Загрузил Aidar Ranis Uly

Intuitsia i matematika 2003 Boss V

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31 3@B0@18426 G37186;26 a = a = 1 .
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m?106; 0?AG00?I@I4k :137?F18B@?7oI13@?a IB3 8 FBII4a
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t@1 G0B8I?I6? IBF1 0?J6@o :17G56@o F8B A10I; x 6 x 
6 :171x6@o
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5@1 6 hGF?@ 1hb62 0?J?I6?2 Ú-.-.-Û. XIB5?I6; A1I3@BI@ c  c
219G@ h4@o 1:0?F?7?I4 6T IB5B7oI4k G37186a (a = a = 1) .
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>1 IB3 6I@?0?3G?@ F0G91?.
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2
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E?081? 5@1 :06k1F6@ 8 91718G – s@1 `6A_6;. {3:1219B
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:0B8FB.
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63:17oT18BI66 A12:7?A3I4k 563?7. S1hB87?I6? A 9B021I65?
3A12G A17?hBI6H U (t) = U cos ωt `6A@68I1a 2I621a 5B3@6
b (t) = U (cos ωt + i sin ωt),
U
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x68B?@3; 35B3@7681? 3@?5?I6? 1h3@1;@?7o3@8. XBA1I4 s7?A@06
5?3@8B 6 :0B867B G2I1x?I6; A12:7?A3I4k 563?7 :01;87;H@
I?1x6FBIIGH 3197B318BII13@o.
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A12:7?A3I4k 563?7 1h›;3I6@o 811hb? I?81T21xI1. Úw 24 :1
5@6 I? G@060G?2.
S0G91a :062?0. n;F4
1 = 1 + x + x2 + . . . ,
1−x
2
3
ln (1 + x) = x − x + x − . . . ,
2
3
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w 84k1F6@ 5@1 2I62B; ?F6I6_B ABA @?Io 83? 80?2; :06
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82?J68B?@3;Ý. g63@6ABW
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760G?@3; 6 5B3@1 :18@10;?@3;. ^71x?I – :1@12G 5@1 301FI6
6IAB0IB_66 Bh3@0BA@I191 5637B. ˜@1h4 G;3I6@o IB7656? 1hb
I13@6 8 36@GB_6;k ™@06 19G0_Bš 6 ™@06 :126F10Bš 5?718?5?
3@8G :1@0?h18B713o 15?Io 2I191 80?2?I6. XF?3o @1x? I?1hk1
F621 80?2; k1@; ?376 0BT1h0B@o3; I?:1I;@I1 – IB 5@1.
{3? IB56IB?@3; 3 IB@G0B7oI191 0;FB 6 88?F?I6; F8Gk 1:?0B
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840B3@B?@ 3B21. ^IB5B7B ?3@?3@8?II42 1h0BT12 :1;87;H@3;
1h0B@I4? 1:?0B_66 8456@BI6? 6 F?7?I6? 87?AGb6? TB 31h1a
I?1hk1F6213@o 88?F?I6; 0;FB
. . . , −3, −2, −1,
:1@12 F01h?a. nB3J60?I6? 63k1FI191 1h›?A@B 2BI6:G7;_6a
0, 1, 2, 3, . . . ,
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2?xFG :01562 :71k1 :13A17oAG AG7o@6860G?@ I? @1@ 63@15
I6A Bh3@0B96018BI6;. E1@0?hI13@o 8 1@06_B@?7oI4k 5637Bk
62??@ 910BTF1 h17?? 8?312GH :0656IG 5?2 TB:63o F179B 31
TIBA12 26IG3. S?71 8 @12 5@1 2BI6:G76018BI6? 5637B26 IB
1:0?F?7?II1a 3@BF66 8481F6@ IB 37?FGHb6a G018?Io Bh3@0BA
_66. S7; 1FI1@6:I4k F?a3@86a IB56IB?@ 63:17oT18B@o3; 362
817oIB; TB:63oä. {8?F?I6? hGA8?II4k 1h1TIB5?I6a :?0?81F6@
B06`2?@6AG IB F0G96? 0?7o34. ƒ102G74 @6:B a − b = c 676
(a − b)(a + b) = a2 − b2
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1:?0B_66 62?76 32437. „@06_B@?7oI4? 6 F01hI4? 5637B IGx
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x + a = b,
ax = b
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èìý + óïõüç
øìçðèüí ëîïûêïôçéçó
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ìçéöçìí ùõð èì ì èêéýûéèëð çéîïð ëèïùóð øêéùôïêóèéõüçì ì ìýûöé
çóð ó øêóô îïçóð î ëøé + óïõüç ò îïèéìêóðò ò úõéçóð èìò ìèçìúé
çóó éìòéèêóöéëîïð óçèéêøêéèï+ óð óòééè èì øêéóòû éëèôìí öèì ìçï ùïçï
îïî ý ïøêóìêóí ó éìòéèêóöéëîó ïîèí îïî øêïôóõìí õéîì ìý ; ðëçóèü
ùï é çìôóöîû
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ì êïççóî ïêïîèéêó ûéèëð é é ó èéòí öèì çéîìèìê é é ì ùôó éçóðí
øìôìêìè
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øìôìêìèìô
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óùéè ì êûøøéí øìèìò
øìôìêìèìôí
èì ìýì çïöïéè øìëõéùìôïèéõüçìé ô øìõçéçóé
øìôìêìèï
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ë
éè êïòôóôïéèëð
øêóòéêçì øì èïîì ë éòé ( êéùó êûøø øìôìêìèìô
éëèéëèôéçç
ìýêï
ìò ô ùéõð è çéîóé êûøø ëóòòéèêó í çïõóöóé îìèì êëèïôóèëð
ôõéöéè ï ëìýì ëû éëèôìôïçóé øêïôóõüç òçì ì êïççóîìô = ïèéò
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îïîóé ô
ëóòòéèêóó êïëëòïèêóôïéòì ì èóøï ôì òì çøêé æçééìîïêéúéçóé
ôïéèëð ôëé ì øðèü êûøøï èéèêï ùêïíîûýï & ) ó øìõûöïéèëð
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ëèôïÿ . éêéýìê ëôìùóèëð î ìøêéùéõéçó + éõìöóëõéçç êéúéçó çéêïôéç øêó*ìöéôóùçìò
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í ïî
ó
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ôïéè ˆ ìîûòéçè øêéù ; ðôóõíýóì êï ó êïëëîï ïõí
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D é îì ôóùéèüí öèì
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< ï ûòééèëðí èì øìîï õóúü ®Ä¯µ±­° ¸­Ã±´®´®µ­Ã½¯±±´®µ 7 îûýìô çéìùó "
çïîìô êï òéêìô & í øêéùëèïôõð ïð ëìýì øéêô úï ô çïøêïôõéçóó
êé ûõüèïèï > ïçï ï Ï *ïêëîì ìí îìèìê ô çïöïõé øêìúõì ì ôéîï ìëû é
ëèôóõ ïûëùìê ˆ ïõéé òì çì øìîï ïèüí öèì ô öóëõì ñîûëîìô÷ òì çì
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y ∈ [0, 1]
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[0, 1]
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3@B8A6 I1 – I? F7; 7Hh4k 690. >BF1 :013@1 86F?@o 5@1 :01
63k1F6@ TB ABF012.
