Algebra Lineal Ejercicios
December 8, 2022 | Author: Anonymous | Category: N/A
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=
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
XLJSHNLEMTNL @C CRÍACECX @C JM GMHSJPM@ @C NEICENCTÁM SAXM MJICDTM MJ ICDTM J JNECMJ NECMJ ^ PCLTNM AM AMPTNHNMJ PTNHNMJ CRMACE?N-46=> VTLDJCAM = 4 R 4
Vmrm uem trmesglramhnøe jnecmj P ? T
↔ V4 `c jm humj sc hlelhce jms sniuncetcs naíicecs?
= 4 4 ∔= = = 4 = ∔ = 4 4 1 4t 4 ∔ t + = 4 7 0 1 + 1 + + ; P ; ; 1 t + t + P t = P 4 t 3 t 9 4 = ∔= = = ∔= ∔= 4
P
Xc pn`c? m) bmjjmr jm gøraujm `c trmesglramhnøe jnecmj, d) dmscs pmrm cj eÿhjcl, namice y vcrngnhmr cj tclrcam `c jm `nacesnøe, h) bmjjmr jm rcprcscetmhnøe amtrnhnmj rcspchtl `c jms dmscs?
= ∔= ∔= = ∔= ∔= ∔= ∔= 4 4 4 , , , y H 1 t + t + =, t + t , t y `) hle jm metcrnlr amtrnz ∔= ∔= ∔= ∔= = ∔= ∔= = 4 = bmjjmr jm namice `c = 4 Xljuhnøe? m) Vmrm hmjhujmr jm gøraujm `c trmesglramhnøe? m d 4 ∔= = = = ∔= = 4 = 4 8 0 1 = ∔= + ∔= 4 + 4 = + ∔= = (=) h ` D 1
Hmjhujmals jls vmjlrcs `c jms hlestmetcs? m d 4= + 4 ∔ 8 + 0 ∔ 4 + 4 ∔ 8 + 4 0
h ` 1 ∔ + 4 ∔ ∔ + 4 + + = 4 8 0 = 4 8 0 = = = m 4= + 4 ∔ 8 + 0 1 m 4 = ∔ 4 + 4 ∔ 8 + 4 0 1 d ∔= = ∔ = 4 4 d 1 4 = = 4 = h ∔ + ∔ 1 h ∔ ∔ = 4 8 0 8 ∔= + 4 4 + 8 + 0 1 ` ∔= 4 = = 0 `
Tcsljvnce`l?
=
=
=
= 1 =3 ( 7m ∔ 8d ∔ 8h ∔ 8` ) ; 4 1 =3 ( 8m ∔ >d ∔ >h + 7` ) ; 8 1 =3 ( ∔ m + 8d + 2h + 8` ) ; = 0 1 ( m + =4d + 9h ∔ 8` ) =3 Ce (=) mpjnhmals P( )? m d 4 ∔= = = = ∔= = 4 1 + + + P P P P P = 4 8 0 = ∔= ∔= 4 4 = ∔= = h ` m
d
=
=
=
1 ( 7m ∔ 8d ∔ 8h ∔ 8` ) ( 4t 4 + 7t + 0 ) + ( 8m ∔ >d ∔ >h + 7` ) ( t 4 + =) + ( ∔m + 8d + 2h + 8` ) ( 4t 4 + 3t + 9 ) + =3 =3 h ` =3
P
+
= =3
( m + =4d + 9 h ∔ 8` ) ( 4t 4 ∔ t + =)
Xnapjngnhme`l?
m P h
d 1 ( m + d + h ) t 4 + ( 4m ∔ d + h ) t + = ( 9m + 0d + =0h + 0` ) ` 3
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
4
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
d) Vmrm bmjjmr dmscs pmrm cj eÿhjcl `cdcals bmhcr huapjnr P ( x ) 1 6 m d = 1 ( m + d + h ) t 4 + ( 4m ∔ d + h ) t + ( 9m + 0d + =0h + 0` ) 1 6 t 4 + 6 t + 6 3 h `
P
m m + d + h 1 6 = = = 6 6 d 4 ∔= = 6 1 6 4m ∔ d + h 1 6 9 0 =0 0 h 6 = 9 + 0 + =0 ( m d =0h + 0` ) 1 6 ` 3 Tcsljvnce`l?
m1∔
4 = h; d 1 ∔ h; ` 1 ∔4h 8 8 m d
Tccapjmzme`l ce
?
h
`
= 4 m d ∔ h ∔ h h ∔4 ∔= D E 1 ∔4 ∔= 8 1 h ` 1 8 (P ) 8 ∔7 ; 8 7 ∔ 8 ∔4h h
@na E (P ) 1 =
P
Vmrm dmscs `c jm namice prnacrl @c jm bmjjmr gøraujm `c trmesglramhnøe jnecmj? hmjhujmals M
= m d P 1 4 h ` 9 3
=
=
∔=
=
0
3
=0
3
m = 6 d 6 M 1 4 h 0 9 3 3 `
=
=
∔=
=
0
3
=0
3
M
Cshmjleme`l MP ? = 4 9 3 6 6 6 P
M 1 6
∔= 9 3 6 6 =
D
N (P )
9 9 1 4 + + ∔ + + t 4t 3 ; t 3 ; =
= 4 9 3 6 = ∔= 0 3 6 MP 1 =0 = = 0 3 3 0 6 6 3
P
1 ( M ) 1 8
; @na N
(P )
Cj tclrcam `c jm `nacesnøe ne`nhm quc? @na E(P ) + @na N (P ) 1 @na T 4 R 4 = + 8 1 0 0 1 0 , cetlehcs sc `caucstrm cj tclrcam `c `nacesnøe.
h) Jm amtrnz Amt DH (P ) cstm `m`m plr? = H Amt D (P ) 1 4 8
=
=
4
4
8
8
=
4
8
= ∔= 1 ∔t 4 + 4t ∔ 8 1 = ( t 4 + t + =) + 4 (t 4 + t ) + 8 (t 4 ) ∔= ∔=
P
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
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OLXSC VM^C HBNVMEM
∔= ∔= ∔= P = ∔= P ∔=
P
=
OLXC VM^C HBNVMEM
4=
1 ∔t 4 ∔ 0t ∔ 1 = ( t 4 + t + =) + 4 (t 4 + t ) + 8 (t 4 ) ∔= 3 ∔= 4 = 1 ∔t ∔ 1 = ( t 4 + t + =) + 4 ( t 4 + t ) + 8 ( t 4 ) ∔= 3 ∔= 4= 1 ∔8t 4 ∔ 4t ∔ 1 = ( t 4 + t + =) + 4 ( t 4 + t ) + 8 ( t 4 ) = 3
Icecrmals cj sniuncetc snstcam? = = = = = = = ∔=
= = 6 4 = 6 6 8
4
4
8
8
∔= ∔0
∔=
∔8 6 4 4 1 4 ∔ 8 ∔8 ∔4= 3 ∔= 3 ∔4= 3
Tcsljvnce`l?
