Mathematisch für Anfänger
Martin Wohlgemuth (Hrsg.)
Mathematisch für Anfänger Die beliebtesten Beiträge von Matroids Matheplanet
Mit Beiträgen von Norbert Engbers, Ueli Hafner, Johannes Hahn, Artur Koehler, Georg Lauenstein, Fabian Lenhardt, Florian Modler, Thorsten Neuschel, Sebastian Stöckl, Martin Wohlgemuth
Herausgeber Martin Wohlgemuth E-Mail:
[email protected] www.matheplanet.de
Wichtiger Hinweis für den Benutzer Der Verlag, der Herausgeber und die Autoren haben alle Sorgfalt walten lassen, um vollständige und akkurate Informationen in diesem Buch zu publizieren. Der Verlag übernimmt weder Garantie noch die juristische Verantwortung oder irgendeine Haftung für die Nutzung dieser Informationen, für deren Wirtschaftlichkeit oder fehlerfreie Funktion für einen bestimmten Zweck. Ferner kann der Verlag für Schäden, die auf einer Fehlfunktion von Programmen oder ähnliches zurückzuführen sind, nicht haftbar gemacht werden. Auch nicht für die Verletzung von Patent- und anderen Rechten Dritter, die daraus resultieren. Eine telefonische oder schriftliche Beratung durch den Verlag über den Einsatz der Programme ist nicht möglich. Der Verlag übernimmt keine Gewähr dafür, dass die beschriebenen Verfahren, Programme usw. frei von Schutzrechten Dritter sind. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Buch berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Der Verlag hat sich bemüht, sämtliche Rechteinhaber von Abbildungen zu ermitteln. Sollte dem Verlag gegenüber dennoch der Nachweis der Rechtsinhaberschaft geführt werden, wird das branchenübliche Honorar gezahlt. Bibliogra¿sche Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliogra¿e; detaillierte bibliogra¿sche Daten sind im Internet über http://dnb.d-nb.de abrufbar. Springer ist ein Unternehmen von Springer Science+Business Media springer.de © Spektrum Akademischer Verlag Heidelberg 2009 Spektrum Akademischer Verlag ist ein Imprint von Springer 09
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Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikrover¿lmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Planung und Lektorat: Dr. Andreas Rüdinger, Barbara Lühker Herstellung: Crest Premedia Solutions (P) Ltd., Pune, Maharashtra, India Satz: Martin Wohlgemuth und die Autoren Umschlaggestaltung: SpieszDesign, Neu–Ulm Titelbild: © Jos Leys ISBN 978-3-8274-2285-9
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0* ## % # # #12 0' 3 / %$ % ""# # ## * "42 0' / % $ #! 3 % 3 "# % $+ 12 5 # % # ! ( ! ! / #"" # #! 6 # ## % % # 1 6 # # / % #% # "" "! 1 6 " $##%/ % ## ! # ! "" 7! ! / $ ! %# (& 5 # ! ! / $" ! / ## * % / #% ! $"" % "! * % % ! %# ( #,/ % ! # ! / % % %/ % ' %# 6 # # % #! 8"+,"! $ , #8 ,## $% 9%/ % * % ! , ", ! : #8 ! , 6 * # % / % # 3 $+ "! #! "! * $# % %! / # $ % 3% ! "! #"" #/ #! % ;! & #! " , #"" ' # % *#! % /
¼
! " # $ %
& % " ' ( )"
# %
&
# $ *
%
" ) +
, -
' %
./ ! /
" ) 0 ' 12 34 5 12 + #34 -
#
6 $ * $ /
) 7 * 2 ) *3 20 " ( 3
- 20 8 /8$
3 "
# $ 8 8 8 - 8 $ # # 5 ) / (8$8! "7 $ # 9 :
*
0" +
: # +; , π(j) π F (π) π
sign(π) = (−1)F (π)
π
!
id ∈ Sn
! "
# $
"
!
n=4
π ∈ S4
π(1) = 2, π(2) = 1, π(3) = 4, π(4) = 3
! %
π
(1,2) (3,4)
& ' '
det
n
π ∈ Sn det(M (π)) = sign(π)
#% ' " #( " $ !
(
%
) * + $
, ' -.$
. ! * ' / 0$ ' % -.( $
K
1 2 3 *
-.$ *
K
4 #
/ 5 '(
% , $ "
, $
½¿
σ ! τ "
sign(τ ◦ σ) = − sign(σ)
# $ % & σ ' ! σ % # & $ % & ( ) ' * σ τ ◦ σ & + # ' ) & ' π Sn ! M (id) = En $, #- ((( " det(M (id)) = sign(id)
$ id $ ! π # ' ) ' . / ' ' ) & # ' ' , 0 + 1 $ 2, # det #- ((( 2 , # # 3 , M (n, K) " ⎛ ⎞ a1,1 ⎜ det ⎜ ⎝ an,1
···
···
a1,n
⎟ ⎟ ⎠
an,n
' ' 3 , 1 ' 1 & # & 4 ' nn ' ' n = 3 ' # 3 , 5 n 6 $, 7 * ' # 3 , ' ( 8 9
½ ¼
⎛
a1,1
···
a1,n
an,1
···
an,n
⎜ det ⎜ ⎝
⎞ ⎟ ⎟= det(M (π))a1,π(1) a2,π(2) · · · an,π(n) , ⎠ π∈Sn
π S ⎛ ⎞ a ··· a ⎜ ⎟⎟⎠ = sign(π)a a · · · a det ⎜ ⎝ n
1,1
1,n
1,π(1) 2,π(2)
an,1
···
an,n
n,π(n)
π∈Sn
! " # $ n ∈ N % !& $ ' ( )* + , # - ) . # / 0 !
K n det : M (n, K) → K A = (ai,j ) ∈ M (n, K) ! sign(π)a1,π(1) a2,π(2) · · · an,π(n)
det A =
π∈Sn
1 . $ (2,2)$2& 2 {1,2} 0 id 0 - ! $ 3 # ! % 4 +
/
det
a1,1
a1,2
a2,1
a2,2
= a1,1 a2,2 − a1,2 a2,1
,# 2 ) , # $ - 5 * ! $
! # - !
( , ) . #
½ ½
K n
A ∈ M (n, K) det A = det AT AT A ¾º λ ∈ K A ∈ M (n, K) det (λA) = λn det (A) ¿º A ∈ M (n, K) det A = 0 Rang(A) < n
! "
! "
# ! "
A
$
! "
B
det (A) =
det (B)
A, B ∈ M (n, K) det (AB) = det (A) det (B).
!
! %
n ∈ N (n, n) A K det A ∈ K ! " # # n ∈ {1,2,3} #! $ % ! n ≥ 4 & ! ! ' (! n! ) *)( ) )!) $ " # ) + (! " ) "
( ' ,,"- ( & ! ! " +
( ! . .! " % / ! ! 0! ) % ! ! ) "
½ ¾
(4,4) M (4, R)
⎛
0
1
0
9
0
3
0
2
⎞
⎜ ⎟ ⎜2 2 3 7⎟ ⎜ ⎟ det (A) = det ⎜ ⎟ ⎝−4 5 −1 −3⎠
⎛
−4
⎜ ⎜2 det (A) = − det ⎜ ⎜ ⎝0
5
−1
2
3
1
0
0
3
0
⎞ −3 ⎟ 7⎟ ⎟ ⎟ 9⎠ 2
! "
#$ %
1 2 & ' ( &
& ) # '' *
⎛
−4
⎜ ⎜0 det (A) = − det ⎜ ⎜ ⎝0 0
5
−1
9 2
5 2
1
0
3
0
⎞ −3 11 ⎟ ⎟ 2 ⎟ ⎟ 9⎠ 2
+ % #$
2 , − 3 & ( -
⎛
−4
⎜ ⎜0 det (A) = − det ⎜ ⎜ ⎝0 0
(−3)
5
−1
9 2
5 2 − 59 − 53
0 0
−3
− 29
& (
⎞
11 ⎟ ⎟ 2 ⎟ 70 ⎟ 9 ⎠ − 53
& , - .
/ & 0 $
⎛ −4 ⎜ ⎜0 det (A) = − det ⎜ ⎜ ⎝0 0
5
−1
−3
9 2
5 2 − 59
11 2 70 9
0
−25
0 0
⎞
⎟
⎟ ⎟ = −(−4) · 9 · − 5 · (−25) = 250 ⎟ 2 9 ⎠
½ ¿
(n, n)
(n − 1, n − 1) ! " # ! $ % &' n n ≥ 2 A = (ai,j ) ∈ M (n, K) $ ( i, j ∈ {1, . . . , n} Ai,j ) (n−1, n−1)
M (n − 1, K)"
A " i j &* K n n ≥ 2 A = (ai,j ) ∈ M (n, K)
j ∈ {1, . . . , n} j ! " #$ det (A) =
n
(−1)i+j ai,j det (Ai,j )
i=1
% i ∈ {1, . . . , n} i!
#$ n det (A) =
(−1)i+j ai,j det (Ai,j )
j=1
* + , ) - . * (5,5)
M (5, R)' ⎛
0
1
0
0
9
⎞
⎜ ⎟ ⎜2 2 3 0 7⎟ ⎜ ⎟ ⎜ ⎟ det (A) = det ⎜−4 5 −1 0 −3⎟ ⎜ ⎟ ⎜−1 17 −3 2 27 ⎟ ⎝ ⎠ 0 3 0 0 2
" &* # &* " ' 5 (−1)i+4 ai,4 det (Ai,4 )
det (A) =
i=1
& )
& " & , / '
½
⎛ det (A) = a4,4
0
1
0
9
0
3
0
2
⎞
⎜ ⎟ ⎜2 2 3 7⎟ ⎜ ⎟ det (A4,4 ) = 2 det ⎜ ⎟ ⎝−4 5 −1 −3⎠
det (A) = 2 · 250 = 500
! " # $ $ % & " ' " ( !
!"
" # $
% & '# ( "' )* # + ' % $#
" # , - # . # / ( 0 # 1 ' # # . ' ! 2 ' !" ( # # 34
$ & # # #
' /# - # 2( 5# 6 " 1 " ' 1 '( !7
# % ! 8 ( #
" # 9 1 ' # / ! # *# ( 1 '( # : " # / '( ' # $ # ;
½
(n, n)
⎛
λ1
⎜ ⎜0 ⎜ ⎜ ⎜ ⎝ 0
0
···
λ2
···
0
···
0
⎞⎛
μ1
⎟⎜ ⎜ 0⎟ ⎟⎜ 0 ⎟ ⎜ ⎟⎜ ⎠⎝ λn 0
0
···
μ2
···
0
···
0
⎞
⎛
0
···
0
λ2 μ 2
···
0
0
···
λ1 μ 1
⎟ ⎜ ⎜ 0⎟ ⎟ ⎜ 0 = ⎟ ⎜ ⎜ ⎟ ⎠ ⎝ μn 0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
λn μn
! " #
$
%
& ' ' ( & " ' ! '"
) '
# * ' + !
,
% ) * % !
'
" "
! ) *
%
- .
-
K
!
n K
" /
A
,
%
A % n n S MS (fA ) .
fA : K → K , x → Ax 0 ! S n - K 1 A
% % .
fA " n 2 K +
' 3
%
'
B = (b1 , . . . , bn ) K ( K n % B MB (fA ) fA " 2
% 2 '!
B
% 4
1 + ! 2 0
5 ' 6
½
V K f : V → V x ∈ V \ {0} λ ∈ K f f (x) = λ x
λ ∈ K f Eigf (λ) := {x ∈ V : f (x) = λ x}
f λ
x λ f x f ! λ! " # f (x) = λ x !$ % λ ∈ K !$ λ f & ' x = 0 f (x) = λ x( Eigf (λ) ") Eigf (λ) ! * + , Eigf (λ) - . V " ! * & ! /( ! 0 !
V K
f : V → V λ, μ ∈ K f
Eig
f (λ)
∩ Eigf (μ) = {0}
¾º Eigf (λ) = Kern(f − λ idV ) ¿º
! V " #
$ %
& ' "
() # ( * + $ , .
!
