0  & 1 &  -1 \\ 
2 & 1 & 1 &  1  & 1 &  3
\end{array}\right].$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 31290
$\displaystyle \left[ \begin{array}{rrrrrr}
1 & 0 & -1 & 2  & 1 &  -3 \\    
0 & 2 & 1 &  1  & 1 &  1 \\ 
1 & 1 & -1 & 0  & 1 &  -1 \\ 
2 & 1 & 1 &  1  & 1 &  3
\end{array}\right]\sim\left[\begin{array}{rrrrrr}
1 & 0 & -1 & 2  & 1   &  -3 \\    
0 & 1 & 0  & -2  & 0  &  2 \\ 
0 & 0 &  1 & 5  & 1   &  -3 \\ 
0 & 0 &  0 & 1  &\frac{1}{4}  &  -1
\end{array} \right]$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 31292
$\displaystyle \sim\left[\begin{array}{rrrrrr}
1 & 0 & 0 & 0 & 0 & 0 \\    
0 & 1 & 0 & 0 & 0 & 0 \\ 
0 & 0 & 1 & 0 & 0 & 0 \\ 
0 & 0 & 0 & 1 & 0 & 0
\end{array} \right],$%
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{\newpage\clearpage
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% latex2html id marker 31299
$\displaystyle A = \left[\begin{array}{rrrr}
1-\lambda^2           & -2\lambda+\lambda^2+\lambda^3    &  0 & -1 \\ 
4-5\lambda-2\lambda^2 & 1-3\lambda+3\lambda^2+2\lambda^3 & -3 & -8 \\ 
\lambda &                               -1 &  1 &  2 \\ 
-2+3\lambda+\lambda^2 &  -2+2\lambda-\lambda^2-\lambda^3 &  2 &  5 
\end{array}\right].$%
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% latex2html id marker 31301
$\displaystyle A \sim \left[\begin{array}{rrrr}
1 &  2 &               \lambda &                               -1 \\  
0 & -1 & 1-\lambda^2           & -2\lambda+\lambda^2+\lambda^3    \\  
-3 & -8 & 4-5\lambda-2\lambda^2 & 1-3\lambda+3\lambda^2+2\lambda^3 \\  
2 &  5 & -2+3\lambda+\lambda^2 &  -2+2\lambda-\lambda^2-\lambda^3 \\  
\end{array}\right].$%
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% latex2html id marker 31307
$\displaystyle A \sim \left[\begin{array}{rrrr}
1 &  2 &               \lambda &                               -1 \\  
0 & -1 & 1-\lambda^2           & -2\lambda+\lambda^2+\lambda^3    \\  
0 & -2 & 4-2\lambda-2\lambda^2 & -2-3\lambda+3\lambda^2+2\lambda^3 \\  
0 &  1 & -2+\lambda+\lambda^2 &  2\lambda-\lambda^2-\lambda^3 \\  
\end{array}\right].$%
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{\newpage\clearpage
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% latex2html id marker 31313
$\displaystyle A \sim \left[\begin{array}{rrrr}
1 &  0 &                     0 &                   0 \\  
0 &  1 & -2+\lambda+\lambda^2 &  2\lambda-\lambda^2-\lambda^3 \\  
0 &  0 & -1+\lambda           &                    0 \\  
0 &  0 &                    0 & -2+\lambda+\lambda^2 \\  
\end{array}\right].$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 31315
$\displaystyle A \sim \left[\begin{array}{rrrr}
1 &  0 &          0 &                    0 \\  
0 &  1 &          0 &                    0 \\  
0 &  0 & -1+\lambda &                    0 \\  
0 &  0 &          0 & -2+\lambda+\lambda^2 \\  
\end{array}\right].$%
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\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 31369
$\displaystyle Q_1Q_2\cdots Q_k\left[ \begin{array}{cc} A & I
\end{array} \right]=\left[ \begin{array}{cc} Q_1Q_2\cdots Q_kA &
Q_1Q_2\cdots Q_k \end{array} \right]=\left[ \begin{array}{cc} I &
A^{-1} \end{array} \right].$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31380}%
% latex2html id marker 31380
$\displaystyle A=\left[\begin{array}{rrrr}
1 & 1 & 1 & 1 \\ 
1 & 2 & 2 & 2 \\ 
1 & 2 & 3 & 3 \\ 
1 & 2 & 3 & 4 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31382}%
% latex2html id marker 31382
$\displaystyle \left[\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 
1 & 2 & 2 & 2 & 0 & 1 & 0 & 0 \\ 
1 & 2 & 3 & 3 & 0 & 0 & 1 & 0 \\ 
1 & 2 & 3 & 4 & 0 & 0 & 0 & 1 
\end{array}\right]\sim\left[\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 
0 & 1 & 1 & 1 & -1 & 1 & 0 & 0 \\ 
0 & 1 & 2 & 2 & -1 & 0 & 1 & 0 \\ 
0 & 1 & 2 & 3 & -1 & 0 & 0 & 1 
\end{array}\right]$%
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{\newpage\clearpage
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% latex2html id marker 31384
$\displaystyle \sim\left[\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 
0 & 1 & 1 & 1 & -1 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 1 & 0 & -1 & 1 & 0 \\ 
0 & 0 & 1 & 2 & 0 & -1 & 0 & 1 
\end{array}\right]\sim\left[\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 
0 & 1 & 1 & 1 & -1 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 1 & 0 & -1 & 1 & 0 \\ 
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 
\end{array}\right]$%
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% latex2html id marker 31386
$\displaystyle \sim\left[\begin{array}{rrrrrrrr}
1 & 1 & 1 & 0 & 1 & 0 & 1 & -1 \\ 
0 & 1 & 1 & 0 & -1 & 1 & 1 & -1 \\ 
0 & 0 & 1 & 0 & 0 & -1 & 2 & -1 \\ 
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 
\end{array}\right]\sim\left[\begin{array}{rrrrrrrr}
1 & 1 & 0 & 0 & 1 & 1 & -1 & 0 \\ 
0 & 1 & 0 & 0 & -1 & 2 & -1 & 0 \\ 
0 & 0 & 1 & 0 & 0 & -1 & 2 & -1 \\ 
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 
\end{array}\right]$%
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{\newpage\clearpage
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% latex2html id marker 31388
$\displaystyle \sim\left[\begin{array}{rrrrrrrr}
1 & 0 & 0 & 0 & 2 & -1 & 0 & 0 \\ 
0 & 1 & 0 & 0 & -1 & 2 & -1 & 0 \\ 
0 & 0 & 1 & 0 & 0 & -1 & 2 & -1 \\ 
0 & 0 & 0 & 1 & 0 & 0 & -1 & 1 
\end{array}\right].$%
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{\newpage\clearpage
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% latex2html id marker 31390
$\displaystyle A^{-1}=\left[\begin{array}{rrrr}
2 & -1 & 0 & 0 \\ 
-1 & 2 & -1 & 0 \\ 
0 & -1 & 2 & -1 \\ 
0 & 0 & -1 & 1
\end{array}\right].$%
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{\newpage\clearpage
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% latex2html id marker 31419
$\displaystyle \left[
\begin{array}{c} a_{11}\,x_1+a_{12}\,x_2+\cdots+a_{1n}\,x_n \\ 
a_{21}\,x_1+a_{22}\,x_2+\cdots +a_{2n}\,x_n \\  \vdots \\ 
a_{m1}\,x_1+a_{m2}\,x_2+\cdots +a_{mn}\,x_n
\end{array} \right]=\left[ \begin{array}{c} 
b_{1} \\  b_{2} \\  \vdots \\  b_{m}
\end{array} \right],$%
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{\newpage\clearpage
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% latex2html id marker 31421
$\displaystyle x_1\left[ \begin{array}{c} 
a_{11} \\  a_{21} \\  \vdots \\  a_{m1}
\end{array} \right]+x_2\left[ \begin{array}{c} 
a_{12} \\  a_{22} \\  \vdots \\  a_{m2}
\end{array} \right]+\cdots+x_n\left[ \begin{array}{c} 
a_{1n} \\  a_{2n} \\  \vdots \\  a_{mn}
\end{array} \right]=\left[ \begin{array}{c} 
b_{1} \\  b_{2} \\  \vdots \\  b_{m}
\end{array} \right].$%
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{\newpage\clearpage
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% latex2html id marker 31451
$\displaystyle \boldsymbol{a}_{\cdot\, 1} = 
\left[\begin{array}{c}
5 \\  1 \\  1
\end{array}\right], \quad 
\boldsymbol{a}_{\cdot\, 2} = 
\left[\begin{array}{c}
0 \\  1 \\  4
\end{array}\right], \quad 
\boldsymbol{a}_{\cdot\, 3} = 
\left[\begin{array}{c}
4 \\  2 \\  2
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31453
$\displaystyle \boldsymbol{b}= \left[\begin{array}{c}
3 \\  1 \\  1
\end{array}\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 31570
$\displaystyle \begin{array}{cccccccc}
x_1 &  &  & +a'_{1\,r+1}\,x_{r+1} & +\;\cdots & +a'_{1n}\,x_n  & = & b'_1 \\ 
& \ddots & & \vdots  & \ddots  & \vdots      &   & \vdots \\            
& & x_r & +a'_{r\,r+1}\,x_{r+1} & +\;\cdots  & +a'_{rn}\,x_n  & = & b'_r 
\end{array}.$%
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{\newpage\clearpage
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% latex2html id marker 31587
$\displaystyle \left[\begin{array}{c}
x_1 \\  \vdots \\  x_r \\  x_{r+1} \\  \vdots \\  x_n 
\end{array}\right] = \left[\begin{array}{c}
b'_1-a'_{1\,r+1}\,x_{r+1}-\cdots-a'_{1n}\,x_n \\ 
\vdots \\  b'_r-a'_{r\,r+1}\,x_{r+1}-\cdots -a'_{rn}\,x_n \\ 
x_{r+1} \\  \vdots \\  x_n \end{array}\right]=\left[\begin{array}{c}
b'_1 \\  \vdots \\  b'_r \\  0 \\  \vdots \\  0 \end{array}\right]+x_{r+1}
\left[\begin{array}{c}
-a'_{1\,r+1} \\  \vdots \\  -a'_{r\,r+1} \\  1 \\  \vdots \\  0
\end{array}\right]+\cdots+x_n\left[\begin{array}{c}
-a'_{1n} \\  \vdots \\  -a'_{rn} \\  0 \\  \vdots \\  1 \end{array}\right].$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 31589
$\displaystyle \boldsymbol{x}= \left[\begin{array}{c}
x_1 \\  \vdots \\  x_r \\  x_{r+1} \\  \vdots \\  x_n 
\end{array}\right],\quad \boldsymbol{b}'=\left[\begin{array}{c}
b'_1 \\  \vdots \\  b'_r \\  0 \\  \vdots \\  0 \end{array}\right],\quad 
\boldsymbol{u}_1=\left[\begin{array}{c}
-a'_{1\,r+1} \\  \vdots \\  -a'_{r\,r+1} \\  1 \\  \vdots \\  0
\end{array}\right],\quad \cdots ,\quad \boldsymbol{u}_{n-r}=\left[\begin{array}{c}
-a'_{1n} \\  \vdots \\  -a'_{rn} \\  0 \\  \vdots \\  1
\end{array}\right].$%
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% latex2html id marker 31618
$\displaystyle \left[
\begin{array}{c}
x_1 \\  x_2 \\  x_3
\end{array}\right] = \left[
\begin{array}{c}
\frac{1}{5} \\  -\frac{4}{5} \\  0
\end{array}\right] + x_3\,\left[
\begin{array}{c}
-\frac{4}{5} \\  \frac{6}{5} \\  1
\end{array}\right].$%
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{\newpage\clearpage
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% latex2html id marker 24729
\includegraphics{m3sustpr1.eps}%
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% latex2html id marker 24735
\includegraphics{m3sustpr2.eps}%
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\stepcounter{paragraph}
{\newpage\clearpage
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% latex2html id marker 31672
$\displaystyle \begin{array}{rrrrrrr}
2\,x_1 & +3\,x_2 & +5\,x_3 & +7\,x_4  & +8\,x_5 & = & 1 \\ 
-x_1 & +2\,x_2 &  +\,x_3 &        & +3\,x_5 & = & 3 \\ 
&  +\,x_2 &  +\,x_3 &  +\,x_4  & +2\,x_5 & = & 1 \\ 
5\,x_1 & -3\,x_2 & +2\,x_3 & +7\,x_4  & -\,x_5  & = & -8.
\end{array} $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31674
$\displaystyle \boldsymbol{x}= \left[ \begin{array}{r}
x_1 \\  x_2 \\  x_3 \\  x_4 \\  x_5
\end{array} \right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 31702
$\displaystyle \boldsymbol{x}=\left[ \begin{array}{r}
-1-x_3-2\,x_4-x_5 \\  1-x_3-x_4-2\,x_5 \\  x_3 \\  x_4 \\  x_5
\end{array} \right]=
\left[ \begin{array}{r}
-1 \\  1 \\  0 \\  0 \\  0
\end{array} \right]+x_3
\left[ \begin{array}{r}
-1 \\  -1 \\  1 \\  0 \\  0
\end{array} \right]+x_4
\left[ \begin{array}{r}
-2 \\  -1 \\  0 \\  1 \\  0
\end{array} \right]+x_5
\left[ \begin{array}{r}
-1 \\  -2 \\  0 \\  0 \\  1
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline31704}%
% latex2html id marker 31704
$ \left[ \begin{array}{r}
-1 \\  1 \\  0 \\  0 \\  0
\end{array} \right]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline31706}%
% latex2html id marker 31706
$ \left[ \begin{array}{r}
-1 \\  -1 \\  1 \\  0 \\  0
\end{array} \right],\;
\left[ \begin{array}{r}
-2 \\  -1 \\  0 \\  1 \\  0
\end{array} \right],\;
\left[ \begin{array}{r}
-1 \\  -2 \\  0 \\  0 \\  1
\end{array} \right]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31730}%
% latex2html id marker 31730
$\displaystyle A_b = \left[
\begin{array}{cccc}
1 & 1 & 1 & 6 \\ 
a & 4 & 1 & 5 \\ 
6 & a+2 & 2 & 13 \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31734}%
% latex2html id marker 31734
$\displaystyle A_b \sim \left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0 & 4-a & 1-a & 5-6a \\ 
0 & 0 & -(3+a) & -6(3+a) \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31744}%
% latex2html id marker 31744
$\displaystyle A_b \sim \left[
\begin{array}{cccc}
1 & 1 & 1 & 6 \\ 
0 & 7 & 4 & 23 \\ 
0 & 0 & 0 & 0 \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31748}%
% latex2html id marker 31748
$\displaystyle A_b \sim \left[
\begin{array}{cccc}
1 & 0 & \frac{3}{7} & \frac{19}{7} \\ 
0 & 7 & 4 & 23 \\ 
0 & 0 & 0 & 0 \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31750}%
% latex2html id marker 31750
$\displaystyle \left[
\begin{array}{c}
x_1 \\  x_2 \\  x_3
\end{array}
\right] = \left[
\begin{array}{c}
\frac{19}{7} \\  23 \\  0
\end{array}
\right] + x_3\,\left[
\begin{array}{c}
-\frac{3}{7} \\  -4 \\  1
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31758}%
% latex2html id marker 31758
$\displaystyle A_b \sim \left[
\begin{array}{cccc}
1 & 1 & 1 & 6 \\ 
0 & 4-a & 1-a & 5-6a \\ 
0 & 0 & -(3+a) & -6(3+a) \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath31788}%
\begin{displaymath}
% latex2html id marker 31788
\left[
\begin{array}{cc}
2 & -1 \\  -1 & 2
\end{array}\right].\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31839}%
% latex2html id marker 31839
$\displaystyle \det(A-\lambda\, I)=\left| \begin{array}{cccc}
a_{11}-\lambda & a_{12} & \cdots & a_{1n}   \\ 
a_{21} & a_{22}-\lambda & \cdots  & a_{2n}   \\ 
\vdots & \vdots & \ddots & \vdots   \\  
a_{n1} & a_{n2} & \cdots  & a_{nn}-\lambda  
\end{array} \right|=0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline31851}%
% latex2html id marker 31851
$ \sigma_1 = {\rm tr\,}A,\sigma_n = \det A.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31858}%
% latex2html id marker 31858
$\displaystyle A=\left[ \begin{array}{ccc}
3 & -2 & -4 \\ 
-1 & 1 & 1 \\ 
1 & -2 & -2
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31860}%
% latex2html id marker 31860
$\displaystyle \det(A-\lambda\, I)=\left|\begin{array}{ccc}
3-\lambda & -2 & -4 \\ 
-1 & 1-\lambda & 1 \\ 
1 & -2 & -2-\lambda
\end{array}\right|=-\lambda^3+2\,\lambda^2+\lambda-2=0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31886}%
% latex2html id marker 31886
$\displaystyle \left[\begin{array}{c}
x_1 \\  -x_1 \\  x_1
\end{array}\right]=x_1\left[\begin{array}{c}
1 \\  -1 \\  1
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31908}%
% latex2html id marker 31908
$\displaystyle \left[\begin{array}{c}
x_1 \\  0 \\  x_1
\end{array}\right]=x_1\left[\begin{array}{c}
1 \\  0 \\  1
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay31930}%
% latex2html id marker 31930
$\displaystyle \left[\begin{array}{c}
-2\,x_2 \\  x_2 \\  -x_2
\end{array}\right]=x_2\left[\begin{array}{c}
-2 \\  1 \\  -1
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{paragraph}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32065}%
% latex2html id marker 32065
$\displaystyle \boldsymbol{x}_j = [x_{ij}] = \left[\begin{array}{c} 
x_{1j} \\  x_{2j} \\  \vdots \\  x_{nj}\end{array} \right],
\hspace{1cm}j=1,2,\ldots, n,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32071}%
% latex2html id marker 32071
$\displaystyle X = \left[\begin{array}{cccc} x_{11} & x_{12} &
\cdots & x_{1n} \\  x_{21} & x_{22} & \cdots & x_{2n} \\  \vdots &
\vdots & \ddots & \vdots \\  x_{n1} & x_{n2} & \cdots & x_{nn}
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32075}%
% latex2html id marker 32075
$\displaystyle A\,X=A\, \left[\begin{array}{cccc} 