= =
= ∔=
∔= ∔0
∔8 = 6 6 ∔8 ∔ 4= 3 ∔ = 3 ∔ 4= 3 = = 6 4 6 =/ 3 =3 == 3 ∔4 6 = 6 3 = 6 6 ∔8 ∔4= 3 ∔= 3 ∔4= 3 6 6 = ∔8 8 ∔= ∔= = 4
=
=
4
4
∔8 ∔ 4= 3 ∔= 3 ∔ 4= 3 =/ 3 =3 == 3 4 1 3
8
8 8 Cetlehcs?
∔=
=
8
∔8
8
∔=
∔=
∔=3 ∔4= ∔= ∔4= = = = = = H H = = == == Amt D (P ) 1 4 4 4 4 Amt D (P ) 1 43 3 ∔=3 =3 ∔3 ∔3 8 8 8 8 4 = 4 = H `) P 1 Amt D (P ) = 4 = 4 H D 4 = Hmjhujmals ? = 4 D 4 = 1 = ∔= + ∔= = + ∔= ∔= + ∔= ∔= = 4 ∔= ∔= ∔= ∔= = ∔= ∔= = Icecrmals?
= ∔= ∔= ∔= 4 ∔= = ∔= ∔= = 1 ∔= ∔= = ∔= 4 ∔= ∔= ∔= = = Tcsljvnce`l?
∔= / 4 ∔= ∔= 4 = ∔4 = 1 1 ∔= = 4 4 ∔4 D ∔= / 4 ∔=
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
0
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
∔= ∔=3 ∔4= ∔= ∔4= 26 4 = 4 4 = ∔ = = = 43 ∔06 H = = == 1 P 1 Amt D (P ) 1 = 4 H = 4 D 3 ∔=3 =3 ∔3 ∔3 4 ∔4 =6 6 ∔= 2 4 = P 1 ∔0 H = 4
6
Vcrl?
4 = 4 = P 1 A m t H P = 4 = 4 H @c jm dmsc H `c `mtl?
= = = H 1 t 4 + t + =, t 4 + t , t 4 AmtH 1 = = 6 = 6 6 = = = 2 0 4 = P 1 = = 6 ∔0 1 0 = 4
= 6 6 6
2 4 = 1 0t 4 + 0t + 2 = 4
P
VTLDJCAM 4 Xcm jm trmesglramhnøe jnecmj P ? T 8 ↔ T 8 `cgnen`m plr
P ( m, d, h ) 1 ( 0m + 4d + 4h, 4h + 0d + (k + =)m, ( 8k ∔ =) m + 4d + 0h ) bmjjmr cj vmjlr `c k `c amecrm quc jm amtrnz cstíe`mr `c P scm `nmilemjnzmdjc lrtlilemjacetc. Jucil bmjjc c Mt .
Xljuhnøe? Jjcvmals jm gøraujm `c trmesglramhnøe m su glram amtrnhnmj?
4 4 m 0 4 4 m 0 P d 1 k + = 0 4 d , `le`c M 1 k + = 0 4 cs jm amtrnz cstíe`mr `c P. h 8k ∔ = 4 0 h 8k ∔ = 4 0 Vmrm quc M scm `nmilemjnzmdjc lrtlilemjacetc cstm `cdc scr snaîtrnhm, cetlehcs sc huapjc quc?
k + = 1 4 1= k 1 8k ∔ = 1 4 0 4 4 1 M 4 0 4 4 4 0
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
3
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Híjhuj Hí jhujl l ` c mutlvmjlrcs M ∔ N 1 6 ? 0 ∔ 4 4 0 4 4 6 6 4 0 4 ∔ 6 6 1 6 4 0 ∔ 4 16 4 4 0 6 6 4 4 0 ∔ 6
∔4
∔ =4 ( 0 ∔ ) + 4
6 4∔ 4
4
∔ ( ( 0 ∔ ) / 4) g 8 + g = ∔ g 8 + g 4
4
∔4 + 0 ∔
4 = 1 6 4 ( ∔ 4 ) ( ∔4 + ) ∔ ( 4 ∔ ) ∔ ( 0 ∔ ) + 4 1 6 4
= = 4 ( ∔ 4 ) ∔4 + ∔ ( 0 ∔ ) + 4 1 6 ( ∔ 4 ) ∔2 + 3 ∔ 4 1 6 ( ∔ 4 ) ( 4 ∔ =6 + =7 ) 1 6 4 4
=,4 1 4 ( ∔ 4 ) ( ∔ 4 ) ( ∔ 2 ) 1 6 1 2 8 Vmrm c Mt sc smdc quc? c Mt 1 V c @ ( V )∔ , pcrl hlal M cs uem amtrnz quc `nmilemjnzmdjc =
lrtlilemjacetc sc huapjc quc? ( V )
∔=
P
1 ( V ) , cetlehcs? c Mt 1 V c @ ( V )
P
Híjhuj Hí jhujl l ` c mutlvchtlrcs M ∔ n N R 1 ? Vmrm =,4 1 4 ? 4 4 4 x 6 4 4 4 y 1 6 rcsljvnce`l? 4 4 4 z 6 4
4
46
4 4 46 4
4
46
4
4
46
∔ g= + g 4 6 6 6 6 4 x + 4 y + 4 z 1 6 x 1 ∔ y ∔ z ∔ g= + g 8 6 6 6 6
x ∔ y ∔ z 1 Tccapjmzme`l ce y y z z
∔= ∔= ∔= ∔= y = + z 6 x= 1 = ; x 4 1 6 6 = 6 = x=
x4
Hlal M cs `nmilemjnzmdjc lrtlilemjacetc `cdcals lrtlelramjnzmr jls mutlvchtlrcs?