−100
−100
200
100
−100
Žb? 1F6I :06I_6:6B7oI4a 63@15I6A 81T21xI4k 0BTI197B
C|
36a – :1I;@6? Z1] Q`WTWm 3TV W  A1@101? 31h3@8?II1 6 84F?7;
?@ m{ 6T 1hb?a @?1066 2?04.
^1h4@6; A 6 B IBT48BH@ I?TB86362426 ?376
P (AB) = P (A)P (B),
Úc.=.-Û
@. ?. 8?01;@I13@o @191 5@1 :016T1aF?@ ™6  6 B š 0B8IB :01
6T8?F?I6H 8?01;@I13@?a A 6 B . t@1 2?xFGA:01562
3183?2 I?
@1 5@1 :1F0BTG2?8B?@ :1F I?TB8636213@oH 6I@G6_6; 6 21x?@
h4@o I? 3183?2 @1 5?91 h4 k1@?713o 1@ @?1066.
E1;3I62 IB :062?0?. j013BH@ F8? 21I?@4. EG3@o 84:BF?
I6? :?081a 21I?@4 9?0h12 1h1TIB5B?@ 31h4@6? A  8@101a – B .
nB3321@062 @BAx? C – ™@17oA1 1FIB 21I?@B 84:B7B 9?0h12š.
S7; 3622?@065I4k 21I?@ 83? @06 31h4@6; :1:B0I1 I?TB
863624 :13A17oAG
1
,
2
1
.
4
Úc.=.=Û
^ I?TB8636213@oH A 6 B 6I@G6_6; 3197B3IB I1 I? 3 I?TB8636
213@oH A 6 C Ú676 B 6 C Û. w G I?? ?3@o 13I18BI6;. >?TB8636
213@o Úc.=.=Û 62??@ ABA h4 ™B06`2?@65?3A6aš kB0BA@?0 ;87;
?@3; 0?TG7o@B@12
5637?II191 318:BF?I6;. ‡B5?3@8?II4? 1@76
56; 8TB62138;T?a 31h4@6a 84;87;H@3; :06 IB0GJ?I66 362
2?@066 21I?@. S7; I?3622?@065I4k 21I?@ Ú3 8?01;@I13@oH
84:BF?I6; 9?0hB 6= 1/2Û 381a3@81 I?TB8636213@6
6 86FB
A B
c.=.-Û
31k0BI;?@3;
B
81@
0B8?I3@8B
Ú
P (A) = P (B) = P (C) =
P (AB) = P (AC) = P (BC) =
P (AC) = P (A)P (C), P (BC) = P (B)P (C)
IB0GJBH@3;.
XF?3o G2?3@I1 TB2?5BI6? 1hb?91 kB0BA@?0B. gB@?2B@65?
3A1? 1:63BI6? 7Hh1a 0?B7oI1a 363@?24 83?9FB I?@15I1 8 0?
TG7o@B@? I?6Th?xI4k 1J6h1A :06 1:0?F?7?I66 :B0B2?@018
:0?I?h0?x?I66 2B71TIB5Bb626 `BA@10B26 6 @. :. E06 s@12
?3@?3@8?II15@1 21F?7o 21x?@ :0?F3@B87;@o :0BA@65?3A6a 6I
@?0?3 76Jo :06 IB76566 1:0?F?7?II1a I?5G83@86@?7oI13@6 A
2B742 ?? 6T2?I?I6;2 81T2Gb?I6;2. { :01@68I12 37G5B? I?@
I6ABA6k 9B0BI@6a 5@1 8481F4 311@8?@3@8GH@ :18?F?I6H 6
381a3@8B2 6TG5B?2191 0?B7oI191 1h›?A@B.
t@1 311h0Bx?I6? ABxF4a 0BT 6T87?AB?@3; IB 38?@ A19FB
@B 676 6IB; @?106; 3@B7A68B?@3; 3 BI12B76;26 86FB ™2B74?
C-
:0656I4 – h17oJ6? :137?F3@86;š. m{ 6I19FB I? 1@87?AB?@3;
IB @BA6? ™2?7156š.
Ì ÎÏâÕÙáÔà ÔÕÔ ØÏ ãÎÏâ }
m0GFI13@6 0?J?I6; TBFB5 h17oJ1a 0BT2?0I13@6 5B3@1 :101x
FBH@ B`106T24 @6:BÜ ™?376 8 TBFB5? 1FIB :?0?2?IIB; – s@1 I?
TBFB5B ?376 h17oJ? F8Gk – 1IB I?0BT0?J62Bš. i ABA 1F17?@o
™:01A7;@6? 0BT2?0I13@6š A19FB 56371 :?0?2?II4k 6T2?0;?@
3; F?3;@AB26 31@I;26 @43;5B26W ‡BA I6 3@0BII1 h17oJB;
0BT2?0I13@o I?0?FA1 ;87;?@3; I? :01A7;@6?2 B :1FB0A12 A1
@104a :1T817;?@ 0?TA1 G:013@6@o 0?J?I6? :?0?k1F; IB B9
0?96018BII1? 1:63BI6?. {1:013 @17oA1 G:60B?@3; 8 IB7656?
:1Fk1F;b?91 6I3@0G2?I@B.
‡7B3365?3A6a 1h0BT?_ ™63:B0?I6; 371xI13@6š FB?@ 3@B@6
3@65?3AB; `6T6AB I1 96:I1@6T60GHbB; 017o s@191 :062?0B
IB3@17oA1 8?76AB 5@1 TB :0?F?7B26 @?021F6IB265?3A1a 3:?
_6`6A6 81T21xI13@6 B90?96018BI6; 13@BH@3; 2B716TG5?II4
26. S0G91a 3@BIFB0@I4a :062?0 – ] QU3Z f3S h W2 kW T1S  9F?
0?5o 6F?@ 1 3@Bh676TB_66 76I?aI1a `GIA_66
Ò Ln (x) =
x1 + . . . + xn
.
n
S1791? 80?2; 356@B713o5@1 3@Bh676TB_6H I?76I?aI4k `GIA
_6a 1h?3:?568BH@ 381a3@8B 3622?@066. >B 3B212 F?7? s@1 I?