½
V K f : V → V f V f V ! " # V $ # % & ' V K f : V → V B V f & B MB (f )
=D
D ( ) # * ) % &
K + * n , A - " ( K " ( A fA : K n → K n, x → Ax ' % ./0 " ( ) &
K n A K A T K
! T AT −1 = D D
!"# Kn
1 ⇒2& ' A ( B 3 ) B MB (fA )
= D,
½
D T : = S MB (idK ) T S ! n
T AT −1 =S MB (idK n ) S MS (fA )
B MS (idK n )
=B MB (fA ) = D
" ⇐#! $ T T AT −1 = D,
D % T −1 & K n B T −1 =
B MS (idK n ),
T =
S MB (idK n ),
! B MB (fA )
=S MB (idK n ) S MS (fA )
B MS (idK n )
= T AT −1 = D
% A ' % $ ( $ & $ ' ( )
( & $ ' * ' $ + ,--! $ $ .'
$ ( .'
& / ' '
$ $ ( $ ' ( !
n V n K B = (b1 , . . . , bn ) f : V → V A := B MB (f ) λ ∈ K λ
! f ⇐⇒ det(A − λ En ) = 0
En n " #
½ ¼
⇒ λ f x0 ∈ V \ {0} f (x0 ) = λ x0 φB : V → K n , x =
n
αi bi
→ (α1 , . . . , αn )
i=1
V
Kn
φB (bi ) = ei "# En ) φB $ &
!
i = 1, . . . , n $ei % ! i& '( φB (! *#
!
! ! ! # !" +( # )
f (x) = φ−1 B (AφB (x)) , x ∈ V '%
x = x0
f (x0 ) = φ−1 B (AφB (x0 )) , "
λ φB (x0 ) = φB (λ x0 ) = φB (f (x0 )) = AφB (x0 )
φB (x0 ) = 0
φB (x0 ) ∈ Kern(A − λ En )
det(A − λ En ) = 0 ⇐ ' det(A − λ En ) = 0 % y0 ∈ K n \ {0} A y0 = λy0 . , ! φB ! ,
−1 −1 f (φ−1 B (y0 )) = φB (A y0 ) = λφB (y0 )
φ−1 B (y0 ) = 0
"
*
λ
det(A − λ En ) -
f
./ 0
n
1 λ 2 " '% 345 , ( ! 6 7 ./ 8 *
K =C
" * %
% ( *
n
K = R
9 ! % "# * %
(%
V ! : det(A−λ En ) ! ! %# & A ! " ! &
8 '% 345 ! , ! ! ./ ! ! ,
6 ./ 7
½ ½
V n K n ∈ N B V f : V → V A :=B MB (f ) det(A − λ En ) f
! " # $ % &
' # K = R # f : R3 → R3 # ( # ) ⎛
⎞
⎛
x1
3 x1 + 4 x2 − 3 x3
x3
3 x1 + 9 x2 − 5 x3
⎞
⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ f⎜ ⎝x2 ⎠ = ⎝2 x1 + 7 x2 − 4 x3 ⎠
# " * " # %*# # f "+ # # , " # , + # - S R3 * * # ( f ) ⎛
3
A :=
S MS (f )
⎜ =⎜ ⎝2
4
3
9
7
⎞ −3 ⎟ −4⎟ ⎠ −5
$ # # # ) ⎛⎛
3
⎜⎜ ⎜ det(A − λ En ) = det ⎜ ⎝⎝2
4
3
9
⎛
3−λ ⎜ = det ⎜ ⎝ 2 3
7
⎞ ⎛ −3 λ ⎟ ⎜ ⎜ −4⎟ ⎠ − ⎝0 −5 0 4
7−λ 9
−3
⎞⎞ 0 λ
0
⎟⎟ ⎟ 0⎟ ⎠⎠
0 λ ⎞
⎟ −4 ⎟ ⎠ −5 − λ
. - # ) det(A − λ En ) = λ3 − 5 λ2 + 8 λ − 4
½ ¾
! " #
$
det(A − λ En ) = (λ − 2)2 (λ − 1) % & f
λ1 = 1 λ2 = 2 % # # Eigf (λ1 ) % $ ' ()) Eigf (λ1 ) = Kern(f − λ1 idR3 ) * f (x) = Ax Kern(f − λ1 idR3 ) = Kern(A − λ1 E3 ) % % λ1 = 1 " + ,
⎛ ⎞⎛ ⎞ ⎛ ⎞ 2 4 −3 x1 0 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜2 6 −4⎟ ⎜x2 ⎟ = ⎜0⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ 3 9 −6 0 x3 ! + - ! " , . % / & $ 0 ⎧⎛ ⎞ ⎫ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ x1 1 ⎪ ⎪ ⎬ ⎨⎜ x 1 ⎟ ⎜ ⎟ ⎜ ⎟ 3 ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ Eigf (λ1 ) = ⎝x2 ⎠ ∈ R : ⎝x2 ⎠ = t ⎝1⎠ , t ∈ R ⎪ ⎪ ⎪ ⎪ ⎩ x ⎭ x 2 3
3
1 % % λ2 = 2 ⎧⎛ ⎞ ⎫ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ x1 1 ⎪ ⎪ ⎬ ⎨⎜ x 1 ⎟ ⎜ ⎟ ⎜ ⎟ 3 ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ Eigf (λ2 ) = ⎝x2 ⎠ ∈ R : ⎝x2 ⎠ = t ⎝2⎠ , t ∈ R ⎪ ⎪ ⎪ ⎪ ⎩ x ⎭ x 3 3
3
* % # & f
2 # 3 R3 % # % & & f
!
R3 % & 4 f
$ ' 5
/ 6 7 % " #
% # ! 8 9 $ 4 :
½ ¿
V K
f : V → V λ ∈ K f λ f
λ ! Eigf (λ)
λ " #$ λ1 = 1 % & λ2 = 2 % ' % & " #$ $ % ( ) * % + !) ( % % ,
n V n K f : V → V λ0 ∈ K f
!
k ∈ N % λ0 k ≤ n - ( # λ0 ! (b1 , . . . , bk ) ( # B V . f B % ) , B MB (f )
=
A1
A2
O
A3
# A2 A3 ( ! O . A1 = λ0 Ek # f . λ0 k
½
n V n K f : V → V f ! " #
f
$
#
$ %
&
⇒ f
f ! "# $ %& ' ( % ) * %
*
& n! + ) * %
*
% ) ,-!. / %! ⇐ 0 % ) * %
*
! / ) ) * %
%! 1 2 1 3 $ / & ! + 4 / 5
* %
* %
6 % 7 + ! / ) f : R3 → R3
⎛
⎞
⎛
x1
1 ⎜ ⎟ ⎜√ ⎟ ⎜ f⎜ ⎝x2 ⎠ = ⎝ 3 x3
0
√ − 3 −1 0
⎞⎛ ⎞ 0 x1 ⎟⎜ ⎟ ⎜ ⎟ 0⎟ ⎠ ⎝x2 ⎠ 1 x3
⎛ ⎞ √ 1−λ − 3 0 ⎜ √ ⎟ 2 det ⎜ 0 ⎟ ⎝ 3 −1 − λ ⎠ = (1 − λ)(2 + λ ) 0 0 1−λ
½
R
f R ! " # #
! $ $
%
& '
n V n K f : V → V f n f ( )
* $ "# f : R2 → R2 + ' x1
f
=
x2
0
−2
x1
1
3
x2
, ' det
−λ
−2
1
3−λ
= λ2 − 3 λ + 2 = (λ − 1)(λ − 2)
f - . "$ λ1 = 1 λ2 = 2 & #
/ $ " ' Eigf (λ1 ) =
Eigf (λ2 ) =
x1 x2
x1 x2
2
∈R :
x1
=r
x2
2
∈R :
x1
x2
−2 1
=r
−1 1
,r∈R
,r∈R
& $ 0 B R2 " f
' B=
−2 −1 , 1 1
$ f 0 $ 1' B MB (f )
=
1
0
0
2
½
T 0 −2 A = !" 1 3 !
# $ T =
S MB (idR2 )
! %
& T −1 ' #& & B $ T −1 =
B MS (idR2 )
=
−2
−1
1
1
! T ( & T −1 $
T =
S MB (idR2 )
=
−1
−1
1
2
!) %
" ) * ) ( & + &, - (2,2), & $
! "$
−1 1
a
b
c
d
−1
−1 0 2
1
d
1 = ad − bc
−c
−2 −2
−1
3
1
−b a
=
1
1
0
0
2
. ' " !" / " "
" 0 1 % #' # " ,/' ( "
" 2 3 "
" , A 1 " / 4" 5%' K 6
T T AT −1 = D,
D !" 7 " $ A = T −1 DT
A
½
A
n
n
n An = T −1 DT = T −1 DT T −1 DT . . . T −1 DT
!
"# # " $ % & # ' ( # # #
An = T −1 Dn T )
n
) &
# # % &
A
n
D
# *
) #' " # "' +,,- # % ' *
)' *% * ' . /*
! " # "$!"
%&
#'
%%
( ! ) " ) " # * + , - . " ! , - ! ! # * ! / ! ! !! , ' - ! 0! 1' ' * +- ! ' . 2" 3"!
4 # " ! " " * 5, ! # ',6 4 ! 7" ! - , '* 8 mx + n = 0, , !" ! , * 8 mx = −n n x=− m
½ ¾
5x + 7 = 0 5x = −7 7 x=− 5
ax2 + bx + c = 0.
a !
a = 0 " # $ %
& ' !$
x2 + px + q = 0
% $
( &
# )
*
(a + b)2 = a2 + 2ab + b2
# √ x2 = a =⇒ x = ± a.
$
a +
!
, (x + a)2 = 0
x + a = ±0 = 0 x = −a
½ ¿
(x + d)2 + e = 0
(x + d)2 = −e √ x + d = ± −e √ x = −d ± −e
! −e " # $
(x + 5)2 − 4 = 0 (x + 5)2 = 4 x + 5 = ±2 x = −5 ± 2
% & " (x + d)2 + e = x2 + 2xd + d2 + e
' ( x + px + q ) 2
p = 2d q = d2 + e
* p q e d p 2 e = q − d2 p 2 =q− 2
d=
& e d p"q "+ ) ,* - . + * "
½
x = −d ±
√
−e p 2 p =− ± −q 2 2
p2 2 − q < 0
x2 + 6x + 8 = 0 x1,2 = −3 ±
√
9−8
x1 = −2 x2 = −4
pq
!
"#$% %&&'( " %&&'(
! )
!
* +! +
! ,+ - !- .- /
n
) 0 1 2
xn + a = 0. 3 + +
x4 − 81 = 0
n
a ≤ 0
½
x4 = 81 √ 4 x = ± 81 x = ±3
x4 + px2 + q = 0
z = x2 z := x2 =⇒ z 2 + pz + q = 0 =⇒ z1,2 =⇒ x1,2
p =− ± 2 √ = ± z1 =±
p 2 2
p − + 2
√ x3,4 = ± z2 =±
−
p − 2
−q
p 2 2
p 2 2
−q
−q
x4 − 6x2 + 8 = 0 z := x2 =⇒ z 2 − 6z + 8 = 0 =⇒ z1,2 = +3 ± =⇒ x1,2 x3,4
√
9−8
= +3 ± 1 √ =± 3+1 = ±2 √ =± 3−1 √ =± 2
! " # $ z = x2 % &
½
x2n + pxn + q = 0
z := xn
z
! x " # $ # x3n + ax2n + bxn + c = 0 x4n + ax3n + bx2n + cxn + d = 0
% z := xn z %
x
& '
x5 + x4 + x3 + x2 + x + 1 = 0
$ ( ) & * + xn + xn−1 + xn−2 + . . . + x2 + x + 1 =
n
xk = 0
k=0
, # * !
n+1 x
−1 =0 x−1
n
(x − 1) · (x + x
n−1
+x
n−2
2
+ . . . + x + x + 1) = (x − 1) ·
n
xk
k=0
=
n
xk+1 −
k=0
= xn+1 − 1
n k=0
xk
½
x x−1−1
xn+1 = 1 xn + xn−1 + xn−2 + . . . + x2 + x + 1 = 0 n+1
x5 + x4 + x3 + x2 + x + 1 = 0 x6 − 1 =0 x−1 x6 − 1 = 0 =⇒ x1 = +1
= !
x2 = −1
" x = −1 #
$ % (a+b)2 # % & & n (a + b)n # $ & ' n
n
(a + b) = a + n · a
n−1
n−1
· b + ... + n · a · b
n n +b = · ak · bn−k k n
k=0
% () * +,# ,# -& - # .&
' (x + a)n = 0 x+a =0 x = −a
x4 − 4x3 + 6x2 − 4x + 1 = 0 (x − 1)4 = 0 x−1=0 x=1
½
p(x)
xk · p(x) = 0
k ! "
#
$ k %& '$
!(
xk xk · p(x) = 0
k "
0
p(x)
)
x4 + 2x3 − 15x2 = 0 x2 (x2 + 2x − 15) = 0 =⇒ x1,2 = 0 2
x + 2x − 15 = 0 x3,4 = −1 ±
1 − (−15)
= −1 ± 4 =⇒ x2 = −5 =⇒ x3 = +3
#
"
)
* ' +,-.(
$ /
*
0 1 $
) 2 3 41 5
3 " x1 , x2 , x3 , . . .