\boldsymbol{x}_1 & \boldsymbol{x}_2&  \ldots & \boldsymbol{x}_n
\end{array} \right]= \left[\begin{array}{cccc} 
A\, \boldsymbol{x}_1 & A\,\boldsymbol{x}_2 &  \ldots & A\, \boldsymbol{x}_n
\end{array} \right]=$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32077}%
% latex2html id marker 32077
$\displaystyle \left[\begin{array}{cccc} 
\lambda_1\, \boldsymbol{x}_1 & \lambda_2\, \boldsymbol{x}_2 & \ldots & \lambda_n\, \boldsymbol{x}_n
\end{array} \right]= [\lambda_j\, x_{ij}]=  [x_{ij}]\,
[\lambda_j\,\delta_{ij}]= X\, [\lambda_j\,\delta_{ij}]. $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32085}%
% latex2html id marker 32085
$\displaystyle X^{-1}A\,X=[\lambda_j\,
\delta_{ij}]=\left[\begin{array}{cccc}
\lambda_1  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{paragraph}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32089}%
% latex2html id marker 32089
$\displaystyle A=\left[
\begin{array}{ccc} 
1 & 0 & 0 \\ 
0 & 2 & -1 \\ 
0 & -1 & 2
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32095}%
% latex2html id marker 32095
$\displaystyle \left[\begin{array}{c}
0 \\  x_2 \\  -x_2
\end{array}\right]=x_2\left[\begin{array}{c}
0 \\  1 \\  -1
\end{array}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32119}%
% latex2html id marker 32119
$\displaystyle \left[\begin{array}{c}          x_1 \\  x_2 \\  x_2 \end{array}\right]=x_1\,\left[\begin{array}{c}         1 \\  0 \\  0 \end{array}\right]+\,x_2\, \left[\begin{array}{c}         0 \\  1 \\  1 \end{array}\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32121}%
% latex2html id marker 32121
$\displaystyle \left[\begin{array}{c}
1 \\  0 \\  0
\end{array}\right], \quad \quad \quad \left[\begin{array}{c}
0 \\  1 \\  1
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{paragraph}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32226}%
% latex2html id marker 32226
$\displaystyle A=X\,[\lambda_j\,
\delta_{ij}]\,X^{-1}=X\,\left[\begin{array}{cccc}
\lambda_1  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n
\end{array}\right]\,X^{-1}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32228}%
% latex2html id marker 32228
$\displaystyle A^2=X\,\left[\begin{array}{cccc}
\lambda_1  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n
\end{array}\right]\,X^{-1}X\,\left[\begin{array}{cccc}
\lambda_1  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n
\end{array}\right]\,X^{-1}=$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 32230
$\displaystyle = X\,\left[\begin{array}{cccc}
\lambda_1^2  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2^2  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n^2
\end{array}\right]\,X^{-1}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32232}%
% latex2html id marker 32232
$\displaystyle A^k= 
X\,\left[\begin{array}{cccc} 
\lambda_1^k  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2^k  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n^k
\end{array}\right]\,X^{-1},$%
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{\newpage\clearpage
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% latex2html id marker 32237
$\displaystyle p(A)=
X\,\left[\begin{array}{cccc} 
p(\lambda_1)  &  0  &  \cdots  &  0  \\ 
0  &  p(\lambda_2)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  p(\lambda_n)
\end{array}\right]\,X^{-1}.$%
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% latex2html id marker 32314
$\displaystyle Q=\left[ \begin{array}{cc}
q_{11} & q_{12}  \\ 
q_{21}  & q_{22}
\end{array} \right]$%
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% latex2html id marker 32316
$\displaystyle \left[ \begin{array}{cc}
q_{11} & q_{12}  \\ 
q_{21}  & q_{22}
\end{array} \right]\,\left[ \begin{array}{c}
1 \\  0
\end{array} \right]=\left[ \begin{array}{c}
q_{11}  \\   q_{21}
\end{array} \right],\hspace{1cm}\left[ \begin{array}{cc}
q_{11} & q_{12}  \\ 
q_{21}  & q_{22}
\end{array} \right]\,\left[ \begin{array}{c}
0 \\  1
\end{array} \right]=\left[ \begin{array}{c}
q_{12}  \\   q_{22}
\end{array} \right]$%
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{\newpage\clearpage
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% latex2html id marker 32335
$\displaystyle Q=
\left[ \begin{array}{cc} 
\cos {\varphi } & -\sin {\varphi } \\ 
\sin {\varphi } &  \cos {\varphi }
\end{array} \right],$%
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{\newpage\clearpage
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% latex2html id marker 32341
$\displaystyle \left| \begin{array}{cc} 
\cos {\varphi }-\lambda & -\sin {\varphi } \\ 
\sin {\varphi } &  \cos {\varphi }-\lambda
\end{array} \right| = {{\lambda}^2} - 2\,\lambda\,\cos \varphi + 1 =
0.$%
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{\newpage\clearpage
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% latex2html id marker 32356
$\displaystyle Q=
\left[ \begin{array}{cc} 
\cos {\varphi } & \sin {\varphi } \\ 
\sin {\varphi } &  -\cos {\varphi }
\end{array} \right],$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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% latex2html id marker 32364
$\displaystyle \left[ \begin{array}{c}
\cos\frac{\varphi }{2} \\    \sin\frac{\varphi }{2}
\end{array} \right] ,\hspace{1cm} \left[ \begin{array}{c}
-\sin\frac{\varphi }{2}  \\    \cos\frac{\varphi }{2}
\end{array} \right].$%
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{\newpage\clearpage
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% latex2html id marker 32368
$\displaystyle X^{-1}Q\,X=\left[\begin{array}{cc}
1  &  0  \\ 
0  &  -1 
\end{array}\right]$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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% latex2html id marker 32370
$ k={\rm tg}\,\frac{\varphi }{2}$%
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% latex2html id marker 32399
$\displaystyle \left[
\begin{array}{ccc} 
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 
\end{array} \right],\hspace{.5cm}
\left[  \begin{array}{ccc} 
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & -1 
\end{array} \right],\hspace{.5cm}
\left[  \begin{array}{ccc} 
1 & 0 & 0 \\ 
0 & -1 & 0 \\ 
0 & 0 & -1 
\end{array} \right],\hspace{.5cm}
\left[  \begin{array}{ccc} 
-1 & 0 & 0 \\ 
0 & -1 & 0 \\ 
0 & 0 & -1 
\end{array} \right].$%
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{\newpage\clearpage
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% latex2html id marker 32416
$\displaystyle Q\,\boldsymbol{x}=\boldsymbol{x},\hspace{.5cm}{\rm ili}\hspace{.5cm}Q\,\boldsymbol{x}=-\boldsymbol{x}.$%
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{\newpage\clearpage
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% latex2html id marker 32434
$\displaystyle \left[
\begin{array}{ccc} 
1 & 0 & 0 \\ 
0 & \cos {\varphi } & -\sin {\varphi } \\ 
0 & \sin {\varphi } &  \cos {\varphi } 
\end{array} \right],\hspace{.5cm}
\left[  \begin{array}{ccc} 
-1 & 0 & 0 \\ 
0 & \cos {\varphi } & -\sin {\varphi } \\ 
0 & \sin {\varphi } &  \cos {\varphi } 
\end{array} \right].$%
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% latex2html id marker 32595
\boldsymbol{y}=\left[
\begin{array}{c}
  y_1(x) \\  y_2(x) \\  \vdots \\  y_n(x) 
\end{array}
\right] ,\;\; \boldsymbol{y}_0=
\left[
\begin{array}{c}
  y_{01} \\  y_{02} \\  \vdots \\  y_{0n}
\end{array}
\right] ,\;\; 
\boldsymbol{f}(x)=
\left[
\begin{array}{c}
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\end{array}
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\begin{displaymath}
% latex2html id marker 32597
  A(x)=\left[
\begin{array}{cccc}
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 a_{21}(x) & a_{22}(x) & \cdots & a_{2n}(x) \\ 
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% latex2html id marker 32636
\boldsymbol{x}(t) = \left[\begin{array}{c}
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\end{array}
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\dot{\boldsymbol{x}}(t) = \left[
\begin{array}{c}
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\end{array}\right] ,\quad 
\boldsymbol{x}_0 = \left[
\begin{array}{c}
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\end{array}\right] ,\quad 
X =\left[
\begin{array}{cccc}
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 0 & 0 & 0 & 1 \\ 
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{\newpage\clearpage
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% latex2html id marker 32638
$\displaystyle A = \left[\begin{array}{cccc}
\frac{\partial\varphi_1(C)}{\partial x_1} & \frac{\partial\varphi_1(C)}{\partial x_2} & \cdots & \frac{\partial\varphi_1(C)}{\partial x_n} \\ 
\frac{\partial\varphi_2(C)}{\partial x_1} & \frac{\partial\varphi_2(C)}{\partial x_2} & \cdots & \frac{\partial\varphi_2(C)}{\partial x_n} \\ 
\vdots   & \vdots   & \ddots & \vdots   \\ 
\frac{\partial\varphi_n(C)}{\partial x_1} & \frac{\partial\varphi_n(C)}{\partial x_2} & \cdots & \frac{\partial\varphi_n(C)}{\partial x_n}
\end{array}\right],$%
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{\newpage\clearpage
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\begin{displaymath}
% latex2html id marker 32798
C_1\,\boldsymbol{u}_1(x_0)+C_2\,\boldsymbol{u}_2(x_0)+\cdots +C_n\,\boldsymbol{u}_n(x_0)=\left[  
\begin{array}{c}
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    C_2 \\ 
    \vdots \\ 
    C_n
\end{array}
\right].\end{displaymath}%
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\begin{displaymath}
% latex2html id marker 32847
\left[
\begin{array}{c}
 y(x) \\ 
 z(x)
\end{array}
\right]=C_1\,\left[
\begin{array}{c}
  e^{2x} \\ 
  -e^{2x}
\end{array}
\right]+C_2\,\left[
\begin{array}{c}
   e^{-3x} \\ 
   \frac{1}{4} \,e^{-3x}
\end{array}
\right]+\left[   
    \begin{array}{c}
     x+x^2 \\   
    -\frac{1}{2} x^2
\end{array}
\right],\end{displaymath}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath32849}%
\begin{displaymath}
% latex2html id marker 32849
\boldsymbol{u}_1=e^{2x}\,\left[
\begin{array}{r}
  1 \\ 
  -1
\end{array}
\right],\;\; \boldsymbol{u}_2=e^{-3x}\,\left[
\begin{array}{r}
   1 \\ 
   \frac{1}{4}
  \end{array}
\right],\;\; \boldsymbol{y}_p=x\left[
\begin{array}{r}
     1 \\ 
     0
\end{array}
\right] +x^2\,\left[
\begin{array}{r}
      1 \\ 
     -\frac{1}{2}
\end{array}
\right]. \end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 25029
\includegraphics{m3prsistdifjed.eps}%
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\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 32864
$\displaystyle \boldsymbol{C}= \left[ \begin{array}{c} C_1 \\  C_2 \\  \vdots \\  C_n
\end{array} \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath32877}%
\begin{displaymath}
% latex2html id marker 32877
\left[
\begin{array}{c}
 y(x) \\ 
 z(x)
\end{array}
\right] '= \left[ \begin{array}{rr}
          -2 & -4 \\ 
          -1 & 1
          \end{array}
      \right] \left[ \begin{array}{c}
              y(x) \\ 
              z(x)
             \end{array}
          \right].\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32879}%
% latex2html id marker 32879
$\displaystyle \left[
\begin{array}{c}
y(x) \\ 
z(x)
\end{array}
\right] = e^{\lambda x} \left[ \begin{array}{c}
C_1 \\ 
C_2
\end{array}
\right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32905}%
% latex2html id marker 32905
$\displaystyle \left[ \begin{array}{r} 1 \\  -1 \end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32913}%
% latex2html id marker 32913
$\displaystyle \left[ \begin{array}{r} 1 \\  \frac{1}{4} \end{array} \right]. $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline32943}%
% latex2html id marker 32943
$ \left[ \begin{array}{cc} 1 & 0 \\  0 & 1
\end{array} \right]$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath32947}%
\begin{displaymath}
% latex2html id marker 32947
\left[ 
\begin{array}{c} y \\  z \end{array} 
\right] =e^x \left[
\begin{array}{c} C_1 \\  C_2 \end{array} 
\right] = C_1\,e^x \left[ 
\begin{array}{c} 1 \\  0 \end{array} 
\right] +C_2\,e^x \left[ 
\begin{array}{c} 0 \\  1 \end{array} 
\right],\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline32956}%
% latex2html id marker 32956
$ \left[ \begin{array}{cc} 1 & 1 \\  0 & 1
\end{array} \right],$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline32970}%
% latex2html id marker 32970
$ \; C_1 \left[ \begin{array}{c} 1 \\  0 \end{array} \right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay32988}%
% latex2html id marker 32988
$\displaystyle \left[ \begin{array}{c} y \\  z \end{array} \right] =C_1\,e^x
\left[ \begin{array}{c} 1 \\  0 \end{array} \right] +C_2\,e^x
\left[ \begin{array}{c} x \\  1 \end{array} \right] .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33029}%
% latex2html id marker 33029
$\displaystyle \boldsymbol{y}= C_1\,e^x \left[ 
\begin{array}{c}
1 \\ 
0 \\ 
0
\end{array}
\right] + C_2\,e^x \left[ 
\begin{array}{c}
x \\ 
1 \\ 
0
\end{array}
\right] + C_3\,e^x \left[ 
\begin{array}{c}
\frac{1}{2}\,x^2 \\ 
x \\ 
1
\end{array}
\right].$%
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\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33105}%
% latex2html id marker 33105
$\displaystyle A=\left[ \begin{array}{ccc}
3 & -2 & -4 \\ 
-1 & 1 & 1 \\ 
1 & -2 & -2
\end{array} \right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33109}%
% latex2html id marker 33109
$\displaystyle \left[\begin{array}{c}
1 \\  -1 \\  1
\end{array}\right],\quad 
\left[\begin{array}{c}
1 \\  0 \\  1
\end{array}\right],\quad 
\left[\begin{array}{c}
-2 \\  1 \\  -1
\end{array}\right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33111}%
% latex2html id marker 33111
$\displaystyle X = \left[\begin{array}{ccc}
1 & 1 & -2 \\ 
-1 & 0 & 1 \\ 
1 & 1 & -1 \\ 
\end{array}\right],\quad 
X^{-1} = \left[\begin{array}{ccc}
-1 & -1 & 1 \\ 
0 & 1 & 1 \\ 
-1 & 0 & 1 \\ 
\end{array}\right],\quad 
\boldsymbol{f}= \left[\begin{array}{c}
x\,\sin x \\  \cos 2\,x \\  x^2-1
\end{array}\right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33113}%
% latex2html id marker 33113
$\displaystyle X^{-1}\,A\,X = \left[\begin{array}{ccc} 
1 & 0 & 0 \\  0 & -1 & 0 \\  0 & 0 & 2 \end{array}\right],\quad 
X^{-1}\,\boldsymbol{f}= \left[\begin{array}{c}
-1 + x^2 - \cos 2\,x - x\,\sin x \\  -1 + x^2 + \cos 2\,x \\  
-1 + x^2 - x\,\sin x \end{array}\right].$%
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{\newpage\clearpage
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% latex2html id marker 25077
\includegraphics{m3prsustdifdijag.eps}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay33220}%
% latex2html id marker 33220
$\displaystyle \left[\begin{array}{ccc}
\lambda & 0 & 0 \\ 
0 & \lambda & 0 \\ 
0 & 0 & \lambda
\end{array}\right],\hspace{.5cm}\left[\begin{array}{ccc}
\lambda & 0 & 0 \\ 
0 & \lambda & 1 \\ 
0 & 0 & \lambda
\end{array}\right],\hspace{.5cm}\left[\begin{array}{ccc}
\lambda & 1 & 0 \\ 
0 & \lambda & 1 \\ 
0 & 0 & \lambda
\end{array}\right].$%
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\lthtmlcheckvsize\clearpage}