@BA. Ž376 :1F 3@Bh676TB_6?a ÚB362:@1@65?3A62 :13@1;I3@812Û
:1F0BTG2?8B@o 3@0?27?I6? A IG7H F63:?0366 Ú30?FI?A8BF0B
@65?3A191 GA71I?I6;ې @1 :062?04 IB6kGFJ?a 3@Bh676TB_66
FBH@ ABA 0BT 3622?@065?3A6? `GIA_66.
w3@6IIB; :0656IB 3@Bh676TB_66 TBA7H5B?@3; 8 190BI65?I
I13@6 3A1013@6 013@B `GIA_66 f (x)  @. ?. 8 0B8I12?0I1a :1 n
190BI65?II13@6 90BF6?I@B
|∇f (x)| 6 γ.
Úc.D.DÛ
n
n
Ö TS Wa Z QR Wm 1 a `1S WkWZ[
Q`Z3m 1 Z3 QTR 1_1S 1Z[ ZQ
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fn (x)
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ZWk1ZZ[ 4 3TV a z U3ZTV QZV [ Ø
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n
- ûëèüëêéùçéò
çéìý ìùóòì
ô öóëõóèü ëêéùçéé
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öóëéõýìõüúìò
øêó ï
ùïççìò
ïêó
òéèóöéëîìò
îï
ôïéèëðí
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ùìëèïèìöçì
ëîìõü û ìùçì èìöçì ô øìõçðéèëð êïôéçëèôì
n
√
n
x1 + . . . + xn
n
ùé ) îìçëèïçèï * ïîóò ìýêï ìò øêó ýìõüúìò çéè çéìý ìùóòìëèó øêìó ôìùóèü èêû ùìéòîìé øéêéòçì éçóé ë øìëõéùû óò ó ôõéöéçóéò îìêçð ëèéøéçó ˆ ìëèïèìöçì ô öóëõóèü ï êé ïè
ó ôìëøìõü ìôïèü
ëð øêóôéùéççì ìêòûõì ëìîïð èìöçìëèü ìýéëøéöóôïéèëð û é øêó
øìêðùîï çéëîìõüîó ùéëðèîìô x1 . . . xn = γ
γ
,
n
n
X = (x1 + . . . + xn )/n
n
CD
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Ù
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g4 F17xI4 FG2B@o I? 1 @12 5@1 21x?@
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GF187?@810?I6? B I? ABA1a@1 @B2 ™G917 :BF?I6; 0B8?I G97G
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8 2B@?2B@6A? :17oTGH@3; h17oJ1a :1:G7;0I13@oH.
Ú ÌË
ÎÏÑáãÙ ÕÏÖ âã
w2??@3; I?A1@10B; `GIA_6; F8Gk :?0?2?II4k z = ϕ(x, y) . nB3
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IG7?8GH @15AG. { 0BT0?T? hGF?@ A068B;. S1:G3@62 5@1 8 7H
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x? 62?@o 26I62G2 8 IG7?W
wI@G6_6; 813:6@BIIB; IB :013@4k :062?0Bk FB?@ :171
x6@?7oI4a 1@8?@. E0B867oI4a 1@8?@ – 1@06_B@?7oI4a.
{1To2?2 :013@GH 36@GB_6H
z = 2x − 3x y + y .
Úp.-.-Û
>B 7Hh1a :0;21a y = ax `GIA_6;
4
2
2
z = 2x4 − 3ax3 + a2 x2
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AG :18?F?I6? `GIA_66 8 IG7? 1:0?F?7;?@3; 3@B0J62 57?I12.
XF?3o s@1 a x . E06 a = 0 62??2 z = 2x – @1x? 26I62G2.
{ 3?5?I66 x = 0 – :B0Bh17B z = y – 1:;@o 26I62G2.
„FIBA1 F7; I?IG7?84k x  y  GF187?@810;Hb6k G37186H y =

9F?
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cx
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IG7? 1IB1 I?< c21x?@
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TB:63B8 z 8 86F?
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2
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z = (y − x2 )(y − 2x2 ),
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CU
ÌÞ ÎÔàÐ ÖÔ O ãÕÔ O 㠔 ÔÔ
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3@8?II4a 71AB7oI4a 26I62G2 6 1@3G@3@8GH@ 71AB7oI4? 2BA
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I42.
g1xI1 76 @1 x? 3B21? 3ABTB@o 1 `GIA_66 F8Gk :?0?2?I
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IB:062?0 ϕ(x, y) :06I62B?@ 8 IG7? 26I62B7oI1? TIB5?I6?
3ABx?2 IG7o. >B56IB?2 0?TB@o 90B`6A z = ϕ(x, y) 9106T1I
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6
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21x?@ :17G56@o3; AB0@6IAB ABA IB 063. ...
{1@ F0G91a 8B06BI@ Ú063. ...Û. XF?3o `GIA_6; 62??@ ?F6I
3@8?II4a 71AB7oI4a 26I62G2
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G `GIA_66 I?@ I6ABA6k F0G96k sA3@0?2B7oI4k @15?A I6 26
I62G218 I6 2BA362G218 I6 3?F7184k @15?A.
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– I? 0BF6 TBFB56 B 0BF6 6I3@0G2?I@B.
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38;To 676 x? IBk1F67 213@ 8?FGb6a IB F0G91a h?0?9 3@BI1
8673; 1h7BFB@?7?2 817J?hI1a :B715A6. t@1 h17oJB; @?2B F7;
1@F?7oI191 0BT91810B.
Žb? 1F6I :062?0. E18?0kI13@o 910 21xI1 0B332B@068B@o
ABA 90B`6A `GIA_66 F8Gk :?0?2?II4k z = ϕ(x, y) . Ž376 ?3@o
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CC
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wI@G6_6; :1F@B7A68B?@ A 24376 5@1 :06 F8Gk 8?0J6IBk
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I1 6 FB 6 I?@. {1@ :062?0 Ú063. K.UÛ. ‡ F8G2 :B0B77?7oI42
JIG0B2 :06A7?62 :7?IAG. { 30?FI?a :1713? :01FB862 ?? IB
63A131A B :1 A0B;2 83:G562 F8? 8?0J6I4.