6 f (x) = a · (x − x1 ) · (x − x2 ) · (x − x3 ) · . . . = a ·
n
(x − xi )
i=1
½
x ! " # #$# # # # % & " x0 # # % ! (x − x0 ) ' % (# ) (# " * ! + # + ! ! ! ' , - ' ! . ' ' / 0! 0 0 1 # '! !,
(x3
−(x3
−6x2
+3x
+10
) : (x − 5) = x2 − x − 2
−5x2 ) −x2 −(−x
+3x 2
+5x) −2x
+10
−(−2x
+10) 0
x2 % 2 3 % x2 2! x2 · (x − 5) = x3 − 5x2 x − 5 (−x) 4 % −x2 + 3x + 10
2 3 % −x 2! (−x) · (x − 5) = −x2 + 5x x − 5 # (−2) 4 % −2x + 10 2 3 % −2 (−2) · (x − 5) = −2x + 10 4 ! % ' ) ## ! 5 ! * # ! # 6 !# #! % ! ! " $ # ( 5 ! + ! # . x−5
½ ¼
!
!
" #
t
2
t
$
#
! % ! % &
' & ( $ # ! ) ' ( &
! %
( * + #
, -
n. - (2n−1).
- ,
( + ) $ !
%
/ 0 ' '+
+ ! " /
x0
1
'+ ' 2 + 3 0
xk+1 = xk −
f (xk ) f (xk )
" " ' + , ! " ' + / ! ! *!
4.
+ / 5 ! 5
(
- ! 6
f (xk )
xk
57
/ + + 0
½ ½
! ! " #$#
% f (x)
f (x) x0 x1
x
f (x) & x0
" ' x1 x1 ( )( " * " & )( xk " ( t(x) = mx + n ( " ( m xk + f (xk ) +
, n " ( (xk , f (xk ))" - &
" % f (xk ) = t(xk ) = f (xk ) · xk + n n = f (xk ) − f (xk ) · xk n ! ( x = − m ' . " '
f (xk ) − f (xk ) · xk f (xk ) f (xk ) =− + xk f (xk ) f (xk ) = xk − . f (xk )
xk+1 = −
! "
' ( /
½ ¾
f (x) := x4 + 0,2x3 − x2 + 5x − 7 =⇒ f (x) = 4x3 + 0,6x2 − 2x + 5 x0 := 1 xn+1 = xn −
x4n + 0,2x3n − x2n + 5xn − 7 4x3n + 0,6x2n − 2xn + 5
=⇒ x1 ≈ 1,2368 x2 ≈ 1,2029 x3 ≈ 1,2020 x4 ≈ 1,2020 x5 ≈ 1,2020
! x x "
f (x ) · f (x ) < 0 # $ 1
2
1
x3 = x1 −
2
x2 − x1 · f (x1 ) f (x2 ) − f (x1 )
x3 (x1 , x2 ) (x1 , x3 ) (x2 , x3 ) f (x1 ) · f (x3 ) < 0 f (x2 ) · f (x3 ) < 0
! "
# $ "% & ' () * " +
,
- #.
f (x) ! $ - s(x) = mx + n /
(x1 , f (x1 )) (x2 , f (x2 ))
0 # 1
f (x1 ) = mx1 + n f (x2 ) = mx2 + n
½ ¿
f (x)
f (x) x1
x3
x
x2
m n n = f (x1 ) − m · x1 m=
f (x2 ) − n x2
m=
f (x1 ) − f (x2 ) Δy = x1 − x2 Δx
x = − mn f (x1 ) − m · x1 m f (x1 ) =− + x1 m x1 − x2 = x1 − · f (x1 ), f (x1 ) − f (x2 )
x3 = −
! " #
$
% & '
( ( ( ) * + ' + , -
. &+
½
!
" f (x) = (x − x1 ) · (x − x2 ) · (x − x3 ) · . . . · (x − xn ) = a
n
(x − xi )
i=1
# $ % !&
x1 ·x2 ·x3 ·. . .·xn ' " ! ! ( " )" ' * √
x2 − 2 = 0 √ − 2
+ + 2 !& , $ " - '
. '& / , 0 . "
f (x) = x3 + 6x2 + 12x + 8
1 ±1, ±2, ±4 ±8" ( 2
" x = 1 f (+1) = 27 = 0
3 " x = −1 # ! 3 " x = +2 3 " x = −2
f (−1) = 1 = 0
f (+2) = 64 = 0
f (−2) = 0
4 56 # ( (x + 2) . ( , * x2 + 4x + 4 = 0
7 ! ( ( ( * x2,3 = −2 ±
√ 4 − 4 = −2
7 8 " 3 -/ x = −2"
½
!"# $%%&'
( )( # ) # * ( # + ) , ) # * (
- .# , / * 0 + # 1 2 ( * # . 3 ( ( 4/ # 5 6 / 78# 9 ( 1 5
f (x) := x3 + 2x2 − 2x − 4 = (x3 − 2x) + (2x2 − 4) = x · (x2 − 2) + 2 · (x2 − 2) = (x + 2) · (x2 − 2)
9 ( , −2, √2 −√2 3 : 1 5 f (x) := x8 + 3x7 + 2x6 − 2x5 − 6x4 − 4x3 + x2 + 3x + 2 = (x8 − 2x5 + x2 ) + (3x7 − 6x4 + 3x) + (2x6 − 4x3 + 2) = x2 · (x6 − 2x3 + 1) + 3x · (x6 − 2x3 + 1) + 2 · (x6 − 2x3 + 1) = (x2 + 3x + 2) · (x6 − 2x3 + 1)
* x + 3x + 2 -0 ) 2 + / , x = −1 x = −2 9 * 4( z := x )( 2 + 0 = z − 2z + 1 = (z − 1) # , z = 1 9 , f (x) x = √1 = 1 2
1
2
3
2
2
3
3
½
! " # " $ x4 + ax3 + bx2 + ax + 1
%"
& ' x2 # ( x = 0 ) * + x4 + ax3 + bx2 + ax + 1 = 0 1 1 =⇒ x2 + ax + b + a + 2 = 0 x x
1 1 2 x + 2 +a· x+ +b=0 x x
1 x + 2 = x 2
2 1 x+ − 2. x
2
1 1 x+ +a· x+ +b−2=0 x x z := x+ x1 z z 2 + az + b − 2 = 0 a =⇒ z1 = − + 2 z2 = −
a − 2
a 2 2 a 2 2
−b+2 −b+2
z = x+ x1 x x
1 x zx = x2 + 1 z =x+
0 = x2 − zx + 1 z1 z1 2 −1 =⇒ x1 = + 2 2
½
z1 x2 = − 2 x3 =
z2 + 2
x4 =
z2 − 2
z 2 1
2
z 2 1
2
z 2 1
2
−1 −1 −1
0 = x6 + ax5 + bx4 + cx3 + bx2 + ax + 1 1 1 1 =⇒ 0 = x3 + ax2 + bx + c + b + a 2 + 3 x x x
1 1 1 3 2 = x + 3 +a x + 2 +b x+ +c x x x
1 x + 2 = x 2
!
1 x + 3 = x 3
2 1 x+ −2 x
3
1 1 x+ −3· x+ , x x
z := x + x1 " # $ % 1 x + 4 = x 4
4
2 1 1 x+ −4· x+ +2 x x
& ' ( )
* " + , - & . "
. /
!
" # $ % &
' ( ) * $ ) $ * ( + * ! $ , ( " * ' " +
*
+ * - + . $ , +
/ (
+ $
0 1
($ * $ ! 2 sin2 (a) + cos2 (a) = 1
& 3 * * # !$ ( $ ! $ & + ( * 0 3 $ $ *
cos(a + b) = cos(a) · cos(b) − sin(a) · sin(b)
' 4 ΔABC cos(b) = AB
sin(b) = BC
½ ¼
C a 1
A
b a
E
ΔBCD
∗
D
B
tan(a) = =⇒
BD BD = BC sin(b) BD = sin(b) · tan(a) = sin(b) ·
ΔACE
cos(a + b) = AE
ΔADE
cos(a) = =⇒
sin(a) cos(a)
AE cos(a + b) = AD AD cos(a + b) AD = cos(a)
AD + BD = AB = cos(b) =⇒ cos(b) =
cos(a + b) sin(a) + sin(b) · cos(a) cos(a)
=⇒ cos(a + b) = cos(a) · cos(b) − sin(a) · sin(b)
sin(a + b) = sin(a) · cos(b) + sin(b) · cos(a)
½ ½
π π sin(c) = cos c − cos(c) = − sin c −
2 2 π sin(a + b) = cos a + b − 2 π = cos a + b − 2 π π = cos(a) · cos b − − sin(a) · sin b − 2 2 = cos(a) · sin(b) + sin(a) · cos(b)
sin(a − b) = sin(a) · cos(b) − sin(b) · cos(a) cos(a − b) = cos(a) · cos(b) + sin(a) · sin(b)
sin(−c) = − sin(c)
! cos(−c) = cos(c) "# $ % ! sin(a − b) = sin (a + (−b)) = sin(a) · cos(−b) + sin(−b) · cos(a) = sin(a) · cos(b) − sin(b) · cos(a)
& cos(a − b) = cos (a + (−b)) = cos(a) · cos(−b) − sin(a) · sin(−b) = cos(a) · cos(b) + sin(a) · sin(b)
& ' ( ) a b " * ( #
tan(a + b) =
tan(a) + tan(b) 1 − tan(a) · tan(b)
tan(a − b) =
tan(a) − tan(b) 1 + tan(a) · tan(b)
½ ¾
sin(c) tan(c) = cos(c)
tan(a + b) =
sin(a + b) sin(a) · cos(b) + sin(b) · cos(a) = cos(a + b) cos(a) · cos(b) − sin(a) · sin(b)
cos(a) · cos(b)
tan(a + b) = =
cos(a) · cos(b) ·
cos(a) · cos(b) ·
sin(a)·cos(b) cos(a)·cos(b)
+
sin(b)·cos(a) cos(a)·cos(b)
cos(a)·cos(b) cos(a)·cos(b)
−
sin(a)·sin(b) cos(a)·cos(b)
tan(a) + tan(b) 1 − tan(a) · tan(b)
b −b tan(c) = − tan(−c)
sin(a) + sin(b) = 2 · sin
sin(a) − sin(b) = 2 · cos
sin(a) = sin
sin(b) = sin
a+b 2
· cos
a+b 2
· sin
a+b a−b + 2 2 a+b a−b − 2 2
a−b 2 a−b 2
! "# $
a+b a−b + 2 2 a+b a−b a−b a+b = sin · cos + sin · cos 2 2 2 2 a+b a−b sin(b) = sin − 2 2 a+b a−b a−b a+b = sin · cos − sin · cos 2 2 2 2
sin(a) = sin
sin(a) + sin(b) = 2 · sin
sin(a) − sin(b) = 2 · cos
a+b 2 a+b 2
· cos
· sin
a−b 2 a−b 2
,
.