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% latex2html id marker 25167
\includegraphics{m3prnapetost.eps}%
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% latex2html id marker 25173
\includegraphics{m3prnapetost1.eps}%
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\lthtmldisplayA{displaymath33942}%
\begin{displaymath}
% latex2html id marker 33942
\begin{cases} \frac{\textstyle{\partial \varphi(x,t)}}{\textstyle{\partial t}} =   \frac{\textstyle{\partial}}{\textstyle{\partial x}} \left(p(x)\,\frac{\textstyle{\partial    u(x,t)}}{\textstyle{\partial x}}\right) + f(x,t),& \\ [5mm]  u(0,t)=0,\hspace{1cm}   \frac{\textstyle{\partial u(\ell,t)}}{\textstyle{\partial x}}=0.   \end{cases}\end{displaymath}%
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\stepcounter{paragraph}
{\newpage\clearpage
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% latex2html id marker 34022
$\displaystyle \begin{array}{l}
\rho\,\frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =
\frac{\textstyle{\partial}}{\textstyle{\partial
x}}\left(p\,\frac{\textstyle{\partial 
u(x,t)}}{\textstyle{\partial x}}\right)+f(x,t),\\ [3mm] 
u(0,t)=a,\hspace{1cm}u(\ell,t)=b. 
\end{array}$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath34050}%
\begin{displaymath}
% latex2html id marker 34050