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Ú Ì
N ÏÐã Ôà O ã×ÏáÏÖâÔ « ÖÏÎ ” ã
x
mg
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{ 260? I? :0163k1F6@ I65?91 8 5?2 h4 I? h47 86F?I 32437
ABA191I6hGFo 2BA362G2B 676 26I62G2B. mBA 9181067 ta7?0
:06:6348B; 97Gh6IG TB2437B {3?84JI?2G. g1xI1 h471 h4
3ABTB@o 6 :1 F0G912G. >?@ I65?91 8 260? 5@1 I?7oT; h471 h4
:?0?8?0IG8 843@B86@o 8 `102? 1:@626TB_66. mBA 676 6IB5?
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6 s@1 FB?@ 3816 :71F4.
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B_61II42 21@68B2 6F?@ :1 :G@6 sA868B7?I@I4k :0?1h0BT1
8BI6a. B9 TB JB912 TBA1I4 >oH@1IB @0BI3`10260GH@3; 8
h17?? GF1hI4? `1024. Œ0B8I?I6; ŠB90BIxB :1@12 VB267o@1
IB. . .XB@?2 318?0JB?@3; I?A1@104a 71965?3A6a 3AB51A 6 :06
FG248BH@3; 8B06B_61II4? TBFB56 0?J?I6?2 A1@104k h476
h4 63k1FI4? G0B8I?I6; F86x?I6;. { 0?TG7o@B@? 81TI6ABH@
I184? AB@?91066 24J7?I6; 81TI13;b6? 2?kBI6AG IB h17??
8431A6? s@Bx6.
E01hGxF?I6? 6F?a 1:@626TB_66 :71F1@810I1 IB 3B212 s7?
2?I@B0I12 G018I?. { @1a x? 2?kBI6A? 1I6 s``?A@68I1 0Bh1
@BH@ Gx? 8 3@B@6A?. ˜?2G IB:062?0 0B8I1 IB@;x?I6? :0G
x6I4 IB 063. K.CW E01b? 83?91 :171x6@o3; IB 26I626TB_6H
:1@?I_6B7oI1a sI?0966
U (x) =
kx2
− 2mgx + const.
2
wT U (x) = 0 30BTG :17G5B?@3; kx = 2mg .
0
B
E
D
A
G
¸º®Æ Û ÆÊ
C
E1I;@6? sI?0966 21x?@ :1219B@o 8 0?J?I66 6 563@1 2B
@?2B@65?3A6k TBFB5. S1:G3@62 8 13@01G917oI4a @0?G917oI6A
IBF1 8:63B@o @0?G917oI6A 26I62B7oI191 :?062?@0B 3 8?0J6
Cp
IB26 7?xBb626 IB 3@101IBk 63k1FI191 ] Q_QkQ ¬ `Q ˜ Q Û. ^F?
7B?2 :0181715IGH 21F?7o 4ABC Ú063.ÚK.cÛ. >B ABX BC  AC
1F?I?2 381h1FI1 3A17oT;b6? A17o_B D  E  G  A1@104? 31?F6
I62 0?T6IAB26. >B@;x?I6? 0?T6I1A :G3@o
0B8I1 6 I? TB8636@
1@ 8?7656I4 0B3@;x?I6;. E1@?I_6B7oIB; sI?096;f @19FB
p.D.=Û
U = f (DE + EG + GD).
Ú
g1F?7o :0?F13@B87?IIB; 3B2B 3?h? 0?JB?@ TBFB5G B8@12B@6
5?3A6. { 0B8I18?366 F13@69B?@3; 26I62G2 Úp.D.=ې @.? . 26I6
2G2 :?062?@0B 4DEG  B 6T G37186a 0B8I18?36; 15?86FI1 5@1
0B8I1F?a3@8GHbB;
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F17xIB h4@o 10@191IB7oIB 311@8?@3@8GHb?a 3@101I? 4ABC
.
t@1 Gx? F0G9B; TBFB5B 6 ?? 0?J?I6? 6T8?3@I1Ü @0?G917oI6A
:17G5B?@3; 31?F6I?I6?2 13I18BI6a 8431@ 4ABC .
4DEG
{1@ h17?? 371xIB; TBFB5B. ˜?0?T @15AG IBF1 :018?3@6
:713A13@o A1@10B; h4 1@3?A7B 1@ @0?k90BII191O G97B :60B26FG
IB62?IoJ?91 1h›?2B Ú063. K.pÛ.
E013@1? 0?J?I6? 1:;@o FB?@ `6T65?3A1? 21F?76018BI6? 6
26I626TB_6; :1@?I_6B7oI1a sI?0966. { @15A? O JB0I60I1
TBA0?:62 :713A13@o ABC  6 83H 21F?7o :12?3@62 8 9?02?
@65I4a 313GF TB:17I?II4a 9BT12. {IG@06 SABC – 8BAGG2.
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øêóôì
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ôì òû éçóð "á # ,& îôïùêïèóöç é öõéç "ó õóçé ç é & ñõéïõó ì
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îìèìêïð "ô ìèõóöóé ìè ñìùóçìîì ì÷ & ëèêûîèûêçì ûëèì öóôïí çì èì"á #û é
çé øêìëèïð èéìêéòï ß ÌÞ
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ò ì êïçóöóòëð õóúü ûøìòóçïçóéò ñ. õìëîìëèü ÷ ô èì òìùéõó ëõû óèíçïøêóòéêíìèîê èïð øìõûøõìëîìëèüí ñøêðò òó÷ ) øìõûìîêû çìëèóí
îïî çï êóëûçîéí ó ôéêèóîïõüç é øìõûøêðò é "îïî øìõûìîêû çìëèó ýéë
îìçéöçì ì êïùóûëï &
( øêïôéùõóôìëèü D „& õé îì øêìôéêðéèëðíöèì é îïëïéèëð ùêû ó ïîëó "
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é ó ð çì øêéìùìõéôï èëðíçì çéëîìõü îì ûëõì çð è òìùéõü ˆ õð øêóùïçóð éëèéëèôéççìëèó çéîìèìê ò ìøêéùé
õéçóðò . ûïçîïêé ó ìýêéõ ùï é ñèéòøéêïèûêç èê î÷í îìèìê óëøìõü
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èì é ô õðùóè øéêëøéîèóôçìí çì ìëèïéèëð øìîï çï êïìôìêçìò ûêìôçé Ë Ì N àáàÎÐÔ ÔÎÏÖã Ò – © ãÏÑ
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çìëèó óëëõéùûéò èêïéîèìêó í ìç çïøóëïõ ëèïèü ñ òì éè õó ô òï
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( ùêû ì ëèìêìç í ìçï ô öóëõóòïí øìëîìõüîû
î ô öóëõðéòìòû
çïöéçó
ôëéùï òì çì øêóýïôóèü ì
çóîïéè ôìøêìëí ïöéò ý õì òûùêóèü ë ûçî + óé ‚# & . êóöó "# # # ûçî + ó
çï ôéëüòï
ëéêüé ìçï
çïð ý Dõïýïðô öóëõóòï
øìø èîï øêó
ìøêéùéõóèü
ôí )öóëõóòû
èïîí
öèìý
õ
ýìò
øêóç
+ óøóïõüçì ìý êéöéçï
çï
øêìôïõ
æõó øìõûöï èëð ëõóúîìò û îóé îõïëë ûçî + ó ùóë "
îêéùóèóêû
óé øìçðèóé ô öóëõóòìëèó & í óõó îïîóé èì
ìýð ëóõìô
ïèéõüçì
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è
ìýõïëèü
çéìøêéùéõéççìëèó
. ìø èîï ùììøêéùéõóèü
ò
òéèìùìò
èì
é
çé
øêì
ìùóè
çéêï
êéúóòìëèü øêìýõéò ìëèïçìôï &
ìêéçü çéøêóðèçìëèé "ùìôìõüçì ìöéôóùéç - øêéùéõéçóé ûçî + ó ë
øìòì
ïôéùìòì ô øìõçóò øêðò ùé ëèôó èóøï ëõì óèüí ûòçì "
óèü& üøêóôìùóè
ô óèì é î ûêïôçéçóðò
g(n) = fn (n) + 1.