x= 1 x= x= 1 x=
,=,6 6) ( ∔4=,=, x= 1 4 4 ( ∔=) + = + 6
= ( ∔=,=, ,=,6 6) 4
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
7
OLXSC VM^C HBNVMEM
x4 1
x 4 ∔ x 4 , x= x= x 4 ∔ x 4 , x= x=
x= 1
( ∔=, 6,=) ∔ ( ∔=, 6,=) , ( ∔=, 6,=) ∔ ( ∔=, 6,=) ,
OLXC VM^C HBNVMEM
= 4 = 4
( ∔=,=, 6 ) ( ∔=,=, 6 )
= 4 = 4
( ∔=,=, 6 )
= x4 1 ( ∔=, ∔=, 4 )
( ∔=,=, 6 )
g 4 + g =
6
7
Vmrm 8 1 2 ? ∔0 4 4 x 6 4 ∔0 4 y 1 6 rcsljvnce`l? 4 4 ∔0 z 6 4
4 6
4 g8 + g=
∔0
4 6
∔ g8 + g 4 6 ∔7
4
∔0 6
∔0 4 4
6
∔7 6
7
4
4
=
∔= 6
7 66
∔=
= 6
∔0 6
=
∔4 6
6 =
6
6 ∔= g 4 + g 8
=
6
6 6 = 6
∔= 6
∔ y + z 1 6 y 1 z x z 6 x z ∔ 1 1 x z Tccapjmzme`l ce y z 1 z z
= = z = x8 1 = = = x8
Lrtlelramjnzme`l?
x8 1
x8 x8
x8 1
(=,=,=) =4 + =4 + =4
= x= 1 (=,=,=) 8
Hmjhujl `c jm amtrnz `c pmsl?
∔= 4 = V 1 x= x4 x8 V 1 4 6 Xc huapjc quc ( V )
∔=
∔= 7 ∔=
= 8 =
7 4
8 =
7
8
P
1 ( V )
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
9
OLXSC VM^C HBNVMEM
∔= 4 P ∔= ( V ) 1 7 = 8
= 4
∔= 7 = 8
OLXC VM^C HBNVMEM
6
4
= 8 7
Jm amtrnz `nmilemj cstmrí `m` plr?
c 4t 6 6 4 6 6 6 @ 1 6 4 6 c @t 1 6 c 4t 6 6 c2t 8 6 6 2 6
= 6 @ 1 6 4 6 6
6
P
Vmrm c Mt c Mt 1 V c @ ( V )
∔= 4 = Mt c 1 4 6 ∔c 4t 4 c 4t Mt c 1 4 6
∔=
=
7 ∔=
8 4t c = 8 6 6 =
7 4
6 4 t
c
8
7
∔c 4 t 7
∔c 4 t 7 4c 4t 7
t c 2 ∔=
8 4 t c2 ∔=
7 t c2 = 8 8 8
6
∔= 4 6 ∔= 6t 7 c2 = 8 = 4 ∔= 7 = 8
= 4 ∔= 7 = 8
6
4
7 = 8
4c 4t + c2t 6 4 t 8 2t ∔c + c Mt 4 c 1 8 7 4 t 2t ∔c + c = 8 8
∔c 4 t + c 2t 8 4t 2t ∔c + c 8 4c 4t + c2t 8
∔c 4t + c2t 8 4c + c 2 t 8 4t ∔ c + c2 t 8 4t
VTLDJCAM 8 = = sc pn`c? m) Cj vmjlr `c jm hlestmetc ‚m‑ smdnce`l quc uel `c sus 4 m = ∔
@m`m jm amtrnz? M 1
vmjlrcs prlpnls cs niumj m =+8n, d) `nmilemjnzc jm amtrnz M y su amtrnz nevcrsm plr Bmanjtle Hmyjcy, h) bmjjc Me . Xljuhnøe? m) Sel `c jls mutlvmjlrcs cs = 1 = + 8n , cetlehcs sc huapjc quc? M ∔ = N 1 6 ?
= = = + 8n ∔ 4 ∔ m
= 6
∔8n 16 + ∔ 4 m = 8n 6
=
∔
4
1 6 ( ∔8n ) ∔ ( ∔ m 4 ) 1 6 > ( n 4 ) + m 4 1 6 m 4 ∔ > 1 6
8n
∔=
( m ∔ 8) ( m + 8) 1 6 `c `le`c sc ldtncec? m 1 ∔8 ; m 1 8
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
2
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
= = d) Jm amtrnz M cstmrí `m`m plr? M 1 ∔ > = Híjhuj Hí jhujl l ` c mutlvmjlrcs? = ∔ 8n = + 8n cetlecs cj ltrl mutlvmjlr scrí 4 1 Hlal ue mutlvmjlr cs = 1
Híjhuj Hí jhujl l ` c mutlvchtlrcs M ∔ n N R 1 Vmrm = 1 = + 8n ? ∔8n = x 1 6 , rcsljvnce`l? ∔> ∔8n y 6 ∔8n = 6 ∔8n = 6 nx ∔8nx + y 1 6 y 1 8nx 6 66 ∔> ∔8n 6 8n g= + g 4 x x = = Tccapjmzme`l ce? 1 1 x x= 1 y 8nx 8n 8n
Vmrm 4 1 = ∔ 8n ? 8n = x 6 ∔> 8n y 1 6 , rcsljvnce`l? 8n
= 6
8n
=6
8nx + y 1 6 y 1 ∔8nx
∔> 8n 6
6 66 ∔8n g= + g 4 x x = = Tccapjmzme`l ce? 1 x x 1 1 4 y ∔8nx ∔8n ∔8n Híjhujl `c jm amtrnz `c pmsl?
= ∔8n ∔= = 8 ∔n ∔= ∔= 1 1 ∔ V V 8n ∔8n ∔7n ∔8n = 7 8 n
=
=
V 1 x= x 4 V 1
Híjhujl `c jm amtrnz `nmilemj? @ 1 V ∔= M V @ 1 ∔
6 = + 8n @ 1 ∔ 6 = ∔ 8n ∔ n > = 8n 8n
= 8 ∔n =
7 8
= =
=
Vmrm cj híjhujl `c amtrnz nevcrsm M ∔= plr Bmanjtle Hmyjcy pmrtnals `cj pljnelanl p ljnelanl hmrmhtcrástnhl? 4
V( ) 1 ( ∔ =) + > 1 6 4 ∔ 4 + =6 1 M4 ∔ 4 M + =6 N 1 / / M∔= M ∔ 4 N + =6 M∔= 1
= = ∔= = 4 6 = = ∔= M 1 M 1 4 N ∔ M M 1 ∔ =6 > = =6 =6 6 4 ∔> = =
∔=
∔=
∔= Vmrm M M 1 V @ V ? e
@ 1
= + 8n 6
e
e
(= + 8n )e @ 1 n = ∔ 8 6 6
e
(= ∔ 8n )e 6
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
>
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
e e 6 = 8 ∔n = = (= + 8n ) = (= + 8n ) ∔ M 1 V @ V 1 1∔7 e e 8n (= + 8n ) 8n ∔8n 6 (= ∔ 8n ) 7 8 n e
e M 1 ∔
e
∔=
= (= + 8n )
e
7 8n (= + 8n ) e
(= ∔ 8n ) 8 ∔n e ∔8n (= ∔ 8n ) 8 n e
(= ∔ 8n ) 8 ∔n e 8 n ∔8n (= ∔ 8n ) e
Aujtnpjnhme`l y snapjngnhme`l ldtcecals?