\begin{cases}
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y(0) = 1,\quad y'(1) = 2.
\end{cases}
\end{displaymath}%
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{\newpage\clearpage
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\begin{displaymath}
% latex2html id marker 34076

\begin{cases}
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\end{cases}
\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 34358
$\displaystyle u(x) =  \begin{cases}   \frac{1}{60}\,x,& \quad \text{za } x\in [0,\frac{\ell}{3}], \\  -\frac{1}{30}\,x + \frac{\ell}{60},& \quad \text{za } x\in [ \frac{\ell}{3},\frac{2\,\ell}{3}], \\  \frac{1}{60}\,x -\frac{\ell}{60},& \quad \text{za } x\in [\frac{2\,\ell}{3},\ell]. \end{cases}$%
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\stepcounter{paragraph}
{\newpage\clearpage
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% latex2html id marker 34419
$\displaystyle G(x,\xi) = \left\{
\begin{array}{ll}
A\,x, & 0<x<\xi, \\ 
C\,(x - \ell), & \xi<x<\ell.
\end{array}\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34433}%
% latex2html id marker 34433
$\displaystyle G(x,\xi) = \left\{
\begin{array}{ll}
\frac{\ell-\xi}{\ell\,p}\,x, & 0<x<\xi<\ell, \\ [2mm]
\frac{\xi}{\ell\,p}\,(\ell-x), & 0<\xi<x<\ell.
\end{array}\right.$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{paragraph}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34454}%
% latex2html id marker 34454
$\displaystyle G(x,\xi) = \left\{
\begin{array}{ll}
C_1\,u_1(x), & 0<x<\xi, \\ [2mm]
C_2\,u_2(x), & \xi<x<\ell.
\end{array}\right.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34471}%
% latex2html id marker 34471
$\displaystyle \left|
\begin{array}{ll}
u_1(\xi) & -u_2(\xi) \\ [1mm]
u'_1(\xi) & -u'_2(\xi)
\end{array}\right| = -\left|
\begin{array}{ll}
u_1(\xi) & u_2(\xi) \\ [1mm]
u'_1(\xi) & u'_2(\xi)
\end{array}\right| = -W(\xi),$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34475}%
% latex2html id marker 34475
$\displaystyle W(x) = \left|
\begin{array}{ll}
u_1(x) & u_2(x) \\ [1mm]
u'_1(x) & u'_2(x)
\end{array}\right|.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34481}%
% latex2html id marker 34481
$\displaystyle G(x,\xi) = \begin{cases} \frac{\textstyle{-u_2(\xi)}}{\textstyle{p(\xi)\,W(\xi)}}\,u_1(x),\quad  0<x<\xi<\ell,& \\ [3mm]  \frac{\textstyle{-u_1(\xi)}}{\textstyle{p(\xi)\,W(\xi)}}\,u_2(x),\quad  0<\xi<x<\ell.    \end{cases}$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath34484}%
\begin{displaymath}
% latex2html id marker 34484

\begin{cases}
\displaystyle \left(\frac{1}{1+x^2}\,u'(x)\right)' = 0,& \\ [2mm]
u'(0) = 0,\qquad u(\ell) + k\,u'(\ell) = 0.
\end{cases}
\end{displaymath}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath34507}%
\begin{displaymath}
% latex2html id marker 34507
G(x,\xi) = 
\begin{cases}
  \displaystyle {\frac{3\,\ell + {{\ell}^3} + 3\,k + 3\,{{\ell}^2}\,k
      - 3\,{\xi} -
      {{{\xi}}^3}}{3}},&  \text{za }0<x<\xi<\ell, \\ [2mm]
  \displaystyle {\frac{3\,\ell + {{\ell}^3} + 3\,k + 3\,{{\ell}^2}\,k
      - x\,\left( 3 + {x^2} \right) }{3}},& \text{za }0<x<\xi<\ell.
\end{cases}
\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 34536
$\displaystyle u(x) \approx \sum_{i=1}^n G(x,\xi_i)\,f(\xi_i)\,\Delta\,\xi_i.$%
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{\newpage\clearpage
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% latex2html id marker 34599
$\displaystyle W(x)=\left|
\begin{array}{ll}
x & \ell{}-x \\ 
1 & -1
\end{array}\right| = -\ell{}.$%
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% latex2html id marker 25459
\includegraphics{m3koncdjelpr.eps}%
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\begin{displaymath}
% latex2html id marker 34625
\begin{cases} \frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}},& \\ [3mm] u(0,t)=0,\hspace{1cm}   u(\ell,t)=0,   \end{cases}\end{displaymath}%
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\begin{displaymath}
% latex2html id marker 34642
\begin{cases}       X''(x) + \lambda^2\,X(x) = 0,& \\  X(0) = 0,\quad X(\ell) = 0,     \end{cases}\end{displaymath}%
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\begin{displaymath}
% latex2html id marker 34682

\begin{cases}
  X''(x) + \lambda^2\,X(x) = 0,& \\  X(0) = 0,\quad X'(\ell) = 0.  
\end{cases}
\end{displaymath}%
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\begin{displaymath}
% latex2html id marker 34705

\begin{cases}
  X''(x) + \lambda^2\,X(x) = 0,& \\  X(0) - X'(0) = 0,\quad X'(\ell) = 0.  
\end{cases}
\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 34717
$\displaystyle {\rm tg}\,\ell\,\lambda = \frac{1}{\lambda}.$%
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{\newpage\clearpage
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% latex2html id marker 34784
$\displaystyle \int_{-\ell}^{\ell}\,\sin
\frac{n\,\pi\,x}{\ell}\,\sin \frac{m\,\pi\,x}{\ell}\,dx = \left\{
\begin{array}{rl} \ell, & \mbox{ako je $m=n$} \\ 
0, & \mbox{ako je $m\neq n$}
\end{array}
\right.,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay34786}%
% latex2html id marker 34786
$\displaystyle \int_{-\ell}^{\ell}\,\cos\frac{n\,\pi\,x}{\ell}\,\cos \frac{m\,\pi\,x}{\ell}\,dx =
\left\{  \begin{array}{rl} \ell, & \mbox{ako je $m=n$} \\ 
0, & \mbox{ako je $m\neq n$}
\end{array}
\right.,$%
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% latex2html id marker 34803
\begin{cases}    \displaystyle    a_0 = \frac{1}{\ell}\int_{-\ell}^{\ell}\,f(x)\,dx,\quad\quad a_n =  \frac{1}{\ell}\int_{-\ell}^{\ell}\,f(x)\,\cos\frac{n\,\pi\,x}{\ell}\,dx,& \\ [8mm]    \displaystyle    b_n = \frac{1}{\ell}\int_{-\ell}^{\ell}\, f(x)\,\sin\frac{n\,\pi\,x}{\ell}\,dx,\quad\quad n=1,2,3,\ldots\  .   \end{cases}\end{displaymath}%
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% latex2html id marker 25505
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% latex2html id marker 34965
\begin{cases} \frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}},& \\ [3mm] u(0,t)=0,\hspace{1cm}   u(\ell,t)=0,& \\ [1mm] u(x,0) =  \alpha(x),\hspace{1cm}   \frac{\textstyle{\partial u(x,0)}}{\textstyle{\partial t}} = \beta(x).   \end{cases}\end{displaymath}%
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% latex2html id marker 25545
\includegraphics{m3fourneppro.eps}%
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\begin{displaymath}
% latex2html id marker 35200
\begin{cases} \frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}},& \\ [3mm]  u(0,t)=a,\hspace{1cm}u(\ell,t)=b,& \\ [1mm]  u(x,0) = \alpha(x),\hspace{1cm}   \frac{\textstyle{\partial u(x,0)}}{\textstyle{\partial t}} = \beta(x).   \end{cases}\end{displaymath}%
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% latex2html id marker 35202
\begin{cases} \frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}} + h(x),& \\ [3mm] u(0,t)=0,\hspace{1cm}   u(\ell,t)=0,& \\ [1mm] u(x,0) =  \alpha(x),\hspace{1cm}   \frac{\textstyle{\partial u(x,0)}}{\textstyle{\partial t}} = \beta(x).   \end{cases}\end{displaymath}%
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% latex2html id marker 35235
\begin{cases} \frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial t^2}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}} + h(x,t),& \\ [3mm] u(0,t)=0,\hspace{1cm}   u(\ell,t)=0,& \\ [1mm] u(x,0) = \alpha(x),\hspace{1cm}   \frac{\textstyle{\partial u(x,0)}}{\textstyle{\partial t}} = \beta(x),   \end{cases}\end{displaymath}%
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% latex2html id marker 35269
$\displaystyle \left.
\begin{array}{l}
D''_n(t) + \left(c\,\frac{n\,\pi}{\ell}\right)^2D_n(t) =
A_n(t),\\ [3mm] D_n(0) = \alpha_n \hspace{1cm}D'_n(0) = \beta_n 
\end{array}\right\}\hspace{3cm} n=1,2,3,\ldots $%
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% latex2html id marker 35276