n = 1
n=2
f1
f2
fn (n)
f (n)
n
f (n)
f (n)
‚ , ,&
+ îìèìê éëèéëèôéççì ìèçìëóèü î ô öóëõóò ò ûçî ó
ðòêéúéçóé
øêìèóôçìò ëõûöïé èéìêóð øêéôêï ïéèëð ô òïõìøìõé çû ó êûúîû
èì éé ôêéòð
‚ , ,& øêóç + óøóïõüçì çé ôëé ùï êï êéúóòì "#
øêóòéê ìùóç øìøûõðêç
u(a, n) = 0,
n = f (a)
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"#
< ïëëòìèêóò ìëò ëõéçç é èéîëè í îï ù ó îìèìê ëìùéê óè çé
ýìõéé
ëèï ëõìôì êïçóöéçìí
ó ìøêéùéõðéè
îïîìéèì öóëõì - öéôóùçìí îìõóöéëèôì èï
îóøìùìýçèéîëèìô
øì
èìòû ì êïçóöéçì òçì éëèôì ìøêéùéõéçç ò ìýêï ìò öóëéõ ( õéùìôïèéõüçìí ô èìò òçì éëèôé ëû éëèôûéè
òïîëóòïõüçìé
öóëõì . êóýïôóò î èìòû öóëõû éùóçó + û
. êéù ùû ó ïý ï + ëìùéê óè òéçéé ëèï ëõìôí çì ìøêéùéõðéè öóëõì çï
éùóçó + û ýìõüúéé òïîëóòïõüçì ì Œ
( ì óëèóîï ùéëü òçì ìøõïçìôï ì ùï ìôìêðè ìý ì êïçóöéççìëèó
òçì
éëèôï öóëéõí
ìøêéùéõðéò êï ïòóí øìùêïûòéôï è öèìèì ôêìùé
îìùóêìôîó
ì îïî ý èü ë êï ïòóí îìèìê é ìøêéùéõð è öóëõï çé ëï òó øì ëéýéíïïêïìøóêïðëü
çï ùêû óé êï þ . ìõûöïéèëð û é çé éëèîïð
îìùóêìôîïí
ôóôï ïð ëóëèéòï êï ˆ ïõéé ƒ óëõì ïèìòìô ôì ëéõéççì í óõó òïîëóòïõüçìé öóëõìí îìèì êìé
îèìçóýûùü
îì ùïçóýûùü ïøóë ôïõ çï ýûòïé ƒ èì èïîóé êï
ìøêéùéõð
èþ
óöé ì . õ ë ôì òì çïð ìøìêï çï ýìõüúû ëìôìîûøçìëèü
öéõìôéöéëîó
çïçó . êìëèé úó øêóòéê ) ñöóëõì òóîêìç ô ù òé÷
ƒ éè êé ëõìôï ïùï è ôøìõçé ìøêéùéõéççìé öóëõìí çì û çïèü é ì òì çìí
õóúü ìêìöé
êïëøìõï ìôìêðí
ïð óç ìêòï + óé í îìèìêïð ô ëïòì êï é çé ëìùéê óèëð çé ìëèïéèëð øìëõé ìöóëèîó ìè çéëûêïçìëèé ìè øïêïùìîëï çóöé ì
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ôìë
ó éçóðí éëõó î èìòû êïëøìõïïéè çïëèêìéçóé ó ùóëøì ó + óð ì ùêû ïð ëóèûï + óð ôì çóîïéè ï îûõóëïòóí ùé çïöóçïéè êïýìèïèü øêïôóõì ñçéè
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( ìôëéò çéøõì ìé øêïôóõìí îëèïèó çì òì éè çé çêïôóèüëð óçì ùï
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4mc = F 4t
4ρSc2 4t = 4P S4t,
5@1 :137? :?0?k1FB A :0?F?7G (4P, 4ρ → 0) FB?@
Ú-C.=.UÛ
ìèéõìëü ý ìýêïèóèü ôçóòïçóé çï ôï ç ûòïçóèïêç ïëøéîèí
õéëìëèïôõéçóé
ï ó ùéëüûêïôçéçó
çï ïùçéò
øõïçé - ý éó ôéëèçìí öèì úîìõüç é ïùïöó çï
ô ôï è çïóýìõüúóé èêûùçìëèó = ïùçóò öóë õìò
ôëé øêìëèìí çì ó çïöïõüçì ô õðùóè çéøêóëèûøçì ƒ èì øêóçðèü ï
çéó
ôéëèçìéí ï îïîóé îìç + û ôïèóèüëðí öèì öéòû øêóêïôçðèü :õï ï
êï ýé’óïóîóèëðíèéìêéèóîó
êûîó ìøûëîïôóêèûìèëðíçìîõûýìî
çéè óëîûëëèôìò
êïëøûè ôïéèëð
ôõïùé
êéúéçóð
èïîì ì ëìê èï ïêð ïùïöéççìëèó
æ çïùì øêó çïèüí öèì øì ôïèîìëèóí ûòéçó ìêóéçèóêìôïèüëð ó
) ìçó ðôçì ïçóòï è õóùóêû éé øìõì é çóé ’ ìêòûõï çï‹ ,êé„&ûõüèïèí
) ìùçï ó ó óöéëîó òóçóïè ê ˆ õð èôéêùì ì
"#
í ùé ) òìùûõü ç ï ëèéê çðëõûöïé
ìçï ùïéè
ï ï ôìøêìë ûøóêïéèëð ô ûêïôçéçóé ëìëèìðçóð ëõó øìõì óèüëð çï
‹ , ‹&
"#
è é í èì
í öèì øêóôìùóè î ïçó éççìòû êéûõü
èïèû
s
c=
c=
p
E/ρ
I
E
PV =
P = ρ RT
µ
dP
.
dρ
m
RT,
µ
dP = RT = P
dρ
µ
ρ
s
P
.