8 (= + 8n )e + (= ∔ 8n )e = e M 1 ∔ 7 >n (= + 8n )e ∔ (= ∔ 8n)e
( (
) ∔n (= + 8n ) ) 8 ((= + 8n )
e + n (= ∔ 8n ) e e + (= ∔ 8n ) e
)
VTLDJCAM 0 Ce cj cspmhnl vchtlrnmj T 4 sc tncec jm rchtm " y 1 kx " y cj lpcrm`lr jnecmj ‚J‑ quc trmesglram tl`l vchtlr ‚m‑ ce cj vchtlr ‚d‑, snaîtrnhl mj prnacrl rcspchtl `c jm rchtm ne`nhm`m. Bmjjmr jm amtrnz `cj lpcrm`lr jnecmj J. Xljuhnøe? Cj lpcrm`lr jnecmj scrí `c jm sniuncetc glram?
P ( m= , m4 ) 1 ( d= , d4 ) @cj irígnhl? j ? y 1 kx ↔ a 1 k j= ? y ∔ y6 1 a= ( x ∔ x6 ) j= ? y ∔ m4 1 a= ( x ∔ m= ) (=) Hlal j ⊣ j= a= 1 ∔ Ce (=)? j= ? y ∔ m4 1 ∔
= k
= a
=
a= 1 ∔ k
=
( x ∔ m= ) y 1 m4 ∔ ( x ∔ m= ) k
y 1 kx icecrm jms hllr`cem`ms `cj puetl A, Jm netcrschhnøe `c jms `ls rchtms = 1 ∔ ∔ y m x m ( ) 4 = k k ( m= + km4 ) m + km4 , cetlehcs rcsljvnce`l cj snstcam ldtcecals x 1 = 4 y1 4 k += k + = m + km4 k ( m= + km4 ) A 1 = 4 , 4 + + k k = = Vlr ltrm pmrtc, cj puetl ac`nl A cstm `m`l plr? A 1
( m= , m4 ) + ( d= , d4 ) 4
( d= , d4 ) 1 4A ∔ ( m= , m4 )
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=6
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
m + km k ( m= + km4 ) 1 4 = 4 4 , ∔ ( m= , m4 ) 4 P ( m ,m ) + + k k = = 4 =∔ k 4 k ∔ = 4k 4k P ( m= , m4 ) 1 4 m= + 4 m4 , 4 m= + 4 m4 k += k += k + = k += ( d= , d4 )
=
4
= ∔ k 4 = 4 P m 1 k + = m4 4k k 4 + =
4k
4 k 4 + = m= M 1 4= = ∔ k 4 k + = 4k k ∔ = m4 4 k + =
4k 4 k ∔ =
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
==
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
CRMACE?NN-46=2 VTLDJCAM = @m`m jm trmesglramhnøe jnecmj?
P ? V 4 ↔ T 8 , quc tncec hlal naíicecs pmrm jls pljnelanls `c jm
(
)
(
)
dmsc D 1 t 4 + t + =, t 4 + t , t 4 jms sniuncetcs naíicecs? P t 4 + t + = 1 (=, =, 4, 8) ; P t 4 + t 1 (=, 4, 0 ) y
P ( t 4 ) 1 ( 4,4,4) . m) Bmjjmr jm rcprcscetmhn rcprcscetmhnøe øe amtrnhnmj `c P rcspchtl rcspchtl `c jms dmscs hmeøenhms `c V 4 y T8 . d) eÿhjcl y rmeil, hlal sus rcspchtnvms dmscs y `nacesnlecs. h) Bmjjmr Wcrngnhmr cj tclrcam `cmsá jm `nacesnøe. Xljuhnøe? m) Vmrm hmjhujmr jm gøraujm `c trmesglramhnøe?
mt 4 + dt + h 1 = ( t 4 + t + =) + 4 (t 4 + t ) + 8 ( t 4 )
(=)
Hmjhujmals jls vmjlrcs `c jms hlestmetcs? = = 4 = m
4 4 4 4 1 d 8 0 4 8 h
Tcsljvnce`l? = 1 h
4 1 d ∔ h 8 1 m ∔ d Ce (=) mpjnhmals P( )?
P ( mt 4 + dt + h ) 1 =P (t 4 + t + =) + 4P (t 4 + t ) + 8P ( t 4 )
P ( mt 4 + dt + h) 1 hP (t 4 + t + =) + (d ∔ h ) P (t 4 + t ) + ( m ∔ d ) P (t 4 ) P ( mt 4 + dt + h ) 1 h (=, 4, 8) + ( d ∔ h ) (=, 4, 0 ) + ( m ∔ d ) ( 4, 4, 4 )
P ( mt 4 + dt + h ) 1 ( 4m ∔ d, 4m, 4m + 4d ∔ h ) Jjcvme`l jm glraujm m su glram amtrnhnmj?
m 4 ∔= 6 m P d 1 4 6 6 d h 4 4 ∔= h 8
Cetlehcs jm rcprcscetmhnøe amtrnhnmj `c P rcspchtl `c jms dmscs hmeøenhms `c V 4 y T scrí?
4 ∔= 6 6 M 1 4 6 4 4 ∔=
d) Vmrm cj eÿhjcl? P ( x ) 1 6 E (P ) 1
( mt
4
+ dt + h ) V 4 / P ( mt 4 + dt + h ) 1
m 4 ∔= 6 m 4 ∔= 6 6 4 ∔= 6 6 = 6 6 6 m 1 6 P d 1 4 6 6 d 1 4 6 6 1 6 4 6 6 6 6 = 6 6 d 1 6 h 4 4 ∔= h 4 4 ∔= 6 4 4 ∔= 6 6 6 = 6 h 1 6
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=4
OLXSC VM^C HBNVMEM
( mt
Vlr tmetl, cj eÿhjcl scrí? E (P ) 1
4
OLXC VM^C HBNVMEM
+ dt + h ) V 4 / m 1 d 1 h 1 6
Vmrm icecrmr jms dmscs `cj eÿhjcl rccapjmzmals jms hle`nhnlecs ce mt + dt + h 1 6t + 6t + 6 1 6 4
4
D E (P ) 1
Jm `nacesnøe `cj eÿhjcl cs? @na E (P ) 1 6 P
Vmrm cj rmeil prnacrl hmjhujmals @c jm bmjjmr gøraujm `c trmesglramhnøe jnecmj? M
4 ∔= 6 4 4 4 M 1 4 6 6 MP 1 ∔= 6 4 4 4 ∔= 6 6 ∔= Cshmjleme`l MP ? ∔= 6 4
M 1 6 P
D N 1 ( ∔=,6,4 ) ; ( 6,4,7 ); ( 6,6 ,6,, ∔=) (P ) 6 ∔= 4
6
7
Jm `nacesnøe `c jm namice l rmeil scrí?