½ ¿
cos(a) + cos(b) =
2 · cos
cos(a) − cos(b) = −2 · sin
a+b 2 a+b 2
· cos
· sin
a−b 2 a−b 2
a+b a−b + 2 2 a+b a−b a+b a−b = cos · cos − sin · sin 2 2 2 2 a+b a−b cos(b) = cos − 2 2 a+b a−b a+b a−b = cos · cos + sin · sin 2 2 2 2
cos(a) = cos
cos(a) + cos(b) = 2 · cos
cos(a) − cos(b) = −2 · sin
a+b 2
a+b 2
· cos
a−b 2
· sin
a−b 2
sin(a + b) · sin(a − b) = sin2 (a) − sin2 (b) cos(a + b) · cos(a − b) = cos2 (a) − sin2 (b)
sin(a + b) · sin(a − b) = (sin(a) · cos(b) + sin(b) · cos(a)) · (sin(a) · cos(b) − sin(b) · cos(a))
½
= (sin(a) · cos(b))2 − (sin(b) · cos(a))2 = sin2 (a) · cos2 (b) − sin2 (b) · cos2 (a)
cos (c) = 1 − sin (c) 2
2
= sin2 (a) · (1 − sin2 (b)) − sin2 (b) · (1 − sin2 (a)) = sin2 (a) − sin2 (a) · sin2 (b) − sin2 (b) + sin2 (b) · sin2 (a) = sin2 (a) − sin2 (b)
cos(a + b) · cos(a − b) = (cos(a) · cos(b) − sin(a) · sin(b)) · (cos(a) · cos(b) + sin(a) · sin(b)) 2
2
= cos (a) · cos (b) − sin2 (a) · sin2 (b) = cos2 (a) · (1 − sin2 (b)) − (1 − cos2 (a)) · sin2 (b) = cos2 (a) − sin2 (b)
sin(a) · sin(b) =
cos(a − b) − cos(a + b) 2
cos(a) · cos(b) =
cos(a − b) + cos(a + b) 2
cos(a + b) = cos(a) · cos(b) − sin(a) · sin(b) cos(a − b) = cos(a) · cos(b) + sin(a) · sin(b)
cos(a − b) − cos(a + b) = 2 · sin(a) · sin(b) =⇒ sin(a) · sin(b) =
cos(a − b) − cos(a + b) 2
cos(a − b) + cos(a + b) = 2 · cos(a) · cos(b) =⇒ cos(a) · cos(b) =
cos(a − b) + cos(a + b) 2
½
sin(2a) = 2 · sin(a) · cos(a)
b = a =⇒ sin(a + a) = sin(a) · cos(a) + sin(a) · cos(a) =⇒ sin(2a) = 2 · sin(a) · cos(a)
cos(2a) =
⎧ 2 2 ⎪ ⎪ ⎨ cos (a) − sin (a) 2 · cos2 (a) − 1 ⎪ ⎪ ⎩ 1 − 2 · sin2 (a)
b = a =⇒ cos(a + a) = cos(a) · cos(a) − sin(a) · sin(a) =⇒ cos(2a) = cos2 (a) − sin2 (a)
sin2 (a) = 1 − cos2 (a) !
cos(2a) = cos2 (a) − 1 − cos2 (a) cos(2a) = 2 · cos2 (a) − 1
" # # cos2 (a) = 1 − sin2 (a) ! cos(2a) = (1 − sin2 (a)) − sin2 (a) cos(2a) = 1 − 2 · sin2 (a)
½
tan(2a) =
2 · tan(a) 1 − tan2 (a)
b = a
sin
a 2
=±
1 − cos(a) 2
a ∈ [0 + 4kπ,2π + 4kπ], cos
a 2
=±
k ∈ Z
1 + cos(a) 2
a ∈ [−π + 4kπ, π + 4kπ],
k ∈ Z
sin
sin(a) · sin(b) =
cos(a − b) − cos(a + b) 2
a
cos
2
· sin
a 2
=
a
=1
2
−
a 2
− cos 2
a 2
+
a 2
cos(0) − cos(a) 2 sin = 2 2 a 1 − cos(a) =⇒ sin =± 2 2 a
cos(a − b) + cos(a + b) cos(a) · cos(b) = 2 a a cos( a − a ) + cos( a + a ) 2 2 2 2 cos · cos = 2 2 2 a 1 + cos(a) cos2 = 2 2 a 1 + cos(a) =± cos 2 2
½
tan
a 2
=
sin(a) 1 − cos(a) = 1 + cos(a) sin(a)
! " tan2
a 2
=
sin2 ( a2 ) 1 − cos(a) = cos2 ( a2 ) 1 + cos(a)
# $ % 1 + cos(a) # " =
1 − cos2 (a) (1 + cos(a))2
& ' sin2 (a) = 1 − cos2 (a) ( % # $ 1 − cos(a)
1 + cos(a) ' $ %
a ∈ [−1,1]
1 − a2 cos(arcsin(a)) = 1 − a2 sin(arccos(a)) =
#
"
cos2 (c) + sin2 (c) = 1
arccos(a) c
cos2 (arccos(a)) + sin2 (arccos(a)) = 1 ⇒ a2 + sin2 (arccos(a)) = 1 ⇒ sin(arccos(a)) = 1 − a2
½
arcsin(a) cos2 (arcsin(a)) + sin2 (arcsin(a)) = 1 ⇒ cos(arcsin(a)) = 1 − a2
sin2 (arctan(a)) =
a2 a2 + 1
cos2 (arctan(a)) =
1 a2 + 1
a = tan(arctan(a)) =
sin(arctan(a)) sin(arctan(a)) = cos(arctan(a)) 1 − sin2 (arctan(a))
⇒ a2 · (1 − sin2 (arctan(a)) = sin2 (arctan(a)) ⇒ a2 = sin2 (arctan(a)) + a2 · sin2 (arctan(a)) ⇒
sin2 (arctan(a)) =
a2 +1
a2
sin2 (arctan(a)) 1 = −1 cos2 (arctan(a)) cos2 (arctan(a)) 1 ⇒ a2 + 1 = cos2 (arctan(a))
a2 = tan2 (arctan(a)) =
sin(a+b) cos(a+b) !"# $ !%
x
½
ei·x = cos(x) + i · sin(x)
x = a + b ei·(a+b) = cos(a + b) + i · sin(a + b)
ei·(a+b) = ei·a · ei·b ei·a · ei·b = (cos(a) + i · sin(a)) · (cos(b) + i · sin(b)) = cos(a) · cos(b) + cos(a) · i · sin(b) + i · sin(a) · cos(b) − sin(a) · sin(b) = cos(a) · cos(b) − sin(a) · sin(b) + i · (sin(a) · cos(b) + sin(b) · cos(a))
cos(a + b) = cos(a) · cos(b) − sin(a) · sin(b) sin(a + b) = sin(a) · cos(b) + sin(b) · cos(a),
! sin(3a) = 3 · sin(a) − 4 · sin3 (a)
" # ! ei·x = cos(x) + i · sin(x) $ ! % ei·3·a = cos(3 · a) + i · sin(3 · a)
ei·3a = (ei·a )3 = (cos(a) + i · sin(a))3
$ & !
!" #$ %% & % ' %
( ) ) * + $" * " & ) , * - .# / # " , #
( !0 & 1
b / # " , " 213+ a f (x) dx 1 4 5 " + ) [a, b] n * ) [xk−1 , xk ] $ , a = x0 < x1 < . . . < xn = b ( 6 4 f 7 * ) !8 ! - 1 Of = =
n k=1 n k=1
sup {f (z) | z ∈ [xk−1 , xk ]} · (xk − xk−1 ) sup {f (z) | z ∈ [xk−1 , xk ]} · Δxk
½ ¾
Uf = =
n k=1 n
inf {f (z) | z ∈ [xk−1 , xk ]} · (xk − xk−1 ) inf {f (z) | z ∈ [xk−1 , xk ]} · Δxk
k=1
I = ab f (x) dx
Uf ≤ I ≤ Of [a, b]
n m !
aij j " n i=1 j=1 aij i " m # $
j i
i j % m n
aij =
i=1 j=1
n m
aij
j=1 i=1
& '
(
) % m n
α · aij = α ·
i=1 j=1
m n
aij
i=1 j=1
% m n
(aij + bij ) =
i=1 j=1
*
m n i=1 j=1
Ω
aij +
m n
bij
i=1 j=1
f (x, y) dx dy
Ω+ Ω ,# - . $ ab f (x) dx / # ,# 0 f [a, b] 1 f 2 ' $ Ω f (x, y) dx dy / 3 f (x, y) $ Ω
& #
f (x, y) dx dy # Ω . Ω R : a ≤ x ≤ b c ≤ y ≤ d # Ω f (x, y) dx dy
½ ¿
y d
R
c x a
b
R R xy ! f " R [a, b] Z1 = {x0 , x1 , . . . , xm } [c, d] Z2 = {y0 , y1 , . . . , yn } # Z = Z1 × Z2 = {(xi , yj ) | xi ∈ Z1 , yj ∈ Z2 } $
R Z % & (xi , yj ) R m n &&
Rij ' ()* Rij : xi−1 ≤ x ≤ xi yj−1 ≤ y ≤ yj y d = yn (xi , yj )
yj Rij yj−1 c = y0
x a = x0
xi−1
xi
xm = b
R # R + ' Rij ) f f " , Rij -. - / Mij
½
z = f (x, y)
Rij
R
f Rij
R m R R A = (x − x ) · (y − y ) = Δx · Δy
m · A ≤ M · A ij
ij
ij
ij
ij
ij
ij
ij
i
i−1
j
j−1
i
j
ij
Mij mij
Rij
Rij
mij Mij
Uf =
n m i=1 j=1
mij · Δxi · Δyj ,
½
Of =
m n
Mij · Δxi · Δyj .
i=1 j=1
I Uf ≤ I ≤ O f
I
R
f (x, y) dx dy
! " " #
d
b
f (x, y) dx dy =
f (x, y) dx dy c
R
a
$ " " % & " #
b
f (x, y) dx dy =
f (x, y) dy dx a
R
d
c
' (
) I ' ) " f $
' f * R ' * ) +
, $ (-
) . R : 1 ≤ x ≤ 4 1 ≤ y ≤ 3 ) f (x, y) = x − y + 2 / 0 12 23
R
f (x, y) dx dy
34 1
"
1
4
(x − y + 2) dx =
1 2 ·x −y·x+2·x 2
4 #
3 1
1
(x − y + 2) dx dy
4 = −3y + 1
3 27 3 2 27 −3y + dy = − · y + · y = 15. 2 2 2 1
(-
$ ) 15 $
27 , 2
½
z
f (x, y)
x
R
y
(x − y + 2) dy dx
f(x, y) = sin(x) · sin(y) ! " sin(x) sin(y) dx dy ≈ 1,827
# f ! R $ # % 4 1
3 1
3 1
4 1
& f (x, y) dx dy# Ω " # % '() * Ω
Ω
½
Ω f (x, y) (x, y) ∈ Ω Ω ! "## Ω f (x, y) dx dy $ % % " &
'( ) f y h(x)
Ω
g(x) x a
b
" * Ω x = a x = b % y = h(x) y = g(x) + g(x) ≤ h(x) a ≤ b , % Ω % - ## . / &
'( 01 y h(x) Ω
g(x)
a
Ω
b
x
½
y
x
! "
# # $% " $
F
&
dx · dy
'#(
)% * )% + #,( ,
f (x, y) · dx · dy
y h(x) dy
F
g(x)
a
x
b
dx
" '# *
+ "
f (x, y)
+
y
Vx
' (
" -
.
y h(x)
g(x) a
xi−1 xi x
b
x
g(x)
h(x)
½
h(x)
Vx =
f (x, y) dy g(x)
Ω Vx a b x !