ρ
"#
‹ , %&
- úóýìöçìëèü ‹ , %& ëìëèìóè ô èìòí öèì øêó ôûîìô îìõéýïçóð
"#
èéòøéêïèûêï
çé ûëøéôïéè
ô êïôçóôïèüëðí øì èìòû ôòéëèì "#‹ , ‹& çïùì
c=
-D<
êïëëòïèêóôïèü ïùóïýïèóöéëîó øêì + éëë
íìèîûùï
í
öèì û é øêïôóõüçì ìøêéùéõðéè ëîìêìëèü ôûîï g?xFG :01562 A 8481FG `102G74 -C.=.UÛ 21xI1 h471
h4 :0?F›;86@o 2I191 :0?@?IT6a IB56IB;Ú 3 I?1:0?F?7?II13@6
3B2191 :1I;@6; 3A1013@6 c . >B:062?0 `102B 62:G7o3B 21x?@
TB863?@o 1@ t  6 @19FB I?:1I;@I1
5@1 @BA1? . >1 24 83? s@1
13@B87;?2 TB ABF012 @?2 h17?? 5@1 8 :?0812c :06h76x?I66
0?TG7o@B@4 8?0I4.
æòééè øêïôì çï ëû éëèôìôïçóé ó ùêû ïð èìöîï êéçóðí èêéýû ïð
ñùì óòïçóð÷ ïùïöó ùì îìç + ï ì èì ùï çïùì ûöóè ôïèüí öèì # çïëé
õéçóð ô øóôïéè áá ïõîì ìõð *ìöçì èïî é # ûëóõó êéúïéè ïùïöû
çï áá Š îìòû ìöéèëð ôëé ëèìí èì çï ìùóç çéùìëèï ó øêì + éçè û ìùóè
ìöéçü òçì ì ëóõ ó ôêéòéçó „h0B@62 @?:?0o 8I62BI6? IB F0G9GH 3@101IG F?7B. Ž376
3@?0x?Io 0B3:171x?I 8F17o 136 x  @1 3@?:?Io 3xB@6; u(x, t) 8
1hb?2 37G5B? hGF?@ `GIA_6?a F8Gk :?0?2?II4k. >BaF?2 F6`
`?0?I_6B7oI1? G0B8I?I6?A1@1012G GF187?@810;?@ u(x, t) .nB3
321@062 F8B 3?5?I6; x 6 x + 4x Ú063. -<.=ې B @BAx? TBA7H
5?II4a 2?xFG I626 1h›?2
. ^67B F?a3@8GHbB;
4V = S4x
P/ργ = const
c=
p
γP/ρ
J
J
J
J
x
¸º®Æ † Æ Æ ¬­®³¸´®µ¸­±¯±º¯ ®p ­µº°
x
x+ x
IB 3?5?I6? x  0B8IB
kS
∂u
,
∂x x
9F? k 1:0?F?7;?@3; `6T6A1a TBFB56 Ú9BT @8?0F4a 3@?0x?IoÛ.
n?TG7o@60GHb?a 3671a F?a3@8GHb?a IB 1h›?2 4V  hGF?@
!
∂u
∂u
∂ u
−
4x,
F = kS
= kS
Ú-C.=.pÛ
2
∂x
x+4x
∂x
x
-DU
∂x2
6 @19FB TBA1I >oH@1IB FB?@ 817I181? G0B8I?I6?
Ú-C.=.KÛ
9F? c = pk/ρ – 3A1013@o 817I4 ?376 918106@o 3 IB@;xA1a
– G3A10?I6?.
∂ u/∂t
. êó øéêé ìùé ô øêìëèêïçëèôì
∂2u
∂2u
= c2 2 ,
2
∂t
∂x
2
2
∂2u
= c2
∂t2
∂2u
∂2u
∂2u
+
+
2
2
∂x
∂y
∂z 2
.
Ë Ë Ì ÎãÖ à Ôà áàØÕÏØÎÏÖÏ ÏÑáÔ
S7; 8481FB G0B8I?I6; 0B3:013@0BI?I6; @?:7B h17oJGH 5B3@o
0B33GxF?I6a :0?F4FGb?91 0BTF?7B 21xI1 13@B86@o h?T 6T2?
I?I6a. {2?3@1 Ú-C.=.KÛ :17G5B?@3;
∂T
∂ T
=a
,
Ú-C.D.-|Û
∂t
∂x
9F? T (x, t) – @?2:?0B@G0B ABA `GIA_6; x 6 t . iIB71912 3674
Ú-C.=.pÛ 8 FBII12 37G5B? hGF?@ A1765?3@81 @?:7B
F
2
2
Q = kS
∂2T
4x,
∂x2
A1@101? 8@?AB?@ 8 1h›?2 4V Ú@?:?0o k – A1s``6_6?I@ @?:71
:0181FI13@6Û.
XB@?2 @1 x? 3B21? Q :1F356@48B?@3; F0G962 3:131h12
Q = ρλS4x
∂T
4t,
∂t
9F? λ – @?:71?2A13@o ∂T 4t – :184J?I6? @?2:?0B@G04 TB
∂t
.
80?2; 4t  5@1 6 FB?@ Ú-C.D.-|ې
a = k/ρλ
mBA62 1h0BT12 :06I_6:6B7oIB;
0BTI6_B 3 :0?F4FGb62 37G
5B?2 TBA7H5B?@3; 8 @12 5@1 8 FBII1a 36@GB_66 I? 0Bh1@B?@
BIB719 TBA1IB >oH@1IB. { 0?TG7o@B@? 8@10B; :016T81FIB; :1
80?2?I6 TB2?I;?@3; IB :?08GH.