@naN
(P )
1
P
( M )
@na N (P ) 1 8
81 6+8 81 8 h) Cj tclrcam `c jm `nacesnøe ne`nhm quc? @naV 4 1 @na E(P ) + @naN (P ) , cetlehcs sc hlaprucdm cj tclrcam `c `nacesnøe.
VTLDJCAM 4 @m`m jm trmesglramhnøe jnecmj P ? T 8 ↔ T 8 , `le`c P ( x, y, z ) 1 ( x + 4 y + 8 z, x + y + z, 4 x ∔ 4 y ) . m) Bmjjmr M jm rcprcscetmhnøe amtrnhnmj ce dmsc hmeøenhm. d) Bmjjmr jls mutlvmjlrcs y mutlvchtlrcs. h) Cvmjumr c M .
Xljuhnøe? m) Jjcvme`l jm gøraujm `c trmesglramhnøe jnecmj m su glram amtrnhnmj? x = 4 8 x P y 1 = = = y , cetlehcs jm rcprcscetmhnøe amtrnhnmj ce dmsc hmeøenhm scrí? z 4 ∔4 6 z = 4 8 M 1 = = = 4 ∔4 6
d) Híjhuj Hí jhujl l ` c mutlvmjlrcs M ∔ N 1 6 ? = =
4 =
8 = ∔ 6
4 ∔4 6 6
6
6 = ∔ 4 6 16 = = ∔
6
4
∔4
8 = 16
∔
4 = ∔ (= ∔ ) g + g
∔4 g 4 + g 8
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=8
OLXSC VM^C HBNVMEM
6 4 ∔ (= ∔ )
= 6
4
4 +
=∔
=
∔0 + 4
∔4 ∔
(
OLXC VM^C HBNVMEM
(
4
)
1 6 4 ∔ (= ∔ ) ( ∔4 ∔ ) ∔ ( ∔0 + 4 ) ( 4 + ) 1 6
)
4
( 4 + ) 4 ∔ (= ∔ ) ∔ 0 + 4 1 6 ( 4 + ) ( 4 ∔ 0 + 8) 1 6 ( + 4 ) ( ∔ =) ( ∔ 8) 1 6 = 1 ∔ 4 4 1 =
8 1 8
Híjhuj Hí jhujl l ` c mutlvchtlrcs M ∔ n N R 1 ? Vmrm = 1 ∔4 ? 8 4 8 x 6 = 8 = y 1 6 rcsljvnce`l? 4 ∔4 4 z 6 8
4
86
=
8
=6
4
∔4 4 6
∔8 g 4 + g =
6
= ∔4 g 4 + g 8
6
∔9 6 6 8
6 = 66
6 =
66
∔8 g= + g 4 = 6 = 6 y 1 6, x 1 ∔ z 6 6 66 ∔ g= + g 8
= 6= 8 = 6
∔2 6 6 6 = 6 6
x ∔ z ∔= ∔= Tccapjmzme`l ce y 6 1 z 6 x= 1 6 z z = = x=
Vmrm 4 1 = ? 6 4 8 x 6 = 6 = y 1 6 rcsljvnce`l? 4 ∔4 ∔= z 6 6
4
8 6
6
4
8 6
=
6
= 6
=
6
= 6
4
∔4 ∔= 6
∔4 g 4 + g8
6
∔4 ∔8 6
6
4
86
= 6 = 6 g= + g 8
6
6
66
8 4
y 1 ∔ z, x 1 ∔ z
∔ z x 4 4 z 8 4 Tccapjmzme`l ce y ∔ 4 z 1 ∔ 4 8 x 1 8 z ∔4 ∔4 z x 4
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=0
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Vmrm 8 1 8 ? ∔4 4 8 x 6 = ∔4 = y 1 6 rcsljvnce`l? 4 ∔4 ∔8 z 6 ∔4 = 4
4
∔4
4
(= / 4) g= + g 4
6
∔= 3 / 4 6
g= + g 8
6
6
8 6
∔4 = 6 ∔4 ∔8 6
8 6
4 g 4 + g =
∔4 6
6 6
6
6
2 6
∔= 3 / 4 6 x 1 0 z, y 1 3 z 6
6 6
4
0 z x 2 2 z 3 Tccapjmzme`l ce y z 1 3 x 8 1 3 4 4 z 4 4 z x 8
Vlr tmetl, jls mutlvchtlrcs scríe jls sniuncetcs?
∔= 4 2 = 4 8 x 1 6 ; x 1 8 ; x 1 3 = ∔4 4
h) Vmrm cvmjumr c M prnacrl hmjhujmals jm amtrnz `c pmsl. Híjhujl `c jm amtrnz `c pmsl?
∔= 4 2 V 1 x= x 4 x 8 V 1 6 8 3 = ∔4 4 Cetlehcs jm amtrnz nevcrsm `c V scrí?
V ∔= 1
= ∔=7 46 =0 ∔3 = =6 6 ∔3 86 8 6 8
Jm amtrnz `nmilemj cstmrí `m` plr?
= 6 @ 1 6 4 6 6
g ( = ) 6 g (@) 1 6 8 6 6
g ( 4 ) 6 6 g ( 8 ) 6
6
c ∔4t 6 6 c∔4 ∔4 6 6 @t t t Ce eucstrl hmsl @ 1 6 = 6 c 1 6 c 6 t = c 6 6 6 c8t 1 1 6 6 6 8
6
6
c 68 6 c
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=3
OLXSC VM^C HBNVMEM
Mt
Vmrm c
OLXC VM^C HBNVMEM
g ( M) 1 V g ( @ ) V ∔= c Mt 1 V c @t V ∔= hle t 1 = c M 1 V c @ V ∔=
∔4 ∔= 4 2 c c Mt 1 6 8 3 6 = ∔4 4 6
6
∔=7 46 =0 = c 6 ∔3 =6 ∔3 86 6 c8 6 8 8
6
Aujtnpjnhme`l?