V =
b
f (x, y) dx dy = a
Ω
b
h(x)
Vx dx =
f (x, y) dy dx a
g(x)
" Ω # $%!$$ y&' ( a b x&' ( g(y) h(y) ! ) )
! * "
V =
b
h(y)
f (x, y) dx dy =
f (x, y) dx dy a
Ω
g(y)
y b g(y)
h(y)
a x
+
, ' - r !
f (x, y) = r2 − x2 − y 2 ) ' ( ! Ω ' ' xy&. ( ! #
Ω : −r ≤ x ≤ r − r2 − x2 ≤ y ≤ r2 − x2 , V ' V =2·
r
√
f (x, y) dx dy = Ω
−r
−
r2 −x2
√
r2 −x2
r2 − x2 − y 2 dy dx
¾¼¼
√
y = sin(u) · r2 − x2 dy = √ cos(u) · r2 − x2 · du √ r 2 − x2 π u1 = arcsin − √ =− 2 2 2 r −x
√ 2 r − x2 π √ u2 = arcsin = 2 r 2 − x2
π 2
−π 2
r2 − x2 · cos(u)2 · r2 − x2 du = (r2 − x2 ) · π/2
V =2·
r −r
π 4 · (r2 − x2 ) dx = π · · r3 2 3
½º
(α · f (x, y) + β · g(x, y)) dx dy = α ·
Ω
f (x, y) dx dy + β ·
Ω
g(x, y) dx dy Ω
Ω f ≥ 0 Ω f (x, y) dx dy ≥ 0 Ω f ≤ g Ω f (x, y) dx dy ≤ Ω g(x, y) dx dy ! "
# $ % Ω n %
$ % Ωi f Ω Ωi
f (x, y) dx dy = Ω
f (x, y) dx dy + . . . + Ω1
f (x, y) dx dy Ωn
& f g Ω g Ω (x0 , y0 ) ∈ Ω Ω
f (x, y) · g(x, y) dx dy = f (x0 , y0 ) ·
g(x, y) dx dy Ω
¾¼½
f Ω
Ω
f m M ! " m · g(x, y) ≤ f (x, y) · g(x, y) ≤ M · g(x, y)
# "
m · g(x, y) dx dy ≤
Ω
f (x, y) · g(x, y) dx dy ≤
Ω
! $" m·
g(x, y) dx dy ≤
Ω
M · g(x, y) dx dy
Ω
f (x, y) · g(x, y) dx dy ≤ M ·
Ω
g(x, y) dx dy Ω
%
g(x, y) ≥ 0
& g(x, y) dx dy ≥ 0 Ω g(x, y) dx dy = 0
" Ω m·0≤
f (x, y) · g(x, y) dx dy ≤ M · 0
Ω
Ω f (x, y) · g(x, y)dx dy = 0
(x0 , y0 ) ∈ Ω Ω g(x, y) dx dy > 0
" m≤
Ω
f (x, y) · g(x, y) dx dy ≤M g(x, y) dx dy Ω
' % ( (x0 , y0 ) ∈ Ω f (x0 , y0 ) =
Ω
f (x, y) · g(x, y) dx dy , g(x, y) dx dy Ω
f Ω ) m M Ω
!
f (x, y) · g(x, y) dx dy = f (x0 , y0 ) ·
Ω
g(x, y) dx dy Ω
(x0 , y0 ) ∈ Ω
g(x, y) = 1
"
(x0 , y0 ) ∈ Ω
IΩ :=
Ω
f (x, y) dx dy = f (x0 , y0 ) · IΩ
Ω
dx dy
Ω.
¾¼¾
Ω f (x, y) dx dy f (x, y) Ω ! " # ! # # $ Ω 1 dx dy % D " uv & # x = x(u, v) y = y(u, v) xy' # ( S ) *+ *, " # p(u, v) = (x, y) = (x(u, v), y(u, v)) - pu pv
$ xu · yv − yu · xv = 0 y
v x = x(u, v) y = y(u, v) ⇐⇒
D
S
u = u(x, y) v = v(x, y) u
x
. / 0$ S 1 ui vi 1 ui & # 1 p(ui , v) = (x(ui , v), y(ui , v)) v ( ui 1 vi p(u, vi ) = (x(u, vi ), y(u, vi )) u ( vi / (ui , vk ) ∈ D / (x(ui , vk ), y(ui , vk )) ∈ S 2 ) *+ *34 5 0$ ) *+ *3
( !6 2 ) *+ *74 . ! xy' dx · dy . ) $ ( u v . # /8 A1 := (x(u1 , v1 ), y(u1 , v1 )) A2 := (x(u2 , v1 ), y(u2 , v1 ))
¾¼¿
y v
v2 v3 u1 , v1
v2
v1 u0
v0
u0
u1
u2 u3
u1
x(u1 ,v1 ) y(u1 ,v1 )
u2
u3
v1 v0 x
u
= (x(u1 + Δu, v1 ), y(u1 + Δu, v1 )) A3 := (x(u1 , v2 ), y(u1 , v2 )) = (x(u1 , v1 + Δv), y(u1 , v1 + Δv)) A4 := (x(u2 , v2 ), y(u2 , v2 )) = (x(u1 + Δu, v1 + Δv), y(u1 + Δu, v1 + Δv))
S A A S A A t = p t = p 1
2
2
1
3
Δu→0
v2
p(u1 + Δu, v1 ) − p(u1 , v1 ) Δu
(x(u1 , v2 ), y(u1 , v2 ))
(x(u2 , v2 ), y(u2 , v2 )) dy
(x(u1 , v1 ), y(u1 , v1 ))
v1
(x(u2 , v1 ), y(u2 , v1 ))
u1
u2 dx
S
2
1
v
t1 = lim
1
u
¾¼
Δu · t1 ≈ p(u1 + Δu, v1 ) − p(u1 , v1 ) ≈ Δu · pu
t2 = lim
Δv→0
p(u1 , v1 + Δv) − p(u1 , v1 ) , Δv
Δv · t2 ≈ p(u1 , v1 + Δv) − p(u1 , v1 ) ≈ Δv · pv .
Δu · pu Δv · pv ! " # $ %& ' $ z = 0 $& ( ) $*
pu × pv · Δu · Δv u1
S2
t2
u1 + Δu
v1 + Δv
t1
S1
v1
+ Δu Δv ( & $ dx · dy = pu × pv · du · dv,
1 dx dy =
, $ *
S
pu × pv du dv.
D
⎛ ⎞ ⎛ ⎞
∂x(u,v) ∂x(u,v)
∂u ∂v
⎜ ∂y(u,v) ⎟ ⎜ ∂y(u,v) ⎟ ⎜ ⎟×⎜ ⎟
pu × pv =
⎝ ∂u ⎠ ⎝ ∂v ⎠
0 0
¾¼
∂x(u, v) ∂y(u, v) ∂x(u, v) ∂y(u, v) = · − · ∂u ∂v ∂v ∂u = |xu · yv − xv · yu |
p : (0, ∞)×(−π, π] → R2 \{0},
p(r, φ) = (x(r, φ), y(r, φ)) = (r·cos(φ), r·sin(φ))
p−1 (x, y) = (r(x, y), φ(x, y)) r(x, y) =
x2 + y 2
φ(x, y) = (sign(y) + 1 − | sign(y)|) · arccos x/ x2 + y 2
xr · yφ − xφ · yr = cos(φ) · r · cos(φ) − r · (− sin(φ)) · sin(φ) = r.
(x, y) | x
1
−1
√
−
2
+ y2 ≤ 1
1−x2
√
1 dy dx 1−x2
!
" D : 0 ≤ r ≤ 1, −π < φ ≤ π # $
1
−1
√
−
1−x2
√
1−x2
ππ
1 dy dx = −
1
r dr dφ = π 0
S " f (x, y) %
D& $
f (x, y) dx dy = S
D
f (x(u, v), y(u, v)) · pu × pv du dv
¾¼
D n D1 , . . . , Dn S n Si ! "# $
f (x(u, v), y(u, v)) · pu × pv du dv =
D
n
f (x(u, v), y(u, v)) · pu × pv du dv
i=1 Di
% ! (ai , bi ) ∈ Di $
f (x(u, v), y(u, v))· pu × pv du dv = f (x(ai , bi ), y(ai , bi ))·
Di
pu × pv du dv
Di
% xi = x(ai , bi ) yi = y(ai , bi ) Si # $
Di
pu × pv du dv =: ISi
f (x(u, v), y(u, v)) · pu × pv du dv =
mi % mi ≤ f (xi , yi ) ≤ Mi $ n
f (xi , yi ) · ISi
i=1
D
Uf =
n
&
mi · ISi ≤
i=1
Mi
%'
f (x(u, v), y(u, v)) · pu × pv du dv ≤
n
f
"
Si
Mi · ISi = Of
i=1
D
& D f (x(u, v), y(u, v))· pu × pv du dv () * " & f S n → ∞ " " $
f (x, y) dx dy ≤
S
f (x(u, v), y(u, v)) · pu × pv du dv ≤
D
f (x, y) dx dy S
$
f (x, y) dx dy = S
f (x(u, v), y(u, v)) · pu × pv du dv
D
+ , $
f (x, y) dx dy = S
f (r · cos(φ), r · sin(φ)) · r dr dφ
D
- . /- /
2 z = 2 − x + y 2 / Ω : (x − 1)2 + y 2 ≤ 1, z = 0 ! $
V = Ω
2−
x2 + y 2 dx dy
¾¼
r=
x2 + y 2
dx · dy = r · du · dv.
Ω (x−1)2 +y2 ≤ 1 x2 +y2 ≤ 2x r2 ≤ 2 · r · cos(φ),
0 ≤ r ≤ 2 · cos(φ) φ π2 − π2 ≤ φ ≤ 0 !
1 1 Γ:− ·π ≤φ≤ ·π 2 2
V =
2−
Ω
=
1 π 2
− 12 π
=
1 π 2
− 12 π
0 ≤ r ≤ 2 · cos(φ)
x2 + y 2 dx dy
2·cos φ
0
(2 − r) · r dr dφ
1 r − · r3 3 2
2·cos(φ) dφ 0
= ... = 2π −
32 9
u = x + y v = x − y !" u # $ v # Γ uv %& '( ) x y ! ( u v & x = y= *" '( Γ " Ω & p(u, v) = , ) ( x · y dx dy Ω Ω x2 + y 2 = 4 x2 + y 2 = 9 x2 − y 2 = 1 x2 − y 2 = 4 2
2
2
2
u+v 2
u−v 2
u+v 2
u−v 2
pu =
1 1 √ √ √ √ , 2· 2· u+v 2· 2· u−v
pv =
2·
√
1 1 ,− √ √ √ 2· u+v 2· 2· u−v
.
¾¼
y x2 + y 2 = 9
3
x2 − y 2 = 1 x2 − y 2 = 4
x2 + y 2 = 4
2
1
1
2
3
x
⎛ ⎞ ⎛ ⎞
1 √ 1√
2·√2·√u+v
2· 2· u+v
⎜ ⎟ ⎜ ⎟ ⎟ × ⎜− √ 1√ ⎟ ⎜ √ 1√
pu × pv =
⎝ 2· 2· u−v ⎠ ⎝ 2· 2· u−v ⎠
0 0 1 1 1 1 1 1 √ √ = − · · √ − · ·√ 4 2 4 2 u+v· u−v u + v · u − v 1 1 √ = ·√ 4 u+v· u−v
Ω
x · y dx dy =
1
4
4
9
u+v · 2
u−v 1 15 1 √ · ·√ du dv = 2 4 8 u+v· u−v
!"
# $ % " & " '
( %) *! + + % ) % " , % ! + % * % ! " -) ! % ., ! %% / 0 ) R2 R3 1 % + % * p : [a, b] → Rn 2 + n ∈ {2,3} [a, b] % , % ( % * % % & + !)3!