-DC
Ë Ë Ìœ ð ÐÔÑáÔÓàÑâÏÐ ØÏ Ï ž ÔÔ
( éò êïëëû ùéçó øêéù ùû ó ùôû êï ùéõìô ô ìöó éççìò ìè ùéèïõé ôóùé
óëöéêø
ôï è ýìõüúû öïëèü ó óöéëîó ïùïö ëé ùï éëèü çéöèì
ôêìùé
ëóõ
íôõóð é çï ùôó éçóéí ó çéöèì ôêìùé îììêùóçïè íìøêé ùéõð
ìý óëîõ éöéçóðìîûë&ôçóòïçóð æ éëèü ôëé ì ùôï ôïêóïçèï "éëõó çé ìôìêóèü
øêó ùôó éçóé ìëèïçïôõó ( óõï øìêì ùïéè ùôó éçóéí
"
ôïéèëðí &
ˆ ôó éçóé øêìóë ìùóè øì óçéê + óóí ëóõï ìøêéùéõðéè ûëîìêéçóé
æ
ë
ëóõì
í ó ë îììêùóçïèì øêìóë ìùðè òéèïòìê ì øêó øéêé ìùé ô
ùêû
û ìýõïëèü ó òé ïçóîóí ëîï éòí ô ìøèóîû & ì øì ëûèó ) çóöé ì
"
çéîêûèóõüç
òéçðéèëð é îìõéýïçóð
çïûêïôçéçóéò
øêû óçé ìøóë ôïéèëð ûêïôçéçóéò
í
? ïðèçóî
) û ìõí ) òìòéçè
" ìý öçì ëóõ & . êó
óçéê
+ óó ñèì ìí öèì îêûèóèëð÷í ñëóõï÷
) òìòéçè
øéêé
ìùé
î
õéîèêóöéëîìòû îìõéýïèéõüçìòû îìçèûêû ùéîìêï + óó òéçð èëð
çïöóèéõüçéé ) ïêðùí ) óçùûîèóôçìëèüí ) éòîìëèü & í ï ûêïôçéçóé
) èì é ëïòìé"
ñëóõï÷
) õéîèêóöéëîìé çïøêð éçóé . ì èìòû êï ìôìê ì õûýóçç ëôð ð êï çìêìùç øêóîõïùç ìý õïëèé
ô ìøêéùéõéççì ëèéøéçó øêìóëèéîï è ó õûîïôëèôï óõó çéìëôéùìò õéççìëèó
’ ó óîï ôì ôëé ëôìó óøìëèïëð ùì ëó øìê ìøóêïõïëü ô ìëçìô çìò
çï
õóçé
ç é ùó éêéç + óïõüç é ûêïôçéçóð . ì èìòû óëèìîó ñòó ëèóöéëîì
ì øìùìýóð÷ êé ûõüèïèìô ùìëèïèìöçì ìöéôóùç ‰ êïôçéçóð ìùçó
ó èé øðèü
éí êïèïîó
çó + ï ) ô çï ôïçóð îì ó + óéçèìôå çïùì ëîï ïèü ì úóêìîìò óëøìõü ìôïçóó øêó èìò îìò øõéîëç
öóëéõ øêóç + óøéí ôëé ëîï ïçì ý õì ô õïôé ? ïèéòïèóîï ëèì #
óè çï ïêó òéèóöéëîó ìøéêï+ óð í ùõð îìèìê îìòøõéîëçïð
øõìëîìëèü
) ó êìôìé øìõé ì èì öéêéëöûê ìý ì éîìèìê é ë éê çïçóð ôëé èïîó
ûëøéúçì
ûçî + óìçóêû è çï îêï èì ì øìõðí ìý ìùðëü ýé òçóòì éùóçó
+ ýû ïõèéêóðí çïøêóòéê & öéò îìçîêéèçì ïîõ öïéèëð øêóöó " ùó éêéç + óïõüç é ûêïôçéçóð ñó êï è øì ôëéòû øìõ ÷þ ˆ ï
çïé éèìôòéëèì
ìí öèì
ëêïû çïöóçï è ë
øêóýéýéïð ôóùóòì
î ûëõû ïòçéìý ïìùð
= ïöéò î êéïõüçìòû îìõéýïçó
ùìýïô
õðèü
ìùóòìëèó
óîèóôçìé
þ
:õïôçïð øêóöóçï ) ô ùó éêéç + óïõüç ëôì ëèôï
í
‹ ‚ &
"# ##
èôïçóé
é ô öóëõéçóé øêìó ôìùçì ëôìùóèëð î ûòçì éçó çï æçèé êóêì ) î ùéõéçó . ìõûöïéèëð ëû éëèôéçç ô ó ê ú ô øêìëèìèé ? ì éè øìîï ïèüëðí öèì èì ôëé é ôìøêìë ûùìýëèôïí çé îïëï ó ëð
ëûô îïöéëèôì
éëèôï ùéõï èì çé èïî ìõóöéëèôì ùé ëèôóèéõüçì óçì ùï øéêé ìùóè
ˆ ï é çé çïöóèéõüç é èé çóöéëîóé ûøêì éçóð òì ûè úï ï
å < ïûòééèëðí ò ïìëèêðéò ôçóòïçóé õóúü çï ìùçì ëèìêìçé ùéõï -Dc
F
z
KL M
F = k ż
F =0
ż = 0
KL LM
F =
kz̈
I ϕ̈ = −kϕ ϕ
kϕ
q
L
I
mẍ = −kx
C
Lq̈ = −
q
,
C
q/C
cos ωt
eiωt = cos ωt + i sin ωt,
cos ωt
i sin ωt
eiωt
(e
iωt 0
) = iωe
iωt
,
iω
úïêì ìòùï øìùçðèü
èû óõó óçû çïûöçû ùóë+ óøõóçû çï ùêû ì èï *ïò
èëð
çìô
é îïèé ìêóó ò úõéçóðí ó êï ôóèóé óùéè ëìôëéò ùêû ì ùìêì
ì í ô ìùð çï ïîìçìòéêçìëèóí ùõð îìèìê ëèïê ð î ôììý é çé
ìùóèëð * ïî óõó óçïöéí çì ëôì ëèôì ‹ ‚ & ô èéìêóó õóçé ç ùó éêéç + ó ## ôêìùé øì ó + óìççì ëóëèéò ëöóë ïõüç
ûêïôçéçó ìîï ôïéèëð "#öéò èì
õéçóð
ùõð
ïêó
òéèóîó
ìçéöçìí òì éè øìîï ïèüëð ëèêïçç òí öèì
óôììý
êïéè çéëìêï òéêçì ýìõüúû êìõü ô ùó ûêï í îìèìê é óòé è ùéõìí
é ìôìêðí ë êï çììýêï ç òó ûçî + óðòó . êóöóç èìòû ùôé ì øéêô
í ïêòìçóöéëîóò îìõéýïçóðò ô õóçé ç ùó ûêï ùé ëèôóèéõüçì
øêóçïùõé
óè + éçèêïõüçïð êìõü ìôèìê í êïõì éçóé ûçî + ó ô êð ùóëøìõü
’ ûêüé øêéùëèïôõéçóé ô ôóùé ëûòò ïêòìçóî & øì ôìõðéè éîèóôçì
" ëôì ëèôì ‹ ‚ & ô ìý éò ëõûöïé ìôïèü
ìçéöçìí öèìý ôëé "#
èì ì +##éçóèü ô øìõçì òéêéí çïùì øêì èó ëìýëèôéç çîïî òîìòøïëí
øûèéò øìçìëììèôéèëèôû
é èéêêóèìêóó - ý óé ëììýêï éçóð ìêìúóí
ìçó
çé
ïòéçð è ìø èïí ìëìýéççì ìèêó + ïèéõüçì ì ? ì çì
ëêïûí çïøêóòéêí ëîï ïèüí öèì ùõð ô öóëõéçóð ëûòò ÿ
eiωt
øìõé çì êïëëòìèêéèü
öèì õéîìí ô öóëõðéèëð îïî ëûòòï éìòéèêóöéëîì øêì êéëëóó ë øìîï ïèé
õéò
S = cos ωt + cos(ωt + ϕ) + . . . + cos(ωt + nϕ)
Ω = eiωt + ei(ωt+ϕ) + . . . + ei(ωt+nϕ) ,
eiϕ
ei(n+1)ϕ − 1 iωt
e .