40c8 ∔ =6c + =7 =7c ∔4 = M =3c8 ∔ =3c c 1 86 ∔4 8 7c + =6c ∔ =7c
46c ∔ 46 4 6c ∔4
40c8 ∔ =6c ∔ =0c ∔4
86c
=3c8 ∔ =3c
46c ∔4 ∔46c + 46
7c8 + =6 c + =0c ∔4
VTLDJCAM 8 @m`m jm trmesglramhnøe P ? T 8 ↔ T , `le`c P ( x, y, z ) 1 x + 4 y + 8z . m) Bmjjmr jm rcprcscetmhn rcprcscetmhnøe øe amtrnhnmj `c P rcspchtl rcspchtl `c jms dmscs D 1 (=,6,6 ), (=,=,6 ), (=,=,=) y
1 = . H 1 d) Bmjjmr eÿhjcl, rm rmeil eil y vcrngnhmr vcrngnhmr cj tclrcam tclrcam `c jm `nacesnøe.
Xljuhnøe? m)
H
Jm amtrnz Amt D (P ) cstí `m`m plr?
Amt D (P ) 1 H
Cetlehcs?
P (=, 6, 6 ) 1 = 1 (=) 1 = P (=,=, 6 ) 1 8 1 (=) 1 8
P (=,=,=) 1 7 1 (=) 1 7 H
Icecrmals Amt D (P ) ? H
Amt D (P ) 1 = 8 7
d) Jjcvme`l jm glraujm `c trmesglramhnøe m su glram amtrnhnmj ldtcecals? x x P y 1 = 4 8 y z z
( )
Vmrm cj eÿhjcl? P x 1 6
E(P ) 1 ( x, y, z ) T 8 / P ( x, y, z ) 1
x = 4 8 y 1 = 4 8 1 6 x + 4 y + 8z 1 6 z
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=7
OLXSC VM^C HBNVMEM
Vlr tmetl, cj eÿhjcl scrí? E(P ) 1
( x, y, z ) T
8
OLXC VM^C HBNVMEM
/ x + 4 y + 8z 1 6
Vmrm icecrmr jms dmscs `cj eÿhjcl rccapjmzmals jm hle`nhnøe ce ,=,6 ) , ( ∔8,6,= ,6,=) ( x, y, z ) ( ∔4 y ∔ 8z, y, z ) y ( ∔4,=, 6 ) + z ( ∔8, 6 ,=) D E (P ) 1 ( ∔4,=,6
Jm `nacesnøe `cj eÿhjcl cs? @na E (P ) 1 4 P
Vmrm bmjjmr cj rmeil prnacrl hmjhujmals M @c jm gøraujm `c trmesglramhnøe jnecmj?
= M 1 = 4 8 MP 1 4 8
= ∔4 g= + g 4 MP 1 6 D N (P ) 1 = ∔8 g= + g 8 6
Jm `nacesnøe `c jm namice l rmeil scrí?
@naN (P ) 1 ( MP ) @naN (P ) 1 =
Cj tclrcam `c jm `nacesnøe ne`nhm quc? @na T 8 1 @na E (P ) + @na N (P ) 8 1 4 += 8 1 8, cetlehcs sc hlaprucdm cj tclrcam `c `nacesnøe.
VTLDJCAM 0= = Xn P ? V ↔ V , sn cj eÿhjcl cstí hlapucstl plr aÿjtnpjls `c ( t + + 4 ) y cj rmeil cstm icecrm`l plr + = , bmjjmr jm rcprcscetmhnøe amtrnhnmj `c P rcspchtl `c jm dmsc D 1 t + =, t . 4t +
Xljuhnøe? Vmrm bmjjmr jm rcprcscetmhnøe amtrnhnmj `cdcals ldtcecr jm gøraujm `c trmesglramhnøe. Cj ceuehnm`l `cj prldjcam ne`nhm quc? ‚cj eÿhjcl cstí hlapucstl plr aÿjtnpjls `c ( t + + 4 ) ‑, pmrm quc huapjm cstm hle`nhnøe jm gøraujm `c trmesglramhnøe `cdcrí scr jm sniuncetc? d P ( mt + d ) 1 ( 4 m ∔ d ) t + m ∔
4
Hlaprldmals hle ( t + + 4 ) y sus aÿjtnpjls?
4 ( t + 4 ) P ( t + 4 ) 1 ( 4 = ∔ 4 ) t + = ∔ 4 1 6 0 ( 4t + 0 ) P ( 4t + 0 ) 1 ( 4 4 ∔ 0 ) t + 4 ∔ 1 6 4 7 ( 8t + 7 ) P ( 8t + 7 ) 1 ( 4 8 ∔ 7 ) t + 8 ∔ 1 6 4
Xc puc`c ldscrvmr quc jm gøraujm sn huapjc hle jm hle`nhnøe pmrm icecrmr cj eÿhjcl. Vlr ltrm pmrtc, cj ceuehnm`l ne`nhm quc cj rmeil cstm icecrm`l plr jm dmsc 4t + + = cetlehcs hlaprldmals cstl? Jjcvmals jm gøraujm `c trmesglramhnøe m su glram amtrnhnmj? m 4 ∔= m ∔= 4 P d 1 = ∔= / 4 d M 1 = ∔= / 4
M
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
OLXSC VM^C HBNVMEM
∔= = 4 MP 1 = ∔= / 4 ∔= ∔= / 4
4
M 1
=9
OLXC VM^C HBNVMEM
4 = MP 1 4 g 4 ∔4 ∔=
4 = MP 1 D 1 4t + = g= + g 4 6 6
Xc puc`c ldscrvmr quc sn huapjc. Jm rcprcscetmhnøe amtrnhnmj `c P rcspchtl `c jm dmsc D 1 t + =, t cstí `m`m plr?
= Amt D (P ) 1 4
=
4
Mblrm hmjhujmals jms hlestmetcs? hlestmetcs?
= 1 = ( t + =) + 4 ( t ) 4 P ( t ) 1 4t + = 1 = ( t + =) + 4 ( t ) P ( t + =) 1 t +
Icecrmals cj sniuncetc snstcam? = = = = = 4 = 6 1 = / 4 = 4 4 Tcsljvnce`l? = = = / 4 =
4
1 4 = / 4 =
^ jm rcspucstm scrí?
= / 4 = = / 4 =
Amt D (P ) 1
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=2
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
CRMACE? N-46=2 VTLDJCAM = Vmrm uem trmesglramhnøe jnecmj P ? T 8 ↔ T 8 `cgnen`m hlal jm prlychhnøe `c ue vchtlr sldrc cj pjmel x ∔ 4 y + 8 z 1 6 . Xc pn`c bmjjmr? m) jm gøraujm `c trmesglramhnøe jnecmj, d) dmscs pmrm cj eÿhjcl, namice y vcrngnhmr cj tclrcam `c jm `nacesnøe, h) jm amtrnz `c jm trmesglramhnøe jnecmj hle rcspchtl m jm dmsc D 1 =,=,= , =,=,6 , =,6,6
)( ) ( ) ( Xljuhnøe? m) Jm gøraujm `c trmesglramhnøe cstmrí `m`m plr uem prlychhnøe lrtlilemj, snce`l jm ansam? P ( x, y, z ) 1 Vr ly D ' v 1 v, u '= u '= + v, u '4 u '4 `le`c? v 1 ( x, y , z ) cs cj vchtlr quc sc prlychtmrm sldrc cj pjmel.