\begin{cases}
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  u(\ell,t)=0,& \\ [1mm] u(x,0) = 0,\hspace{1cm}
  \frac{\textstyle{\partial u(x,0)}}{\textstyle{\partial t}} = 0.
\end{cases}
\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 35297
\begin{cases} \frac{\textstyle{\partial u(x,t)}}{\textstyle{\partial t}} =   c^2\,\frac{\textstyle{\partial^2    u(x,t)}}{\textstyle{\partial x^2}},& \\ [3mm] u(0,t)=0,\hspace{1cm}   u(\ell,t)=0,& \\ [1mm] u(x,0) = g(x).   \end{cases}\end{displaymath}%
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% latex2html id marker 35366

\begin{cases}
\frac{\textstyle{\partial u(x,t)}}{\textstyle{\partial t}} =
c^2\,\frac{\textstyle{\partial^2 u(x,t)}}{\textstyle{\partial x^2}},& \\ [3mm] 
u(0,t)=0,\qquad u(\ell,t)=a\,t,& \\ [1mm] 
u(x,0) = 0.
\end{cases}
\end{displaymath}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath35372}%
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% latex2html id marker 35372

\begin{cases}
\displaystyle  \frac{{\partial v(x,t)}}{{\partial t}} =
  c^2\,\frac{{\partial^2 v(x,t)}}{{\partial x^2}} - 
\frac{a\,x}{\ell},& \\ [3mm]
  v(0,t)=0,\qquad v(\ell,t)=0,& \\ [1mm]
  v(x,0) = 0.
\end{cases}
\end{displaymath}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath35378}%
\begin{displaymath}
% latex2html id marker 35378

\begin{cases}
\displaystyle  c^2\,w''(x) - \frac{a\,x}{\ell} = 0,& \\ 
w(0) = 0,\qquad w(\ell) = 0.
\end{cases}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath35384}%
\begin{displaymath}
% latex2html id marker 35384

\begin{cases}
  \displaystyle \frac{\textstyle{\partial z(x,t)}}{\textstyle{\partial
      t}} = c^2\,\frac{\textstyle{\partial^2
      z(x,t)}}{\textstyle{\partial x^2}} -
  \frac{a\,x}{\ell},& \\ [3mm]
  z(0,t)=0,\qquad z(\ell,t)=0,& \\ [1mm]
  \displaystyle z(x,0) = \frac{a\,x^3}{6\,c^2\,\ell} - \frac{\left(
      a\,\ell\,x\right) }{6\,c^2}.
\end{cases}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline35517}%
% latex2html id marker 35517
$ x\in [0,\ell]\setminus I,$%
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{\newpage\clearpage
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% latex2html id marker 25627
\includegraphics{m3lmstr103.eps}%
\lthtmlpictureZ
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\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35715}%
% latex2html id marker 35715
$\displaystyle f(x) = \left\{
\begin{array}{ll}
0, & \mbox{za  }0 \leqslant{} x < \frac{\ell{}}{2}, \\ [2mm]
1, & \mbox{za }\frac{\ell{}}{2} \leqslant{} x \leqslant{} \ell{}.
\end{array}\right.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35749}%
% latex2html id marker 35749
$\displaystyle \Phi(x) = \int_x^{\ell} f(\xi)\,d\xi = \left\{
\begin{array}{ll}
\int_{\frac{\ell{}}{2}}^{\ell} d\xi, & \mbox{za  }0 \leqslant{} x <
\frac{\ell{}}{2}, \\ [2mm] 
\int_x^{\ell} d\xi, & \mbox{za }\frac{\ell{}}{2} \leqslant{} x
\leqslant{} \ell{} 
\end{array}\right. = \left\{
\begin{array}{ll}
\frac{\ell{}}{2}, & \mbox{za  }0 \leqslant{} x < \frac{\ell{}}{2}, \\ [2mm]
\ell{} - x, & \mbox{za }\frac{\ell{}}{2} \leqslant{} x \leqslant{} \ell{}.
\end{array}\right.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35757}%
% latex2html id marker 35757
$\displaystyle u(x) = \frac{1}{p}\,\int_0^x \Phi(\xi)\,d\xi = \left\{
\begin{array}{ll}
\frac{1}{p}\,\int_0^x \frac{\ell}{2}\,d\xi, & \mbox{za  }0 \leqslant{}
x < \frac{\ell{}}{2}, \\ [2mm] 
\frac{1}{p}\,\int_0^{\frac{\ell}{2}} \frac{\ell}{2}\,d\xi +
\frac{1}{p}\,\int_{\frac{\ell}{2}}^x (\ell - \xi)\,d\xi, & \mbox{za
}\frac{\ell{}}{2} \leqslant{} x \leqslant{} \ell{} 
\end{array}\right.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35759}%
% latex2html id marker 35759
$\displaystyle = \left\{
\begin{array}{ll}
\frac{\ell{}\,x}{2\,p}, & \mbox{za  }0 \leqslant{} x <
\frac{\ell{}}{2}, \\ [2mm] 
\frac{8\,\ell{}\,x - \ell^2 - 4\,x^2}{8\,p}, & \mbox{za
}\frac{\ell{}}{2} \leqslant{} x \leqslant{} \ell{}. 
\end{array}\right.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath35883}%
\begin{displaymath}
% latex2html id marker 35883

\begin{cases}
  w''(t) - w(t) = 0,& \\ 
w(0)=w_0,\qquad w'(T) = 0.
\end{cases}
\end{displaymath}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35885}%
% latex2html id marker 35885
$\displaystyle w(t) = w_0\,\frac{{\rm ch}\,(t - T)}{{\rm ch}\,T}.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay35889}%
% latex2html id marker 35889
$\displaystyle \Phi(u) = \frac{w_0^2\,{\rm sh}\,2\,T}{2\,{\rm ch}\,^2T}.$%
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\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36054}%
% latex2html id marker 36054
$\displaystyle u' = {\rm ctg}\,\psi{},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36058}%
% latex2html id marker 36058
$\displaystyle 1+{u'}^2 = 1+{\rm ctg}\,^2\psi{} = \frac{1}{\sin^2\psi{}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36066}%
% latex2html id marker 36066
$\displaystyle du = u'\,dx = {\rm ctg}\,\psi{}\,dx.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36141}%
% latex2html id marker 36141
$\displaystyle {\cal P}(x,y) = \left[     \begin{array}{cc}   p_{11}(x,y) &  p_{21}(x,y) \\    p_{12}(x,y) & p_{22}(x,y)     \end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36149}%
% latex2html id marker 36149
$\displaystyle \left[
\begin{array}{cc}
p_{11}(x,y) &  p_{21}(x,y) \\ 
p_{12}(x,y) & p_{22}(x,y)
\end{array}\right]\,\left[
\begin{array}{c}
n_x \\ 
n_y
\end{array}\right] = \left[
\begin{array}{c}
p_{11}(x,y)\,n_x + p_{12}(x,y)\,n_y \\ 
p_{21}(x,y)\,n_x + p_{22}(x,y)\,n_y
\end{array}\right].$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36168}%
% latex2html id marker 36168
$\displaystyle \int_{\partial\Omega}\vec{a}\cdot \vec{n}\,ds =
\int\!\!\int_{\Omega} {\rm div\,}\vec{a}\,dxdy,$%
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\stepcounter{paragraph}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline36256}%
% latex2html id marker 36256
$ \psi_z = p\,{\rm tg}\,\alpha.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline36258}%
% latex2html id marker 36258
$ {\rm tg}\,\alpha = \frac{\partial
u}{\partial\vec{n}},$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36269}%
% latex2html id marker 36269
$\displaystyle \psi = p\,\frac{\partial u}{\partial\vec{n}} = p\,{\rm grad\,}u\cdot
\vec{n} = 
p\,\left(\frac{\partial u}{\partial x}\,n_x + \frac{\partial
u}{\partial y}\,n_y\right),$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36271}%
% latex2html id marker 36271
$\displaystyle \iint_{D}\,\rho\,\frac{\partial^2 u}{\partial t^2}\,dS = \int_{\partial D}\,
p\,{\rm grad\,}u\cdot \vec{n}\,ds + \iint_{D}\,f\,dS.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay36273}%
% latex2html id marker 36273
$\displaystyle \iint_{D}\,\rho\,\frac{\partial^2 u}{\partial t^2}\,dS = \iint_{D}\,
p\,{\rm div\,}({\rm grad\,}u)\,dS + \iint_{D}\,f\,dS.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline36275}%
% latex2html id marker 36275
$ {\rm div\,}({\rm grad\,}u) = \Delta\,u,$%
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% latex2html id marker 36287
$\displaystyle \rho\,\frac{\partial^2 u}{\partial t^2} = {\rm div\,}(p\,{\rm grad\,}u) + f.$%
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% latex2html id marker 36326
$\displaystyle \varphi = \gamma\,u,\hspace{1cm}\psi = p\,\frac{\partial
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% latex2html id marker 36377
$\displaystyle \varphi = \gamma\,\rho,\hspace{1cm}\psi = p\,\frac{\partial
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% latex2html id marker 36385
$\displaystyle \frac{\partial{}\gamma\,\rho(u)}{\partial{}t} = {\rm div\,}(p\,{\rm grad\,}u) + f.$%
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% latex2html id marker 36475
$\displaystyle {\rm div\,}(u\,{\rm grad\,}v) = {\rm grad\,}u\cdot{\rm grad\,}v + u\,\Delta v.$%
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% latex2html id marker 36479
$\displaystyle u\,\Delta u = {\rm div\,}(u\,{\rm grad\,}u) - {\rm grad\,}u\cdot{\rm grad\,}u,$%
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% latex2html id marker 36481
$\displaystyle \iiint_{\Omega} u\,\Delta u\,dV = \iiint_{\Omega} {\rm div\,}(u\,{\rm grad\,}u)\,dV - \iiint_{\Omega} {\rm grad\,} u\cdot{\rm grad\,}u\,dV.$%
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% latex2html id marker 36483
$\displaystyle \iiint_{\Omega} {\rm div\,}(u\,{\rm grad\,}u) \,dV = \iint_{\partial\Omega}
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% latex2html id marker 36485
$\displaystyle \iiint_{\Omega}\,({\rm grad\,}u)^2\,dV = \iiint_{\Omega}\,f\,u\,dV + \iint_{\partial\Omega}\,
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% latex2html id marker 36510
$\displaystyle \iiint_{\Omega}\,({\rm grad\,}w)^2\,dV = 0.$%
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% latex2html id marker 36512
$\displaystyle {\rm grad\,}w = \vec{0},$%
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% latex2html id marker 36543
$\displaystyle -\iiint_{\Omega}\,{\rm div\,}({\rm grad\,}u)\,dV = \iiint_{\Omega}\,f\,dV.$%
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% latex2html id marker 36545
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$\displaystyle \iiint_{\Omega}\,({\rm grad\,}w)^2\,dV = 0.$%
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% latex2html id marker 36567
$\displaystyle {\rm grad\,}w = \vec{0},$%
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% latex2html id marker 36581
$\displaystyle \begin{array}{l}
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u|_{\partial K} =\alpha,
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% latex2html id marker 36600
$\displaystyle r = \sqrt{x^2 + y^2},\hspace{1cm}\varphi = {\rm Arctg}\,\frac{y}{x}.$%
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% latex2html id marker 36656
$\displaystyle \begin{array}{l}
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% latex2html id marker 25919
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% latex2html id marker 36787