% 2 + ( , %% 4 p 2 p(a) p(b) 4 2 "% * " %3 ! ) ) $ 3 ! %)) $ % %% 5 2 "% % " ) ( [a, b) , 6! %2 +% 2 %% % % -% * " %, $ 3 !/ %
. C := {p(t) | t ∈ [a, b]} + p -% % 6 2 73 %% ) ) 2 + ! %2 +% ) %/ " + 73 % 8 !+%92 % ) $ , 3 ! % 0 * " %3 !2 % + ) :) % p : [a, b] → Rn , p(t) = p(b + a − t)
¾½¼
p1 : [a, b] → Rn p1 p2
p2 : [b, c] → Rn
n
p : [a, c] → R , p(t) =
C1 + C2
p1 (t)
t ∈ [a, b]
p2 (t)
t ∈ [b, c]
Ci
pi
! "
t ∈ [a, b]
p (t) = 0
#
$ % & " $ # ' & # & (
f : [a, b] → R ) *
" +
p : [a, b] → R2 , p(t) = (t, f (t))
L * " + f (x) # [a, b] ) ( b L = a 1 + f (x)2 dx - p(t) = (x(t), y(t)) a ≤ t ≤ b b % ( x (t)2 + y (t)2 dt a
, % (
z
t4
t5 t3 t2
tm−1
t6
t1 = a
y
tm = b
x R3 C = {p(t) | t ∈ [a, b]} a ≤ t ≤ b R3 #
-
p(t) = (x(t), y(t), z(t))
#
¾½½
L L = ab x (t)2 + y (t)2 + z (t)2 dt m ! t1 < t2 < · · · < tm t1 = a" tm = b # ! $ # ti ti+1 % &' ( li ! ( #! si "
ti+1 si
Δzi li
ti
Δyi Δxi
C
(Δxi )2 + (Δyi )2 + (Δzi )2
C L = l1 + . . . + lm−1 ≈ s1 + . . . + sm−1 = (Δx1 )2 + (Δy1 )2 + (Δz1 )2 + . . . + (Δxm−1 )2 + (Δym−1 )2 + (Δzm−1 )2 =
m−1 i=1
=
m−1
(Δxi )2 + (Δyi )2 + (Δzi )2
i=1
Δxi Δti
2
+
Δyi Δti
2
+
Δzi Δti
2 Δti
! "# $ %& max
i=1,...,m−1
Δti ,
Δti = ti+1 − ti
' ( [a, b] " # $ si ) $ li # b dx 2 dy 2 dz 2 * "$ ( a dt + dt + dt dt (
$
L=
b
a
dx dt
2
+
dy dt
2
+
dz dt
2
dt.
¾½¾
! C " # $ %& ' ! $ '! f (x, y) ( )
'! * f (x, y, z) + f ! f (x, y) = f0 $ f0 ·L L "
! , - + , # ! , n = 3 f C p : [a, b] → R3 , p(t) = (x(t), y(t), z(t)) a ≤ t ≤ b . / [a, b] m ! t1 < t2 < · · · < tm t1 = a tm = b m − 1 0 0 % & Bi = f (x(τi ), y(τi ), z(τi )) τi ∈ (ti , ti+1 )
B ' 1li " 0 [ti , ti+1 ]23 B ≈ B1 · l1 + . . . + Bm−1 · lm−1 ≈
m−1
Bi ·
i=1
=
m−1
(Δxi )2 + (Δyi )2 + (Δzi )2
Bi ·
i=1
Δxi Δti
2
+
Δyi Δti
2
+
Δzi Δti
2 Δti
" , . ! ! ' +
b
a
f (x(t), y(t), z(t)) ·
dx dt
2
+
dy dt
2
+
dz dt
2 dt,
, ' ( 4 * 5
C
f (x, y, z) ds :=
b
a
f (x(t), y(t), z(t)) ·
dx dt
2
+
dy dt
2
+
dz dt
2 dt.
¾½¿
Rn
! " # $"
# g:
x2 y2 + = 1, x, y ≥ 0, 4 9
# d(x, y) = xy % %
# & $'
# (
# )
p(ϕ) =
x(ϕ) y(ϕ)
=
2 cos(ϕ) 3 sin(ϕ)
% ) M=
π 2
dx dϕ
= −2 sin(ϕ)
M= 0
π 2
dy dϕ
x(ϕ) · y(ϕ) ·
0
= 3 cos(ϕ)
2 cos(ϕ) · 3 sin(ϕ) ·
,
dx dϕ
2
π ϕ ∈ 0, . 2
+
dy dϕ
2 dϕ
38 (−2 sin(ϕ))2 + (3 cos(ϕ))2 dϕ = 5
! ! " #
# " $ % &! '&!! ()"*+" $ , P K - r r = 1 . l !
# &! W = (K · r) · l = K · (r · l) = K · x ' $ +"
¾½
K x
r
P
K·
r
l
K ⎛
⎞
u(x, y, z)
⎜ ⎟ ⎟ K(x, y, z) = ⎜ ⎝ v(x, y, z) ⎠ w(x, y, z)
C
p : [a, b] → a ≤ t ≤ b ! "#$$
%&'(
R3 , p(t) = (x(t), y(t), z(t))
z
t1 = a
y
tm = b
x
) K(x, y, z)
$ ) * #$ $ +
¾½
a = t1 < t2 < ... < tm = b ! li " # # K(p(ti )) ·
p (ti ) li , p (ti )
$ % & '"& ' ( # W ! & ' # ! ) W ≈
m−1
K(p(ti )) ·
i=1 m−1
p (ti ) li p (ti )
p (ti ) (Δxi )2 + (Δyi )2 + (Δzi )2 p (ti ) i=1 i 2 i 2 i 2 m−1 ( Δx ) + ( Δy ) + ( Δz ) Δti Δti Δti = K(p(ti )) · p (ti ) Δti p (ti ) ≈
K(p(ti )) ·
i=1
! * & + )
b
a
dx 2 dt
K(p(t)) · p (t)
+
dy 2 dt
+
dz 2 dt
dt =
p (t)
b a
K(p(t)) · p (t) dt
, # - ( + h . C : p(t) = (x(t), y(t), z(t)), t ∈ [a, b] % . h . C - )
C
h d x :=
b
a
h(p(t)) · p (t) dt
/ 0 & ! 1 . ! . ! " ' ( * ! % (
& % h h(x, y, z) = (xy, yz, xz) . p p(t) = (t, t2 , t3 ), t ∈ [−1,1] ⎛
⎞
1
⎜ ⎟ ⎟ h(p(t)) · p (t) = h(t, t2 , t3 ) · ⎜ ⎝ 2t ⎠ 3t2
¾½
⎛
⎞ ⎛
t · t2
⎞ 1
⎜ ⎟ ⎜ ⎟ 2 3⎟ ⎜ ⎟ =⎜ ⎝t · t ⎠ · ⎝ 2t ⎠ t · t3
3t2
= t · t2 · 1 + t2 · t3 · 2t + t · t3 · 3t2 = t3 + 5t6
3 10 t + 5t6 dt = . 7 −1 1
C C1 , C2 , . . . , Cn = + +... + C
C1
C2
Cn
C2 C3 C1
C4
g h C a
C
a · (g + h) d x=a·
C
g d x+
C
h d x
C C
C
h d x=−
C
h d x
p(t) C r(t) = p(a + b − t) C r (t) = −p (a + b − t)
¾½
z
C1 C2
C3
y
x
f (x, y, z)
h h = ∇f
! "
p(t)
#
C
∇f d x = f (p(b)) − f (p(a))
$% &' ( ' )%
C
∇f d x=
b
a b
= ∇f (p(t)) · p (t)!
∇f (p(t)) · p (t) dt
=
df (p(t)) dt
a
df (p(t)) dt dt
= f (p(b)) − f (p(a)). " % # *
p(a) = p(b)
%
f (p(b)) − f (p(a)) = 0!
) +,!, !
C1
( '
C1
C2
% * ' !
C3
-
¾½
h=
P (x, y)
Q(x, y)
=
y2 2xy −
y
p(t) = (cos(t), sin(t)), t ∈ 0, = π 2
∂P (x,y) ∂y
2y '
∂Q(x,y) ∂x
! " # $ %% & f ( !
∂f ∂f = y2 , = 2xy − y ∂x ∂y
) * ' x f (x, y) = xy2 + g(y) y * ' ∂f = 2xy + g (y) = 2xy − ∂y
y
f (x, y) = xy 2 − ey +c
π f p − f (p(0)) = f (0,1) − f (1,0) = − + c − (−1 + c) = 1 − 2
! " ## $ % & ' & ( & ! ) ! "
& * + , ' % & - &
. ) ! ( / % 0 ! R3 & + , % ) ! ' .' &( ! 1 ! 2 % & , * 3% ) P : B → R3 ' & B 4 (! 5! R2 ' & ! / & 0 ' $ 1 % !! ' ! 6 ⎛ ⎞ xu (u, v) xv (u, v) ⎜ ⎟ ⎟ JP (u, v) = P (u, v) = (Pu , Pv ) = ⎜ ⎝ yu (u, v) yv (u, v)⎠ zu (u, v) zv (u, v) (u, v) ∈ B 0 * " / P (B) & P . ' & 7 ! ! ' * & / P (B) * &
¾¾¼
B R2 P : B1 → R3 r : B2 → B1 ! " # $ P ◦ r
P v
R3
R2 u
% & ' ( E = Pu 2 = x2u + yu2 + zu2 = Pu , Pu G = Pv 2 = x2v + yv2 + zv2 = Pv , Pv F = xu · xv + yu · yv + zu · zv = Pu , Pv g = E · G − F2 = det P T · P = Pu × Pv 2
) P * E = G F = 0 + g = 1 , '
-, z = f (x, y) . /0 1 B ⊆ R2 f : B → R $ -, f {(x, y, z) ∈ B × R | z = f (x, y)} # P : B → R3 P (x, y) = (x, y, f (x, y))
¾¾½
z z = f (x, y)
y
B
B → R2
x
⎛
⎞ 1
⎜ P = ⎜ ⎝0 fx
0
⎟ 1⎟ ⎠, fy
E = 1 + fx2 G = 1 + fy2 F = fx · fy g = E · G − F2 = 1 + fx2 + fy2
P
M ! " # # # $ % $ &' ( ( B !" )*'+&'
, u0 v0 P - $ P (u0 , v) P (u, v0 )' .
M $ % →a = → Pv (u0 , v0 )Δv b = Pu (u0 , v0 )Δu /
' ( $ % 0
¾¾¾
M a
P v B
P (u, v0 )
A
b
v0 + Δv P (u0 , v) v0
A
A = (u0 , v0 )
u
u0 u0 + Δu
A = Pu (u0 , v0 )
a = Pv (u0 , v0 ) · Δv b = Pu (u0 , v0 ) · Δu
→ →
a × b = Pu (u0 , v0 )Δu × Pv (u0 , v0 )Δv = Pu (u0 , v0 ) × Pv (u0 , v0 )·ΔuΔv
!
" #
F $ " M Fi % & ' ( )
n
F = F1 + . . . + Fn ≈
Pu (ui , vi ) × Pv (ui , vi ) · ΔuΔv
i=1
! $ *+ ,-./ " 0
1 )
F =
Pu × Pv du dv
B
2 + ! 3 )
F P
: B → R3 3 45
F =
Pu × Pv du dv
B
6
0
7 ! 1
3 "
¾¾¿
P v
R3 u
F =
√
g du dv =
B
E · G − F 2 du dv B
√ g = M = B 1 du dv B ! " # $ z = f (x, y) %
&$!
1
F =
1 + fx2 + fy2 dx dy B
' " ( ) * ) + r , - ⎛
r · cos(u) · cos(v)
⎞
⎜ ⎟ ⎟ P : B → R3 , P (u, v) = ⎜ ⎝ r · sin(u) · cos(v) ⎠ , r · sin(v)
B := (u, v) | 0 ≤ u < 2π, , ⎛
− 21 · π ≤ v ≤
⎞ −r · sin(u) · cos(v) ⎜ ⎟ ⎟ Pu = ⎜ ⎝ r · cos(u) · cos(v) ⎠ , 0
1 2
·π
⎛ ⎞ −r · cos(u) · sin(v) ⎜ ⎟ ⎟ Pv = ⎜ ⎝ −r · sin(u) · sin(v) ⎠ . r · cos(v)
¾¾
⎛ ⎞ r2 · cos(u) · cos(v)2 ⎜ ⎟ ⎜ r2 · sin(u) · cos(v)2 ⎟ . ⎝ ⎠ r2 · sin(v) · cos(v) Pu × Pv = r2 · cos(v).