D ' 1 u '= , u '4 cs uem dmsc lrtlelramjnzm`m `cj pjmel.
Híjhu Hí jhu jl ` c jm dmsc D `cj pjmel? @cj pjmel x ∔ 4 y + 8 z 1 6 x 1 4 y ∔ 8 z rccapjmzmals ce ( x, y, z ) 1 ( 4 y ∔ 8z , y, z ) 1 y ( 4,=, 6 ) + z ( ∔8, 6,=) , ldtcence`l jm dmsc u=
u4
D 1 u= , u4 D 1 ( 4,=, 6) , ( ∔8, 6,=) .
Híjhujl `c jm dmsc lrtlelramjnzm`m D ' `cj pjmel? Cstm dmsc scrí D ' 1 u '= , u '4 Lrtl elramjnzmals jls vchtlrcs?
u= 1
u4 1
u= u=
u= 1
( 4,=,6) 4
( 4 ) + =4 + 64
u 4 ∔ u 4 , u= u= u 4 ∔ u 4 , u= u=
u4 1
= u= 1 ( 4,=,6) 3
( ∔8, 6,=) ∔ ( ∔8, 6,=) , ( ∔8, 6,=) ∔ ( ∔8, 6,=) ,
= 3 = 3
= Jm dmsc lrtlelramjnzm`m scrí? D ' 1 u '= , u '4 D ' 1
=
( 4,=, 6 )
3 =
( 4,=, 6 )
3
3
( 4,=, 6 ) ,
(4,=, 6 ) (4,=, 6 )
= u4 1 ( ∔8,7,3 )
96
= ( ∔8, 7, 3 ) 96
Híjhuj Hí jhujl l `c jm gøraujm `c trmesglramhnøe? P ( x, y, z ) 1 Vr ly D ' v 1 v, u '= u '= + v, u '4 u '4 P ( x, y , z ) 1 ( x, y , z ) ,
= 3
( 4,=, 6 )
= 3
( 4,=, 6 ) + ( x, y, z ) ,
= 96
( ∔8, 7, 3 )
= 96
( ∔8, 7, 3 )
Tcmjnzme`l lpcrmhnlecs sc ldtncec?
=8 = 8 x + 4 y ∔ 8 z 4 x + =6 =6 y + 7 z ∔8 x + 7 y + 3 z , , = 0 = 0 =0
P ( x, y, z ) 1
d) Jjcvmals jm glraujm m jm glram amtrnhnmj?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
=>
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
x =8 4 ∔8 x =8 x + 4 y ∔ 8z 4 x + =6 y + 7 z ∔ 8x + 7 y + 3 z = , , P ( x, y , z ) 1 P y 1 4 =6 7 y =0 =0 =0 z =0 ∔8 7 3 z M
( )
Vmrm cj eÿhjcl? P x 1 6
1 ( x, y, z ) T 8 / P ( x, y, z ) 1
E (P )
x =8 4 ∔8 x 6 = P y 1 4 =6 7 y 1 6 z =0 ∔8 7 3 z 6 =8 4 ∔8 6 6 ∔78 ∔04 6 ∔=8 g 4 + g '= = 3 8 6 8 6 = 3 8 g 4 + g '8 6 4= 4= =0 6 ∔8 7 3 6 6 6
6
6
z
4
8
8
8 g 8 + g '=
6 6 66
= 3 8 6
(= / 9 ) g '8
∔ ( 3 / 8) g8 + g '4
6 8 46
= 6 ∔= / 8 6 x 1 , y 1 ∔ z 6
8
4
6
z z 4 , ∔ z , z (= , ∔4, 8) D E (P ) 1 (=, ∔4,8 ) 8 8 8
Tccapjmzme`l ce ( x, y, z )
Jm `nacesnøe `cj eÿhjcl cs? @na E (P ) 1 = P
Vmrm bmjjmr jm dmsc `c jm namice hmjhujmals M . P Hlal jm amtrnz M cs snaîtrnhm cetlehcs jm amtrnz trmespucstm scrí jm ansam M 1 M ?
=8 4 = 4 =6 P M 1 =0 ∔8 7
∔8 7 3
Vmrm jm dmsc `c jm namice dmstmrm hle cshmjlemr jm amtrnz MP . =0 g '= =8 4 ∔8 =8 4 ∔8 ∔=8 g 4 + g '=
= P 4 =6 M 1 =0 ∔8 7
3 7
(=0 / 4 ) g '4 = =0 g '8
3
∔8 7
3 8
8 g 4 + g '8
6 ∔78 ∔04 8 = 3 6 4= =0
8 g8 + g '=
(= / 9 ) g '8
6 6 6 = 3 8 D N (P ) 1 (=,3,8) ,( 6,8,4 ) 6 8 4
Jm `nacesnøe `c jm namice cs? @na N (P ) 1 4
Cj tclrcam `c jm `nacesnøe ne`nhm quc? @na T 8 1 @na E (P ) + @na N (P ) 8 1 =+ 4 8 1 8, cetlehcs sc hlaprucdm cj tclrcam `c `nacesnøe.
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
46
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
h) Jm amtrnz Amt D (P ) cstí `m`m plr? = Amt D (P ) 1 4 8
= 4 8
=
4 8
Hmjhujmals jms hlestmetcs?
D 1 (=,=,=) ,( =,=,6) ,(=,6,6 )
7 > 0 1 = (=,=,=) + 4 (=,=, 6 ) + 8 (=, 6, 6 ) 9 9 9 =3 7 8 P (=,=, 6 ) 1 , , 1 = (=,=,=) + 4 (=,=, 6 ) + 8 (=, 6, 6 ) =0 9 =0
P (=,=,=) 1 , ,
=8 = ∔8 1 = (=,=,=) + 4 (=,=, 6 ) + 8 (=, 6, 6 ) =0 9 =0
P (=, 6, 6 ) 1 , ,
Icecrmals cj sniuncetc snstcam? = = = = = = 7 / 9 =3 / =0
= = 6 4 = 6 6 8
4
4 1 > / 9
7/9
8
8
8 / =0
0 / 9
=8 / =0
=/ 9
∔8 / =0
Tcsljvnce`l? = = =
4 8
4 8
/ =0 ∔8 //= =0 0 / 9 8 /= 4 1 3 / 9 > / =0 3 /= / =0 8 ∔8 / 9 8 / =0 =0 == / =0 =0
Cetlehcs?