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% latex2html id marker 36808
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% latex2html id marker 36816
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% latex2html id marker 36836
\begin{cases}     X''(x) + \lambda^2\,X(x) = 0.& \\  X(0) = 0,\quad X(a) = 0.   \end{cases}\end{displaymath}%
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% latex2html id marker 36842
$\displaystyle Y_n(y) = A_n\,{\rm ch}\,\frac{n\,\pi}{a}\,y +
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% latex2html id marker 36844
$\displaystyle u(x,y) = \sum_{n=1}^{\infty} X_n(x)\,Y_n(y) = \sum_{n=1}^{\infty}
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% latex2html id marker 36848
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% latex2html id marker 36850
$\displaystyle B_n = - A_n\,{\rm cth}\,\frac{n\,\pi\,b}{a}.$%
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% latex2html id marker 36863

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% latex2html id marker 36897

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% latex2html id marker 36901

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% latex2html id marker 36905

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% latex2html id marker 36909

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% latex2html id marker 36962
$\displaystyle \iiint_{\Omega} v\,\Delta v\,dV = \iint_{\partial\Omega} v\,\frac{\partial
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\iiint_{\Omega} ({\rm grad\,}v)^2\,dV,$%
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% latex2html id marker 36968
$\displaystyle a = -\frac{\iiint_{\Omega} ({\rm grad\,}v(P))^2\,dV}{\iiint_{\Omega}\,v(P)^2\,dV} <
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% latex2html id marker 36985