iϕ
e −
1
ì ìêï ùì øìõé çééí éëõó îìòû èì øìôé õì
- îìçöïèéõüçì
øêéùôïêóèéõüçì
øìòûöóèüëð
ë ô öóëõéçóéò *ì ùï ìîûë ùìýïôõéçóð
óîèóôç
òçóò öïëèé øìõûöóè ëìôëéò ùêû ìé ìëôé éçóéí ï øêéìý éìòéèêóöéëîì øêì êéëëóó ýûùéè
ôêï ìôïçóé
õðùéèü çéûùìýçì
òïõéçüîóò ëûòò
ìèîê èóéòô ëûòòû
Ω=
S = Re(Ω)
S
S
Ë Ë Ì± — à Ï âÖã áÏÖÏ – Ðà © ã Ôâà
XB:6J?2 G0B8I?I6? 0?F6I9?0B Ú-C.-.<Û F7; 1FI12?0I191 37G
5B;
i~ ∂ ψ
∂ψ
=
.
Ú-C.U.-=Û
∂t
2m ∂x
g1xI1 3ABTB@o 5@1 -C.U.-=Û 8 ABA12@1 32437? 848?F?I1 6T
:0?F:171x?I6; 1 86F?Ú 817I181a `GIA_66 Ú-C.-.=Û.m?:?0o ?3@?
3@8?II1 :1321@0?@o ABA6? `GIA_66 GF187?@810;H@ Ú-C.U.-=Û.
„ABT48B?@3; 6k 2I191 Ú@1 ?3@o h17oJ? 5?2 TBFG248B713oÛ.
ÿ ì çóîï é ô ïùïöé ì ùó êïî + óìççì êéúéèîé N
2
2
-Dp
SB7?? IB56IB?@3; ™`G@h17š :0?F3@B87;Hb6a 31h1a G87?AB
@?7oI1? TBI;@6? 8 1h7B3@6 96:1@?@65?3A6k G21TBA7H5?I6a.
^0?F6 0?J?I6a Ú-C.U.-=Û ?3@o
ψ(x, t) = ψ(x) e
.
Ú-C.U.-DÛ
E1F3@B87;; Ú-C.U.-DÛ 8 Ú-C.U.-=Û 6 31A0BbB; IB e  :17G
5B?2 G0B8I?I6?
F7; B2:76@GF4 ψ(x)
∂ ψ 2mE
+
ψ(x) = 0.
Ú-C.U.-<Û
∂x
~
S63A0?@I4? 3:?A@04 81TI6ABH@ :062?0I1 @BA. n?J?I6?2
-C.U.-<Û
37Gx6@
Ú
−iEt/~
−iEt/~
2
2
2
ψ = a sin(kx + b),
√
2mE
k=
.
~
Ž376 5B3@6_B TBxB@B IB 1@0?TA? [0, l] A0B?8426 G37186;26
 @1 s@62 G37186;2 21xI1 GF187?@8106@o @0?
ψ(0) = ψ(l) = 0
hG;
kl = nπ; n = 1, 2, . . . ,
5@1 FB?@ F63A0?@I4a IBh10 81T21xI4k sI?096a
π ~
-C.U.-UÛ
E =n
,
Ú
2ml
B TIB56@ 6 5B3@1@ 8 367G Ú-C.-.DÛ.
>?IB2I191 371xI?? 1:0?F?7;?@3; F63A0?@I4a 3:?A@0 B@1
2B 81F101FB. ^11@8?@3@86? sA3:?062?I@B7oI42 FBII42 TB24
AB?@ A0G9 6 @?106; :17G5B?@ 1`6_6B7oI4a 3@B@G3 ABA 0Bh1
@13:131hI4a 6I3@0G2?I@. ŠHh6@?76 6I@?0:0?@B_6a :06FG24
8BH@ 26`4 6 :6JG@ G5?hI6A6.
t@1 3ABTBI1 1@IHFo I? 8 GA10 `6T6AB2 A1@104? IB 3B212
F?7? TB37Gx68BH@ 813k6b?I6;. ‡7B3365?3A6a 2B@?2B@6A IB
6k 2?3@? :101kG h4 I? 84FG2B7 hGFG56 :18;TBI 6I42 G3@B
812. Ž376 :1321@0?@o IB s3A6T A101@A1a _?:15A6 8?FGb?a
1@ Ú-C.-.=Û A Ú-C.U.-Uې @1 7196AG TF?3o 3:131hI4 0B3321@0?@o
ìôìêðè ô èìò ëõûöïéí öèì öïëèó + ï çï ìùóèëð ô øìèéç + óïõü çì ’ðòéó óîó
n
2
2 2
2
-DK
@17oA1 `6T6A6. ^ABx?2 8 s7?A@01F6IB26A? 0?J?I6?2 817
I18191 G0B8I?I6; 37Gx6@ A17?hBI6? F?a3@86@?7oI1a 6T2?0;?
21a 8?7656I4 ÚIB:0;x?II13@6 :17;ې B 63:17oT18BI6? A12
:7?A3I4k 0?J?I6a
– 81:013 GF1h3@8B. XF?3o IB1h101@. ƒ6
T65?3AB; 6I@?0:0?@B_6; A17?hBI6a 1@3G@3@8G?@ 6 817I18B;
`GIA_6; 6TIB5B7oI1 1:0?F?7;?@3; ABA A12:7?A3IB; 5@1h4 I?
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R
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:013š.
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µ
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3
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dt
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R
3
2
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