F =
π 2
−π 2
0
2π
r2 · cos(v) du dv = 4π · r2
f (x, y, z) !" "
""# $ % P "&
' ()#*+
, ' + $ % , - " " $ % # . !" " G -& " / " % 0 1 2" $"" /
G= B
f (x(u, v), y(u, v), z(u, v)) · Pu × Pv du dv
¾¾
f dσ :=
P
f (x(u, v), y(u, v), z(u, v)) · Pu × Pv du dv
B
f P
S
l ! w z " b #
$% &'
( )
⎛ ⎞ u · cos(w · v) ⎜ ⎟ 2π ⎟ P (u, v) = ⎜ ⎝ u · sin(w · v) ⎠ , 0 ≤ u ≤ l, 0 ≤ v ≤ w b·v
* f (x, y, z) = - ) G=
)
0
2π w
0
l
x2 + y 2 + ,
f (x(u, v), y(u, v), z(u, v)) · Pu × Pv du dv
⎛ ⎞ ⎛ ⎞
−uw · sin(wv)
cos(wv)
⎜ ⎟ ⎜ ⎟
⎟ ⎜ ⎟
⎜ Pu × Pv =
⎝ sin(wv) ⎠ × ⎝ uw · cos(wv) ⎠
0 b
¾¾
⎛ ⎞
b · sin(wv)
⎜ ⎟
⎜ ⎟
=
⎝−b · cos(wv)⎠
uw
G= = =
0
2π w
0
2π w
0
l
0
u·
b2 + u2 · w2 du dv
l 1 2 2 2 3/2 · (b + w · u ) dv 3 · w2 0 2π w · (b2 + w2 · l2 )3/2 − b3 · dv
1 3 · w2 2·π 2 2 2 3/2 3 = · (b + w · l ) − b 3 · w3
=
b2 + u2 · w2
l (u · cos(w · v))2 + (u · sin(w · v))2 · b2 + u2 · w2 du dv
2π w
0
=
0
z
y
x
! !" # $ ! % &" ' (x, y, z) ! ! () n )! " () −n * (x, y, z) → n(x, y, z) +
¾¾
(x, y, z) → n(x, y, z) k ! " ! # $ " %
n k
$ & ' n ( ! )! * k dO ! +
dO
k
n
' ! , & ! ! & .! ! $ .! & ' #. , k · n ! /0 dO ! * k · n dO 1 ! / ! 0 2
& ! 3 k · n dO
S
¾¾ S
S
k
⎛ ⎞ ⎛ ⎞ x x(u, v) ⎜ ⎟ ⎜ ⎟ ⎟ = ⎜y(u, v)⎟ P (u, v) = ⎜ y ⎝ ⎠ ⎝ ⎠ z z(u, v)
(u, v) ∈ D
Pu × Pv
n = dO
!
Pu × Pv · du dv " k · n dO = k · S
1 · (Pu × Pv ) Pu × Pv
D
=
!
dO =
1 · (Pu × Pv ) · Pu × Pv du dv Pu × Pv
k · (Pu × Pv ) du dv
D " #
$%%
⎛ ⎞ ⎛ ⎞ ∂x(u,v) k1 ⎜ ⎟ ⎜ ∂u ⎟ k = ⎜k2 ⎟ , Pu = ⎜ ∂y(u,v) ⎟ ⎝ ⎠ ⎝ ∂u ⎠ ∂z(u,v) k3 ∂u & '
() !
⎛
Pv =
⎞
∂x(u,v) ∂v ⎜ ∂y(u,v) ⎟ ⎜ ⎟ ⎝ ∂v ⎠ . ∂z(u,v) ∂v
!
k
S
P : B → R3
k · n dO :=
S
k · (Pu × Pv ) du dv.
B
! " z = f (x, y)
P (x, y) = (x, y, f (x, y)) (x, y) ∈ D ! # Px = (1, 0, fx ) Py = (0,1, fy ) $
⎛
⎞ k1
⎜ k · (Px × Py ) = det ⎜k2 ⎝
1
0
0
⎟ 1⎟ ⎠ = −k1 · fx − k2 · fy + k3
k3
fx
fy
¾¾
−k1 · fx − k2 · fy + k3 dx dy
D
k = (x, y,0)
x2 + y 2 + z 2 = r2
⎛ ⎞ ⎛ ⎞ x r · cos(u) · cos(v) ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ P (u, v) = ⎜ ⎝y ⎠ = ⎝ r · sin(u) · cos(v) ⎠ z r · sin(v)
0 ≤ u < 2π,
· π ⎛ ⎞ ⎛ ⎞ x r · cos(u) · cos(v) ⎜ ⎟ ⎜ ⎟ k = ⎜y ⎟ = ⎜ r · sin(u) · cos(v) ⎟ , ⎝ ⎠ ⎝ ⎠ z 0
− 12 · π ≤ v ≤
1 2
⎛
⎞ −r · sin(u) · cos(v) ⎜ ⎟ ⎟ Pu = ⎜ ⎝ r · cos(u) · cos(v) ⎠ , 0
⎛
⎞ −r · cos(u) · sin(v) ⎜ ⎟ ⎟ Pv = ⎜ ⎝ −r · sin(u) · sin(v) ⎠ r · cos(v)
⎛
r · cos(u) · cos(v)
⎜ det ⎜ ⎝ r · sin(u) · cos(v)
−r · sin(u) · cos(v) r · cos(u) · cos(v)
0
0
⎞ −r · cos(u) · sin(v) ⎟ 3 3 −r · sin(u) · sin(v) ⎟ ⎠ = r ·cos (v). r · cos(v)
1 ·π 2
− 12 ·π
0
2π
r3 · cos3 (v) du dv =
8 · π · r3 . 3
! " # $
% & ' $ %
( # ) * + , - . # * # + + / # 0 # #)1, 2 2
3 + 4 56" * 7 8 "* . + 0 y(x) 4 6" 9 y (x). y (x) + - . 4 y (x) = 0 " ) * "* :# 4 6 ; 1 * . # 0 7 4 *+ 0 #
0 m ∈ N j ∈ J , ∀j ≥ j ∀ i = 1...m : fj − f Ki ≤
∞ ε ε ∧ 2−i ≤ 4 2 i=m+1
$ ∀j ≥ j : d(f, fj ) ≤
m i=0
2−i ·
∞
2−i · 1 ≤ ε2 + 2ε H(U )
ε 4
+
i=m+1
!"
H(U )
H(U )
!" # $ %
&
H(U )
' ( %
) ! # &
X X
) %
%
*+ ,-./0
¾
H(U ) ! " # $ Y M > 0 % ∀y ∈ Y : y ≤ M
& ' ! "( ' & ) * + H(U ) , Y ⊆ H(U ) H(U ) ' Y , - ./ K ⊆ U M > 0 0 % ∀y ∈ Y : sup |y(x)| ≤ M x∈K
& ) #
/ ' / . 1 +
U ⊆ C U = ∅ A ⊆ H(U ) A
& f|K |f ∈ A , + . K = Br (x) ⊆ BR (x) ⊆ U r < R < d(x, ∂U ) f B (x) ≤ M + f ∈ A ' u, v ∈ Br (x) f ∈ A
R
|f (u) − f (v)| =
v
u
f (z) dz ≤ |u − v| · f K
γ ∂BR (x) z ∈ K 1 f (ξ) f (z) = dξ 2πi · 2 γ (ξ − z) 2π 1 |f (z)| · ≤ · γ (t) dt 2 2π 0 |γ(t) − z| 2π M 1 · R dt ≤ · 2π 0 (R − r)2
¾
=
1 M · 2πR · 2π (R − r)2
R, r M δ
∀f ∈ A ∀u, v ∈ Br (x) : |u − v| < δ =⇒ |f (u) − f (v)| < ε
f|K |f ∈ A
f|K |f ∈ A C(Br (x)) M ∀f ∈ A : f Br (x) ≤ M
! C(Br (x)) ! " # $ ! $ % & ' C(Br (x)) (
f|K |f ∈ A C(K) & ) K ⊆ U # (fn ) ! ! ' A ( (fnk ) ! K f ∈ H(U ) " K d(K, ∂U ) > 0 0 < r < R < d(K, ∂U ) ! K r$) % x1 , . . . , xn ∈ K * K⊆
n
i=1
Br (xi ) ⊆
n
i=1
Br (xi ) ⊆
n
BR (xi ) ⊆ U
i=1
! (fn ) & Br (xi ) ( $ " ) ( (fn ) + K * A H(U ) ! ,-. ' ( K0 ⊆ K1◦ ⊆ K1 ⊆ K2◦ ⊆ K2 ⊆ . . . ⊆ U i Ki = U
/ ! ' (fn )n∈N A ( fn(0)
K0 " ( fn(1)
n∈N
K1 & k ∈ N ' fn(k)
n∈N
n∈ N
Kk i ≤ k " " (gn ) :=
(n)
fn
n∈N
& Kk ,-. !
(gn ) 0 # 1 2 3 4 g (gn ) H(U ) # ! A g ∈ A A
¾
! " # $ %
# mf gnk → Gockel
#
! ! " !# # $% % &! ! ! % &! ! '( ) % *% * % % ! ! +# # ) '! ! , # * &#- &% ζ(2) + ) % ζ(s) . / ζ(s) =
∞ 1 ns
01234
n=1
+ +. % # % 5 ) s Re(s) > 1 % % # ) s > 1 ζ(2) * ! % 6 / ζ(2) =
∞ 1 1 1 = 1 + 2 + 2 + ... n2 2 3
01214
n=1
+ 6 ! &% 7 % "% 8 39:2 !% $ " ' ! ; 3 0 ? % % 4 ! 7% !
# ' ) % % % +% ' % + ? % sin(x) ! @(% ! /
¾
sin(x) = x −
x3 x5 x7 + − + ... 3! 5! 7!
0 ±π ±2π
±3π
?
sin(x) = x(x2 − π 2 )(x2 − 4π 2 )(x2 − 9π 2 ) . . .
x ∈ R x2 x2 x2 sin(x) = x 1 − 2 1− 2 2 1 − 2 2 ... π 2 π 3 π ?
!
" # $ % & ' ( % $ sin(x) ? = x
x2 x2 x2 1− 2 1− 2 2 1 − 2 2 ... π 2 π 3 π
)!
* x → 0 + ' ) $ ' & sin(x) lim = 1, ,! x→0 x -% ' & % . /% 0 / % . ! 2π1+ * ' 2 3+ * 1' 10 / % $ 2! y 2
1
0 1
π
x 2π
−1
−2
¾
! " # x3 x5 x7 sin(x) = x − + − + ... 5! 7! 3! 2 x x2 x2 =x 1− 2 1− 2 2 1 − 2 2 ... π 2 π 3 π
$ %&
' ( )*+, -./ 0 1 2 # −
#
1 1 1 1 1 = − 2 − 2 2 − 2 2 − 2 2 − ... 3! π 2 π 3 π 4 π
x3
$ 3&
1 1 1 π2 + 2 + 2 + ··· = 2 1 2 3 6
$ 4& 1 ' + / 5 60 7 6/ ! + ζ(2) # 1,64493406684822643647 . . . 6 8 + 0 9 / : / 2
" ; 1 0 # 1 < = !
! ! " # $ $ %& &' # '& ( &) #* % # & + # & , # - , . & "#
" # / $
√ 2 2 0 , &1 2 #
. " #
& # #3 1 0 , 4 √ 4& & 0 # / $ 2 2 $
#- - / - & & # 5- 6
, / 1 - 0 . , $ 5 &
$ + 7 #3 8 9& # %& , : (, ; - , 1 ' 0 #
" # # ) " 2 $ + # +&
" # % = & 4 ?@ $ - " ' 9&A # B & # C# % # &1 " # #3 & 8 # & %> #1 61 D E% = + F1 G -
0 ' EH =1 F 1 $ 6
/ 1 " # & 1 & &# # 1 - = 1 0 # ( ! 1 ( # s = 1
! + # 2 ! ! 3 2 ζ(s) # A & &/ 4 "5 $
¾
ζ(s) ! " # # $ % # % & ' " ( f : R → R
|x| → ∞
0
fˆ : R → R; r → fˆ(r) =
1 |x|
R
f (x) e2πirx dx,
!" !