\begin{array}{l}
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% latex2html id marker 37071
$\displaystyle \iint_{\Omega{}} \left({\rm grad\,}v\cdot{\rm grad\,}u + v\,\Delta
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% latex2html id marker 37073
$\displaystyle \iint_{\Omega{}} v\,\Delta u\,dxdy = -\iint_{\Omega{}} {\rm grad\,}
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% latex2html id marker 37079
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v\cdot{\rm grad\,}u\,dxdy = \iint_{\Omega}\,f\,v\,dxdy.$%
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% latex2html id marker 37097
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v\cdot{\rm grad\,}u\,dxdy = \int_{\partial\Omega{}}
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% latex2html id marker 37101
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v\cdot{\rm grad\,}u\,dxdy =  - \iint_{\Omega}\,v\,\Delta u\,dxdy.$%
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% latex2html id marker 37111
$\displaystyle F(w) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}w)^2dxdy - \iint_{\Omega}\,f\,w\,dxdy.$%
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% latex2html id marker 37127
$\displaystyle F(w) = F(u+v) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}u + {\rm grad\,}v)^2dxdy - \iint_{\Omega}\,
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% latex2html id marker 37129
$\displaystyle = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}u)^2dxdy +
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% latex2html id marker 37131
$\displaystyle + \frac{1}{2}\,\iint_{\Omega}\,
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% latex2html id marker 37133
$\displaystyle = F(u) + \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}v)^2dxdy \geqslant F(u).$%
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% latex2html id marker 37139
$ {\rm grad\,}v=\vec{0},$%
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% latex2html id marker 37176
$\displaystyle h(\lambda{}) = F(u+\lambda{}\,v) = \frac{1}{2}\iint_{\Omega}\,({\rm grad\,}u +
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% latex2html id marker 37178
$\displaystyle = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}u)^2dxdy - \iint_{\Omega}\,f\,u\,dxdy$%
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% latex2html id marker 37180
$\displaystyle +
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% latex2html id marker 37182
$\displaystyle + \frac{1}{2}\,\lambda{}^2\,\iint_{\Omega}\,({\rm grad\,}v)^2dxdy.$%
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% latex2html id marker 37198
$\displaystyle \iint_{\Omega}\,{\rm grad\,}u\cdot{\rm grad\,}v\,dxdy - \iint_{\Omega}\,f\,v\,dxdy = 0$%
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% latex2html id marker 37207
\begin{cases}       -\Delta\,u = f,\hspace{1cm}\mbox{na }\Omega{},& \\  u|_{\partial\Omega{}} = 0,&     \end{cases}\end{displaymath}%
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% latex2html id marker 37211
$\displaystyle F(w) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}w)^2\,dxdy - \iint_{\Omega}\,f\,w\,dxdy.$%
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% latex2html id marker 37239
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% latex2html id marker 37256
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% latex2html id marker 37260
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% latex2html id marker 37264
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% latex2html id marker 37634
$\displaystyle \frac{1}{2^{17}} \approx 7.62939\times 10^{-6}, \qquad
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% latex2html id marker 37637
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% latex2html id marker 37641
$\displaystyle f(\lambda{}) = {\rm tg}\,\lambda - \frac{1}{\lambda}.$%
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% latex2html id marker 38230
$\displaystyle \lambda{} = \frac{1}{{\rm tg}\,\lambda{},}$%
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% latex2html id marker 38232
$\displaystyle \varphi(\lambda) = \frac{1}{{\rm tg}\,\lambda},$%
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% latex2html id marker 38239
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$\displaystyle \varphi(\lambda) = {\rm Arctg}\,\frac{1}{\lambda},$%
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% latex2html id marker 38254
$ R\setminus\{0\}.$%
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$\displaystyle \lim_{\lambda{}\rightarrow{}0+} {\rm Arctg}\,\frac{1}{\lambda{}} =
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% latex2html id marker 38272
$\displaystyle {\rm Arctg}\,1 = \frac{\pi}{4},\qquad {\rm Arctg}\,\frac{\pi}{4} = 0.905023 <
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$\displaystyle {\rm Arctg}\,x = 0$%
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% latex2html id marker 26267
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\begin{displaymath}
% latex2html id marker 38720
x_l = \left\{
\begin{array}{ll}
x_{k-1}, & \mbox{ako\  je\  }f(x_{k-1})\,f(x_k)<0, \\ 
x_{k-2}, & \mbox{ako\  je\  }f(x_{k-2})\,f(x_k)<0. \\ 
\end{array}\right.\end{displaymath}%
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{\newpage\clearpage
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\begin{displaymath}
% latex2html id marker 38749
A = \left[
\begin{array}{cccc}
        a_{1\,1} & a_{1\,2} & \cdots & a_{1\,n}   \\ 
        a_{2\,1} & a_{2\,2} & \cdots  & a_{2\,n}  \\ 
        \vdots & \vdots & \ddots & \vdots \\ 
        a_{n\,1} & a_{n\,2} & \cdots  & a_{n\,n}
\end{array}\right],\hspace{1cm}
\boldsymbol{b}=\left[
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b_1 \\  b_2 \\  \vdots \\  b_n
  \end{array}
\right],\hspace{1cm}\boldsymbol{x}=\left[
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x_1 \\  x_2 \\  \vdots \\  x_n
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\right].\end{displaymath}%
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% latex2html id marker 38786
L = \left[
\begin{array}{ccccc}
        0 & 0 & \cdots & 0 & 0  \\ 
        a_{2\,1} & 0 & \cdots  & 0 & 0  \\ 
        \vdots & \vdots & \ddots & \vdots & \vdots \\ 
        a_{n-1\,1} & a_{n-1\,2} & \cdots  & 0 & 0  \\ 
        a_{n\,1} & a_{n\,2} & \cdots  & a_{n\,n-1} & 0
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% latex2html id marker 38788
D = \left[
\begin{array}{cccc}
        a_{1\,1} & 0 & \cdots & 0   \\ 
        0 & a_{2\,2} & \cdots  & 0  \\ 
        \vdots & \vdots & \ddots & \vdots \\ 
        0 & 0 & \cdots  & a_{n\,n}
\end{array}\right],\hspace{1cm}
R = \left[
\begin{array}{ccccc}
        0 & a_{1\,2} & \cdots & a_{1\,n-1} & a_{1\,n}   \\ 
        0 & 0 & \cdots  & a_{2\,n-1} & a_{2\,n}  \\ 
        \vdots & \vdots & \ddots & \vdots \\ 
        0 & 0 & \cdots  & 0 & a_{n-1\,n} \\ 
        0 & 0 & \cdots  & 0 & 0
\end{array}\right].\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 38813
$\displaystyle \boldsymbol{x}^{(0)}=\left[
\begin{array}{c}
x_1^{(0)} \\  x_2^{(0)} \\  \vdots \\  x_n^{(0)}
\end{array}
\right],$%
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{\newpage\clearpage
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% latex2html id marker 38847
$\displaystyle \boldsymbol{x}^{(k)}=\left[
\begin{array}{c}
x_1^{(k)} \\  x_2^{(k)} \\  \vdots \\  x_n^{(k)}
\end{array}
\right]$%
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{\newpage\clearpage
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% latex2html id marker 38901
$\displaystyle \boldsymbol{x}^{(0)}=\left[
\begin{array}{c}
x_1^{(0)} \\  x_2^{(0)} \\  \vdots \\  x_n^{(0)}
\end{array}
\right],$%
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{\newpage\clearpage
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% latex2html id marker 39048
$\displaystyle \boldsymbol{x}^{(0)}=\left[
\begin{array}{c}
x_1^{(0)} \\  x_2^{(0)} \\  \vdots \\  x_n^{(0)}
\end{array}
\right],$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39109}%
% latex2html id marker 39109
$\displaystyle \boldsymbol{x}^{(0)}=\left[
\begin{array}{c}
x_1^{(0)} \\  x_2^{(0)} \\  \vdots \\  x_n^{(0)}
\end{array}
\right],$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39275}%
% latex2html id marker 39275
$\displaystyle G^k = X\,\left[\begin{array}{cccc} 
\lambda_1^k(G)  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2^k(G)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n^k(G)
\end{array}\right]\,X^{-1},$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39278}%
% latex2html id marker 39278
$\displaystyle \lim_{k\rightarrow{}\infty{}} G^k= \lim_{k\rightarrow{}\infty{}}
X\,\left[\begin{array}{cccc} 
\lambda_1^k(G)  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2^k(G)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n^k(G)
\end{array}\right]\,X^{-1} =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39280}%
% latex2html id marker 39280
$\displaystyle = X\,\left(\lim_{k\rightarrow{}\infty{}}
\left[\begin{array}{cccc} 
\lambda_1^k(G)  &  0  &  \cdots  &  0  \\ 
0  &  \lambda_2^k(G)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lambda_n^k(G)
\end{array}\right]\right)\,X^{-1} = $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39282}%
% latex2html id marker 39282
$\displaystyle = X\,
\left[\begin{array}{cccc} 
\lim_{k\rightarrow{}\infty{}}\lambda_1^k(G)  &  0  &  \cdots  &  0  \\ 
0  &  \lim_{k\rightarrow{}\infty{}}\lambda_2^k(G)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lim_{k\rightarrow{}\infty{}}\lambda_n^k(G)
\end{array}\right]\,X^{-1} = O.$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39288}%
% latex2html id marker 39288
$\displaystyle \left[\begin{array}{cccc} 
\lim_{k\rightarrow{}\infty{}}\lambda_1^k(G)  &  0  &  \cdots  &  0  \\ 
0  &  \lim_{k\rightarrow{}\infty{}}\lambda_2^k(G)  &  \cdots  &  0  \\ 
\vdots     &  \vdots     &  \ddots  &  \vdots     \\ 
0  &  0  &  \cdots  &  \lim_{k\rightarrow{}\infty{}}\lambda_n^k(G)
\end{array}\right] = O.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39354}%
% latex2html id marker 39354
$\displaystyle A = \left[
\begin{array}{rrr}
1.00 & 2.51 & 3.72 \\ 
3.15 & - 0.20 & - 1.97 \\ 
4.43 & 5.84 & 1.79 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39356}%
% latex2html id marker 39356
$\displaystyle G_J = \left[\begin{array}{rrr}
0 & -2.51 & -3.72 \\ 
15.75 & 0 & -9.85 \\ 
-2.47486 & -3.26257 & 0
\end{array}\right].$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39362}%
% latex2html id marker 39362
$\displaystyle G_S = \left[\begin{array}{rrr}
0 & -2.51 & -3.72 \\ 
0 & -39.5325 & -68.44 \\ 
0 & 135.189 & 232.497
\end{array}\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39368}%
% latex2html id marker 39368
$\displaystyle G_J = \left[
\begin{array}{rrr}
0 & 0.0634921 & 0.625397 \\ 
-0.758562 & 0 & -0.306507 \\ 
-0.268817 & -0.674731 & 0
\end{array}
\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39374}%
% latex2html id marker 39374
$\displaystyle G_S = \left[
\begin{array}{rrr}
0 & 0.0634921 & 0.625397 \\ 
0 & -0.0481626 & -0.780909 \\ 
0 & 0.0154291 & 0.358786
\end{array}
\right],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39385}%
% latex2html id marker 39385
$\displaystyle A = \left[
\begin{array}{rrrr}
2 & -1 & 0 & 0 \\ 
-1 & 2 & -1 & 0  \\ 
0 & -1 & 2 & -1  \\ 
0 & 0 & -1 & 2  \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39387}%
% latex2html id marker 39387
$\displaystyle G = - D^{-1}(L+R) = \left[
\begin{array}{rrrr}
0 & 1/2 & 0 & 0 \\ 
1/2 & 0 & 1/2 & 0  \\ 
0 & 1/2 & 0 & 1/2  \\ 
0 & 0 & 1/2 & 0  \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39393}%
% latex2html id marker 39393
$\displaystyle G = -(L + D)^{-1}\,R = \left[
\begin{array}{rrrr}
0 & 1/2 & 0 & 0 \\ 
0 & 1/4 & 1/2 & 0  \\ 
0 & 1/8 & 1/4 & 1/2  \\ 
0 & 1/16 & 1/8 & 1/4  \\ 
\end{array}
\right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39461}%
% latex2html id marker 39461
$\displaystyle L_k(x) = \frac{(x-x_0)(x-x_1)\cdots (x-x_{k-1})(x-x_{k+1})\cdots
(x-x_n)}{(x_k-x_0)(x_k-x_1)\cdots (x_k-x_{k-1})(x_k-x_{k+1})\cdots
(x_k-x_n)} = \prod^n_{\substack{i=0\\ i\neq k}}
\frac{x-x_i}{x_k-x_i}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39463}%
% latex2html id marker 39463
$\displaystyle P(x) = \sum_{k=0}^n y_k\,\prod^n_{\substack{i=0\\ i\neq k}}
\frac{x-x_i}{x_k-x_i}.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39491}%
% latex2html id marker 39491
$\displaystyle P(x) = \sum_{k=0}^n f(x_k)\,\prod^n_{\substack{i=0\\ i\neq k}}  \frac{x-x_i}{x_k-x_i}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39779}%
% latex2html id marker 39779
$\displaystyle \int_a^b f(x)\,dx \approx{} \int_a^b P(x)\,dx.$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39782}%
% latex2html id marker 39782
$\displaystyle \int_a^b f(x)\,dx \approx{} \int_a^b \sum_{k=0}^n  f(x_k)\,\prod^n_{\substack{i=0\\ i\neq k}}   \frac{x-x_i}{x_k-x_i}\,dx = \sum_{k=0}^n  f(x_k)\,\int_a^b \prod^n_{\substack{i=0\\ i\neq k}}   \frac{x-x_i}{x_k-x_i}\,dx.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39803}%
% latex2html id marker 39803
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{b-a}{2}\,\sum_{k=0}^n w_k\,
f\left(\frac{b-a}{2}\,t_k + \frac{b+a}{2}\right),$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39805}%
% latex2html id marker 39805
$\displaystyle w_k = \int_{-1}^1 \prod^n_{\substack{i=0\\ i\neq k}}   \frac{t-t_i}{t_k-t_i}\,dt,\hspace{1cm}k=0,1,2,\ldots,n.$%
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{\newpage\clearpage
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% latex2html id marker 39864
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{b-a}{2}\,\left[f(a)+f(b)\right],$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39878}%
% latex2html id marker 39878
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{h}{2}\,\left[f(a)+f(x_1)\right] +
\frac{h}{2}\,\left[f(x_1)+f(x_2)\right] + \cdots{} +
\frac{h}{2}\,\left[f(x_{m-1})+f(b)\right],$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39880}%
% latex2html id marker 39880
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{h}{2}\,\left\{f(a)+2\,\left[f(x_1) + f(x_2) + \cdots{} + f(x_{m-1})\right]+f(b)\right\}.$%
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{\newpage\clearpage
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% latex2html id marker 39906
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{b-a}{6}\,\left[f(a) +
4\, f\left(\frac{a+b}{2}\right) + f(b)\right],$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39920}%
% latex2html id marker 39920
$\displaystyle \int_a^b f(x)\,dx \approx{} \frac{b-a}{6\,m}\,\left[f(a) +
4\, f(x_1) + f(x_2)\right]+$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39926}%
% latex2html id marker 39926
$\displaystyle \int_a^b f(x)\,dx\approx\frac{h}{3}\{f(a)+2\,[f(x_2)+f(x_4)+\cdots 
+f(x_{2m-2})]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline39936}%
% latex2html id marker 39936
$ {\rm Arctg}\,1=\frac{\pi}{4},$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline39937}%
% latex2html id marker 39937
$ {\rm Arctg}\,0=0.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay39939}%
% latex2html id marker 39939
$\displaystyle \frac{\pi}{4} = {\rm Arctg}\,1 - {\rm Arctg}\,0 = \int_0^1 \frac{d\,{\rm Arctg}\,x}{d\,x}\,dx = \int_0^1 \frac{1}{1 + x^2}\,dx.$%
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{\newpage\clearpage
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% latex2html id marker 39998
$\displaystyle \pi \approx 3.141592652.$%
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{\newpage\clearpage
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% latex2html id marker 40001
$\displaystyle \int_{a}^{b}\,f(x)\,dx\approx \frac{b-a}{2}\,\sum_{j=1}^n c_j\,f(x_j).$%
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{\newpage\clearpage
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% latex2html id marker 40062
$\displaystyle \int_{-1}^{1}\,\varphi(t)\,dt \approx{} \sum_{j=1}^n
c_j\,f\left(\frac{b-a}{2}\,t_j + \frac{b+a}{2}\right) = \sum_{j=1}^n
c_j\,\varphi(t_j).$%
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{\newpage\clearpage
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% latex2html id marker 40286
$\displaystyle y_{i+1} \approx y_i + y'_i\,h.$%
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{\newpage\clearpage
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% latex2html id marker 40288
$\displaystyle y_{i+1} \approx y_i + h\,f(x_i,y_i).$%
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% latex2html id marker 26583
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\stepcounter{subsubsection}
{\newpage\clearpage
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% latex2html id marker 40383
$ \frac{1}{6}\,\sum\approx{}y_1-y_0$%
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% latex2html id marker 40419
$ \frac{1}{6}\,\sum\approx{}y_2-y_1$%
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{\newpage\clearpage
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% latex2html id marker 40669
$\displaystyle u_{i+1} \approx u_i + u'_i\,h.$%
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{\newpage\clearpage
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% latex2html id marker 40671
$\displaystyle u'_i \approx \frac{u_{i+1}-u_i}{h}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay40681}%
% latex2html id marker 40681
$\displaystyle u'_i \approx \frac{u_i-u_{i-1}}{h}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay40697}%
% latex2html id marker 40697
$\displaystyle u'_i \approx \frac{u_{i+1}-u_{i-1}}{2\,h}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay40713}%
% latex2html id marker 40713
$\displaystyle u''_i \approx{}\frac{u_{i+1}-2\,u_i+u_{i-1}}{h^2}.$%
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\begin{displaymath}
% latex2html id marker 40746
u = \left[
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\end{array}
\right]^T,\end{displaymath}%
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% latex2html id marker 40748
A(h) = \left[
\begin{array}{cccccc}
 (2+h^2\,q_1) & -1 & 0 & \cdots{} & 0 & 0 \\ 
-1 &  (2+h^2\,q_2) & -1 &  \cdots{} & 0 & 0 \\ 
0 & -1 &  (2+h^2\,q_3) & \cdots{} & 0 & 0 \\ 
\vdots{} & \vdots{} & \vdots{} & \ddots{} & \vdots{} & \vdots{} \\ 
0 & 0 & 0 & \cdots{} &  (2+h^2\,q_{n-1}) & -1 \\ 
0 & 0 & 0 & \cdots{} & -1 &  (1+h^2\,q_{n-1}) 
\end{array}
\right],\end{displaymath}%
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{\newpage\clearpage
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\begin{displaymath}
% latex2html id marker 40750
b(h) = \left[
\begin{array}{cccc}
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\end{array}
\right]^T\end{displaymath}%
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{\newpage\clearpage
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% latex2html id marker 40756
$\displaystyle \sum_{\substack{i=0\\ i\neq k}}^n|a_{i\,j}|< |a_{i\,i}|,\qquad
i=1,2,\ldots,n.$%
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% latex2html id marker 40958
$\displaystyle {{u(x)} = {{\frac{x}{2}} - {\frac{{x^2}}{6}} - 
{\frac{2\,{\rm Arctg}\,x}{3}} + {\frac{\ln (1 + {x^2})}{6}}}}.$%
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{\newpage\clearpage
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% latex2html id marker 40985
$\displaystyle v_i(x) =    \begin{cases}     \frac{x}{h} - i + 1,& \text{ako je       }x_{i-1}\leqslant{}x\leqslant{}x_i \\ [1mm] -\frac{x}{h} + i + 1,&     \text{ako je }x_i\leqslant{}x\leqslant{}x_{i+1} \\ [1mm] 0,&     \text{inače},   \end{cases}$%
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% latex2html id marker 40993
$\displaystyle v_0(x) =    \begin{cases}     -\frac{x}{h} + 1,& \text{ako je       }x_{0}\leqslant{}x\leqslant{}x_1  \\ [1mm] 0,&     \text{inače},   \end{cases}$%
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% latex2html id marker 40995
$\displaystyle v_n(x) =    \begin{cases}   \frac{x}{h} - n + 1,& \text{ako je       }x_{n-1}\leqslant{}x\leqslant{}x_n \\ [1mm] 0,&     \text{inače},   \end{cases}$%
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% latex2html id marker 41010
$\displaystyle v_i'(x) = 
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-\frac{1}{h},& \text{ako je }x_i\leqslant{}x\leqslant{}x_{i+1} \\ [1mm] 
0,& \text{inače}.
\end{cases}$%
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% latex2html id marker 41012
$\displaystyle K_{i\,j} = \int_0^{\ell}\,p\,v'_i\,v'_j\,dx =    \begin{cases}     -\frac{1}{h^2}\,\int_{x_{i-1}}^{x_i} p(x)\,dx,& \text{ako je       }j=i-1 \\ [2mm] \frac{1}{h^2}\,\int_{x_{i-1}}^{x_{i+1}} p(x)\,dx,&     \text{ako je }j=i \\ [2mm] -\frac{1}{h^2}\,\int_{x_{i}}^{x_{i+1}}     p(x)\,dx,& \text{ako je }j=i+1 \\ [2mm] 0,& \text{inače}.   \end{cases}$%
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% latex2html id marker 41235
$\displaystyle \frac{\partial{}^2u(x_i,y_j)}{\partial{}x^2} \approx{}
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\frac{\partial{}^2u(x_i,y_j)}{\partial{}y^2} \approx{} 
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% latex2html id marker 26695
\includegraphics{m3mkd2dimporedak.eps}%
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% latex2html id marker 41264
$\displaystyle u_{\text{\footnotesize\ding{192}}} = u_{1\,1},\quad
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u_{\text{\footnotesize\ding{194}}} = u_{1\,2},\quad
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% latex2html id marker 41265
$\displaystyle -4\,u_{\text{\footnotesize\ding{192}}} + u_{\text{\footnotesize\ding{193}}} + u_{\text{\footnotesize\ding{194}}}$%
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% latex2html id marker 41267
$\displaystyle u_{\text{\footnotesize\ding{192}}} -4\,u_{\text{\footnotesize\ding{193}}} + u_{\text{\footnotesize\ding{195}}} +   u_{\text{\footnotesize\ding{196}}}$%
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% latex2html id marker 41269
$\displaystyle u_{\text{\footnotesize\ding{192}}} -4\, u_{\text{\footnotesize\ding{194}}} + u_{\text{\footnotesize\ding{196}}} +   u_{\text{\footnotesize\ding{197}}}$%
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% latex2html id marker 41271
$\displaystyle u_{\text{\footnotesize\ding{193}}} -4\,u_{\text{\footnotesize\ding{195}}} + u_{\text{\footnotesize\ding{198}}} +   u_{\text{\footnotesize\ding{199}}}$%
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% latex2html id marker 41273
$\displaystyle u_{\text{\footnotesize\ding{193}}} + u_{\text{\footnotesize\ding{194}}} -4\, u_{\text{\footnotesize\ding{196}}} +   u_{\text{\footnotesize\ding{199}}} + u_{\text{\footnotesize\ding{200}}}$%
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% latex2html id marker 26749
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% latex2html id marker 41307
$\displaystyle \frac{\textstyle{\partial u(x_i,t_j)}}{\textstyle{\partial t}}
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\frac{\partial{}^2u(x_i,y_j)}{\partial{}x^2} \approx{} 
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% latex2html id marker 26751
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% latex2html id marker 41389
$\displaystyle F(u) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}u)^2dxdy - \iint_{\Omega}\,f\,u\,dxdy.$%
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% latex2html id marker 41411
$\displaystyle F(u_n) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}\sum_{i=1}^n c_i\,v_i)^2dxdy -
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% latex2html id marker 41413
$\displaystyle = \frac{1}{2}\,\iint_{\Omega}\,
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f\,v_i\,dxdy$%
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% latex2html id marker 41415
$\displaystyle = \frac{1}{2}\,\sum_{i=1}^n\sum_{j=1}^n c_i\,c_j\,\iint_{\Omega}\,
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% latex2html id marker 41425
$\displaystyle \frac{\partial \Phi}{\partial c_j} = \sum_{i=1}^n c_i\,\iint_{\Omega}\,
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% latex2html id marker 41437
$\displaystyle K_{ij} = \iint_{\Omega}\,{\rm grad\,}v_i\cdot{\rm grad\,}v_j\,dxdy,\hspace{1cm}b_j = \iint_{\Omega}\,
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% latex2html id marker 26801
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% latex2html id marker 41501
$\displaystyle F(w) = \frac{1}{2}\,\iint_{\Omega}\,({\rm grad\,}w)^2dxdy - \iint_{\Omega}\,w\,dxdy.$%
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% latex2html id marker 41569
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_i\cdot{\rm grad\,}\sum_j c_j\,v_j\,dxdy = \iint_{\Omega}\,
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% latex2html id marker 41571
$\displaystyle \sum_j c_j\,\iint_{\Omega}\,{\rm grad\,}v_i\cdot{\rm grad\,}v_j\,dxdy = \iint_{\Omega}\,
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% latex2html id marker 41575
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_j\,dxdy,\quad j=6,7,10,11,$%
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% latex2html id marker 41580
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_6\,dxdy = \iint_1 + \iint_2 + \iint_4 +
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% latex2html id marker 41582
$\displaystyle {\rm grad\,}v_6^1 = \{0,3\},\;{\rm grad\,}v_6^2 = \{3,0\},\;{\rm grad\,}v_6^4 =
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% latex2html id marker 41584
$\displaystyle {\rm grad\,}v_6^7 = \{3,-3\},\;{\rm grad\,}v_6^9 = \{-3,0\},\;{\rm grad\,}
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% latex2html id marker 41588
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_6\,dxdy = 4.$%
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% latex2html id marker 41590
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_7\,dxdy = \iint_4 + \iint_9.$%
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% latex2html id marker 41594
$\displaystyle {\rm grad\,}v_7^4 = \{3,0\},\quad {\rm grad\,}v_7^9 = \{3,-3\},$%
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% latex2html id marker 41598
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_7\,dxdy = -1.$%
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% latex2html id marker 41600
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_{10}\,dxdy = \iint_7 + \iint_{10}.$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay41604}%
% latex2html id marker 41604
$\displaystyle {\rm grad\,}v_{10}^7 = \{0,3\},\quad {\rm grad\,}v_{10}^{10} = \{-3,3\},$%
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% latex2html id marker 41608
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_{10}\,dxdy = -1.$%
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% latex2html id marker 41610
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_{11}\,dxdy = \iint_9 + \iint_{10}.$%
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% latex2html id marker 41614
$\displaystyle {\rm grad\,}v_{11}^9 = \{0,3\},\quad {\rm grad\,}v_{11}^{10} = v_7^4 = \{3,0\},$%
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% latex2html id marker 41618
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}v_{11}\,dxdy = 0.$%
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% latex2html id marker 41632
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_i\cdot{\rm grad\,}v_i\,dxdy = \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}
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% latex2html id marker 41634
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_7\cdot{\rm grad\,}v_{11}\,dxdy = \iint_{\Omega}\,{\rm grad\,}v_6\cdot{\rm grad\,}
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% latex2html id marker 41640
$\displaystyle \iint_{\Omega}\,{\rm grad\,}v_{10}\cdot{\rm grad\,}v_{11}\,dxdy = \iint_{\Omega}\,{\rm grad\,}
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{\newpage\clearpage
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% latex2html id marker 41688
$\displaystyle a_k = \left\{
\begin{array}{ll}
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0, & \text{ako je $N_k$ paran}.
\end{array}\right.$%
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{\newpage\clearpage
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% latex2html id marker 41864
$\displaystyle n = 127\,\log 2 \approx 127\times 0.30103 \approx 38.$%
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% latex2html id marker 41886
a_{-k} = \left\{
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% latex2html id marker 42011
$ \sqrt{b^2-4ac}\approx
b,$%
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% latex2html id marker 42022
$ a\approx
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\end{document}