R
# $ % &
f (n) =
n∈Z (
θ(s) =
θ(it) =
fˆ(n)
!" '
n∈Z 2
f (x) = e−πtx
# )
2 2 1 e−πtx e−2πirx dx = √ e−πr /t , t R
fˆ(r) =
1 2
eπisn
2
* +
&
n∈Z
´¾½º¿µ 1 −πtn2 1 e = f (n) = 2 2
n∈Z
!" ,
n∈Z
=
2 ´¾½º¾µ 1 ˆ 1 −πn 1 i √ f (n) = e t = √ θ 2 t 2 t t n∈Z n∈Z
∞ Γ(s) = 0 ts−1 e−t dt 0 Γ(s) = (s − 1)Γ(s − 1) C 1
. # / 0
2 3
/14
π −s Γ(s)ζ(2s) = π −s
0
∞
ts−1 e−t dt ·
∞ 1 n2s
n=1
∞ 2 −s = (πn ) n=1
!" -
∞ 0
t
s−1 −t
e
dt
Re(s) >
1 2&
¾ ¼
t = πn2 v =
∞
(πn2 )−s
∞
2
(πn2 v)s−1 e−πn
0
n=1
v
πn2 dv
=
∞ n=1
∞
2
v s−1 e−πn
v
dv
0
1 = v e dv = v θ(iv) − dv 2 0 0 n=1 1 ∞ 1 1 s−1 1 v s−1 θ(iv) dv − v dv + v s−1 θ(iv) − dv = 2 0 2 0 1
∞
s−1
∞
−πn2 v
∞
s−1
v = 1t
v t ∞ i 1 1 s−1 = t θ dt − + t θ(it) − dt t 2s 2 1 1 ∞ ∞ 1 1 1 = t−s− 2 θ (it) dt − + ts−1 θ(it) − dt 2s 2 1 1 ∞ ∞ 1 1 1 1 1 = (t−s− 2 + ts−1 ) θ(it) − dt + t−s− 2 dt − 2 2 2s 1 1 ∞ 1 1 1 1 = (t−s− 2 + ts−1 ) θ(it) − dt − − . 2 1 − 2s 2s 1
∞
−s−1
s π
− s2
Γ
s 2
s/2
∞
ζ(s) =
(t
− s+1 2
+t
s −1 2
1
1 1 1 ) θ(it) − dt − − 2 1−s s
s 1−s s
π
− s2
Γ
s 2
ζ(s) = π
! " # #
− 1−s 2
Γ
1−s 2
ζ(1 − s)
$ % !
# & s Re(s) > 1 ζ(s) ζ(1− s) % ' " # # ζ(s) ( " # C # ) # s = 1 #%
* + + , # # . / % " (#
¾ ½
( ) < 1
! " # $ −2n, n ∈ N 0 ≤ ( ) ≤ 1 % & # a = 2 $ ' $( ( ( ζ(ζ) = ζ(ζ) & ) *
( +,- . ' ) # { 2 + b } . *
/ 0 1 2
! " # $3
$ 4 -5 $ ' 0 ! 6 *
7 .
¾ ¾
!
!"# $ $ # % & ! ' ( ) * + !" ,-.-/0 ( 1 2 -.34 5 ( -6 1 ( ( 7 ,-.340 8 , 9::6 ##0 ;2% ## , 0 : π(x) − Li(x) = O(x 2 +ε ), .
log(x)
!
ε=0
),--,*
ε > 0 ! x : xε > √ π(x) − Li(x) = O( x) / !
0
) * 1+ '
sin(γ log(x)) Li(x) − π(x) √ +O =1+ γ π( x)/2 0≤γ
γ
1 log(x)
# + 1
ρ=
1 2
, + iγ
),--2*
'
" $ $
3
½º
$ " $
1
( 4 "
¾º
$ 4 " + / ' 5 % "
γ
)
* $ ! ' "
4 6
¾
x Li(x)
! " # $ % " " π(x) & ' Li(x) # π(x)
( ) * ' k > 0 + π(x)− √ ¿º
k x log(x)
Li(x) >
10
x < 10
log(log(log(x)))
1034
π(x) < Li(x)
x
!" #$ % & '
() * + , -
. & / " - * &
1034
x < 1010
π(x) < Li(x) + x
. & %
π(x) < Li(x)#
* / 0
1222$
3 *
& * 4 % 55 6 * / 0
!
7 & & * 4 8 & 9 & : ; & - , )& & 8 ? = !
F (x) [1,390821; 1,398244]·10316
!" 7 122@$
" !
* A/& 0 ) ? , "
F (x) =
& ) A9 &
Li(x)−π(x) √
! π( x)/2
π(x)
√ Li( x) . ) 2 = :
x = 10316
1
F (x)
& B
≈
√ x log(x)
*
6
2
6 = ;
13 O( 1000 )
? * = , - C* = *
!
: 8 /
-
¾
1,0 0,0
0,5
F (x)
1,5
2,0
√ F (x) = 2(Li(x) − π(x))/π( x)
1,3 · 10316
1,4 · 10316
1,5 · 10316
1,6 · 10316
1,3970 · 10316
1,3975 · 10316
1,3980 · 10316
1,3985 · 10316
1,3990 · 10316
0,03 0,02 0,01 −0,01 0,00
F (x)
0,04
1,2 · 10316
x Li(x) < π(x)
½º
! "
s Re(s) =
1 2 # $
% & ! '
¾º
! $ ( )*+,# - ! . '/ ! ! "
- 0 12 # $
3
! ! 4 . 5 #
¿º
- '/ 4 / / 0 6 2/ 7 !
- 4 4
¾
! " #$ %&&' () * + , - . /0 , - () + 1 , ( ,2 3 4 $ , 5 6 3 7 $ 2 4 * ) *- ½ 2¾ $ 5 , ( . 8 9
½ ¾
!
" # #$ % & '() )) %"
! "
# $ %$ ! & '( !
' )*+,** %% * ) ) + $ , )) # () ) , ))) ' + ( . ) ' ) / #( ) + $ )
) ' ) 0 , #$ 0 1
) 0 2 0 3 ) ) ) 0) ' )) + $, 0 0 - . (, ) ) 4 0 # 2 5
¾ ¾
B := {(x, y, z) | x + y2 + z 2 ≤ 1} 2
A1 , . . . , An , B1 , . . . , Bm ⊆ B
R3 ) σ1 , . . . , σn τ1 , . . . , τm
B =
n i=1
Ai ∪
m j=1
Bj =
n i=1
σi (Ai ) =
m
τj (Bj ).
j=1
M f : M → M ∈M M M ∈ M ! f (M ) ∈ M " M ∈ M #$% f (M ) #$% R := f (M) " & ' %( ) * + ,- ./" M #$% " 0 ) & & ) % " 1 2 ) 3 1 1 ) " 4 % ,2. ) 0
3$$ " 5 % 2 6% S 2 \D % 7 D "
8 9"
3$$ SO(3) 3$$
R3 " : ) & ) SO(3) &
S 2 " SO(3) 3$$ ; 3 SO(2) * 3 0 4 2 /
¾ ¿
! Q " #
$ % " $ "$$ $ ! Q & # '$ " Q ($ ) " $ $ $ $ "$$
$ * ) $ "# $$ $ $ + ξ . . . ξ ω ω 1
n
−1
Q
ωi ωi−1 ωi−1 ωi
!"
#
"
$ %
!
A = {ai , a−1 i | i ∈ I}, &
' ( )
∀i ∈ I : ai a−1 = ; a−1 i i ai = *
◦ : (w1 , w2 ) → w1 w2 +
,
A
- , . "
/ ' ( ) 0
1
$ .
aa
−1
−1
Fa,b
−1
= a a = bb a b
A = {a, b, a−1 , b−1 } = b−1 b =
.
0
.
¾
S x ρ φ q φ = 2π · p/q p/q ∈ Q 2
φ
φ = r · 2π r ∈ R\Q ρ := ρφ A = {ρ, ρ−1 } ρ ρ−1 ! ζ := ζγ z "# γ = s · 2π s ∈ R\Q $ SO(3) % ρφ , ζγ φ = γ = arccos(1/3) " $ B SO(3) &
' ⎛ ⎞ 1 ⎜ ⎜ ρ := ⎝ 0 0 ⎛
⎜ ζ := ⎜ ⎝
2 3
0 1 3
2 3 1 3
√ 2
√ 2
0 √ ⎟ 2 ⎟ ⎠,
− 32 1 3
√ − 23 2
0
1 3
0
0
⎞
⎟ 0 ⎟ ⎠. 1
% ( )" * ( w ∈ B, w = ) (1,0,0)t w (1,0,0)t ) (0,0,1)t w (0,0,1)t '
ρ ζ ∈ SO(3)
+ % k ∈ N (
w ∈ B ) " w(0,0,1) = (0,0,1) w ∈ {ρ, ρ } w(1,0,0)t = (1,0,0)t
w ∈ {ζ, ζ −1 }. , k ≥ 2 ζ ±1 ) χn w = χ1 χ2 . . . χn (1,0,0)t - ρ±1 ) ,. (0,0,1)t ) t
t
−1
+ * ( w % k ) w(1,0,0)t √ 31k (a, b 2, c)t / a, b, c + k = 1 0 +
¾
w = ζw , w = ζ −1 w , w = ρw , w = ρ−1 w
√w 1 w (1,0,0)t = 3k−1 · (a , b 2, c )t a b c χ1 = ζ ±1 a = a ∓ 4b , b = b ± 2a , c = 3c ,
χ1 = ρ±1 a = 3a , b = b ∓ 2c , c = c ± 4b .
! a, b, c w ∈ B ζ ζ −1 w(1,0,0)t = (1,0,0)t
"# b $ w % ζ ζ −1 % b = ±2 & $ ' () w = χ1 χ2 v ' χ1 χ2 = ζ ±1 ρ±1 , χ1 χ2 = ρ±1 ζ ±1 , χ1 χ2 = ζ ±2 , χ1 χ2 = ρ±2 .
a $ b = b ± 2a $
# c # $ b = b ∓ 2c '
# √ 1 # v(1,0,0)t = 3k−2 · (a , b 2, c )t ' b = b ± 2a = b ± 2(a ∓ 4b ) = b ± 2a − 8b = b + (b − b ) − 8b = 2b − 9b .
b b $ #
¾
Bρ ρ B Bρ−1 Bζ Bζ −1
B = Bρ ∪ Bρ−1 ∪ Bζ ∪ Bζ −1 ∪ {}.
ρBρ−1 B
ρ ! ρ−1 ρ ! ! ρ−1 " # ρ $ % & ! ' & (
q qqq q qq r q qqq q
q qqq q q qq r qq q q ra
q qqq q qq r q qqq q
−1
q q qq r qq q q qqq q
q qqq q q qq r qq q q r b
q qqq q r qq q qqq q
r
q qqq q qq r q qqq q
r q q qq r qq q q qqq q
q qqq q r qq q qqq q
q qqq q q qq r qq q q
q qqq 2 ar a r qq q q qqq q q q qq r qq q q qqq q
! " # $ %& ' ' $ ( " # '
ρBρ−1 = Bρ−1 ∪ Bζ ∪ Bζ −1 ∪ {},
ζBζ −1 = Bζ −1 ∪ Bρ ∪ Bρ−1 ∪ {}.
) & * + , ! - (( * + , *+ , - ((
¾
X := S 2 \D
B ! " S 2 \D D # $ % ! & ' B\{}( D = {x ∈ S 2 | ∃ g = ∈ B g(x) = x}. ) ! & B D * % !
∼ ⊂ {(x, y) | x, y ∈ S 2 } : x ∼ y ⇐⇒ ∃g ∈ B
y = g(x)
+ & ", -&
+ &( x = (x) ∀x ∈ S 2 ",( ) y = g(x) x = g−1 (y) ∀x, y ∈ S 2 , g ∈ B -&( ) y = g(x) z = h(y) z = hg(x) ∀x, y, z ∈ S 2 , ∀g, h ∈ B ./& ! ∼ g(x) = y}, *! 0 S 2 " 1 B $ 2! z S 2 % ! w ∈ B ! & B & ( Ox := {y ∈ S 2 | ∃g ∈ B
∀w ∈ B
w(D) = D w(S 2 \D) = S 2 \D.
$ ( B D S 2 \D
¾
z∈D
B(D) ⊂ D
w ∈ B v ∈ B w vwv −1
v(z)
B(D) := {b(d) | b ∈ B, d ∈ D} #
D
! "
$ %
D
S 2 \D
&"
B(S 2 \D) ⊃ S 2 \D. '
B(S 2 \D) ⊂ S 2 \D
z w(z) = z
(&" ) *
B " ∀w ∈ B v(z) (&" )
v∈B
&"
∀w ∈ B
vwv −1 (v(z)) = v(w(z)) = v(z)
+
, %
w -.
&"
B(S 2 \(S 2 \D)) = B(D) ⊃ D % &""
* '
Bζ ,
Bζ −1 ,
Bρ ,
B
Bρ−1 ,
{}
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