= Amt D (P ) 1 4 8
=
=
4
4
8
8
=0 ∔8 //= =0 0 / 9 8 //= =0 3 //= =0 4 Amt D (P ) 1 3 / 9 > //= =0 == / =0 =0 8 ∔8 / 9 8 / =0 =
VTLDJCAM 4 8 6 m @m`m jm amtrnz M 1 8 ∔= d sc pn`c? m) bmjjmr jls vmjlrcs `c jms hlestmetcs m, d y h `c glram ∔4 6 h t
quc ( 4,6, ∔=) scm ue mutlvchtlr huyl mutlvmjlr hlrrcsple`ncetc cs 1 1 ∔= , d) bmjjc jls `caís mutlvmjlrcs y mutlvchtlrcs, h) `c scr plsndjc `nmilemjnhc jm amtrnz M y sn el jl gucrm cehucetrm su glram Olr`me.
Xljuhnøe? 4 Pcecals `c `mtl? = 1 ∔ = , x= 1 6 ∔ = m) Vmrm bmjjmr jms hlestmetcs bmhcals usl `cj mutlvchtlr `c `mtl. M ∔ = N x = 1
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
4=
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
8 6 m ∔= 6 6 4 6 0 6 m 4 6 2 ∔ m 6 ∔ 6 ∔= 6 6 1 6 8 6 d 6 1 6 7 ∔ d 1 6 ∔ 8 = d ∔4 6 h 6 6 ∔= ∔= 6 ∔4 6 h + = ∔= 6 ∔3 ∔ h 6 @c jm niumj`m` `c amtrnhcs ldtcecals? m 1 2 , d 1 7 , h 1 ∔3
d) Híjhuj Hí jhujl l ` c mutlvmjlrcs M ∔ N 1 6 ? 8 6 2 Jm amtrnz M scrí? M 1 8 ∔= 7 ∔4 6 ∔3 8∔ 8 6 2 6 6 8 ∔= 7 ∔ 6 6 1 6 8 ∔4 6 ∔3 6 6 ∔4 6
=
6
4
(= / 4 )( ) ( 8 ∔ ) g 4 +
6
2
∔= ∔
7
6
∔3 ∔
g = '
16
( 8 ∔ ) ( ∔3 ∔ ) + 2
8 ∔= ∔ ∔4 6
= 1 6 ∔4 6 ∔ ( ∔= ∔ ) ( 8 ∔ ) ( ∔3 ∔ ) + 2 1 6 4
7
∔3 ∔
8 ( + =) ( ( ∔ 8) ( + 3) + =7 ) 1 6 ( + =) ( 4 + 4 + =) 1 6 ( + =) 1 6 =,4,8 1 ∔ =
Híjhuj Hí jhujl l ` c mutlvchtlrcs M ∔ n N R 1 ? 1 ∔ = ? Vmrm =,4,8
8 6 2 ∔= 6 6 x 6 0 6 2 x 6 8 ∔= 7 ∔ 6 ∔= 6 y 1 6 8 6 7 y 1 6 rcsljvnce`l? ∔4 6 ∔3 6 6 ∔= z 6 ∔4 6 ∔0 z 6 0
6
2 6
8 6 7 6 ∔4 6 ∔0 6
(= / 0) g = = (= / 8) g 4 = ∔ (= / 4) g 8 = x
6 4 6 6 46
= 6 46
∔ g= + g 4 6 6 6 6 x + 4 z 1 6 x 1 ∔4 z ∔ g= + g 8 6 6 6 6
6 ∔4 = 4 1 y z x x 1 = + 6 1 = ; 6 6 = 6 =
∔4 z
Tccapjmzme`l ce y y z z
6 46
6
∔4
x=
x4
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
44
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Hlal tcecals 8 mutlvmjlrcs `cdcals icecrmr 8 mutlvchtlrcs, pcrl sc ldscrvm quc el cs msá y pmrm icecrmr cj tcrhcr mutlvchtlr trmdmomals hle cj mutlvchtlr x= y cj snstcam?
(= / 0) g = = (= / 8) g 4 = ∔ (= / 4) g 8 =
0 6 2 6 0 6 2 x 6 8 6 7 y 1 = 8 6 7 = ∔4 6 ∔0 z 6 ∔4 6 ∔0 6
6 4 6 6 4 = / 8 6 4 6
= 6 4 6
∔ g= + g 4 6 6 6 = / 8 ∔ g= + g 8 6 6 6 6
Cj snstcam cs nehlesnstcetc, cetlehcs el sc pl`rí bmjjmr bm jjmr cj tcrhcr mutlvchtlr, tmaplhl sc pl`rí `nmilemjnzmr en jjcvmr m jm glram `c Olr`me.
VTLDJCAM 8 @c uem trmesglramhnøe jnecmj P ? V4 ↔ V4 `cgnen`m plr?
P ( mx 4 + dx + h ) 1 ( 3m + 4d + 4h ) x 4 + ( ma + 4d ∔ 4ah ) x + ( ma + de + ha ) t
Xn sc hlelhc quc ( ∔=,4,4 ) cs ue mutlvchtlr pmrm cj mutlvmjlr 1 1 ∔8 . Xcpn`c? m) bmjjmr jls vmjlrcs `c a y e, d) `nmilemjnhc jm amtrnz M , h) bmjjc jm amtrnz Me , `) bmjjc M∔= plr Bmanjtle Hmyjcy.
Xljuhnøe? m) Vrnacrl jjcvmals jm glraujm `c trmesglramhnøe m jm glram amtrnhnmj? 4 m m 3 4 P d 1 a 4 ∔4a d h a e a h ∔= Pcecals `c `mtl? = 1 ∔8 , x= 1 4 4 Vmrm bmjjmr jms hlestmetcs bmhcals usl `cj mutlvchtlr `c `mtl. M ∔ = N x = 1
4 ∔8 6 4 ∔= 6 3 4 ∔= 6 2 4 ∔ 6 ∔8 6 4 1 6 ∔3a6+ =6 1 66 ∔ 1 ∔ a 4 4 a 6 4 6 a 3 4 a a e a 6 6 ∔8 4 6 a e a + 8 4 6 a + 4e + 7 6
∔3a + =6 1 6 a 1 4 , e 1 ∔0 a + 4e + 7 1 6
@c jm niumj`m` `c amtrnhcs ldtcecals?
d) Jm amtrnz M quc sc `mnmilemjnzmrm cs? 3 4 4 M 1 4 4 ∔0 4 ∔0 4 Hlal sc puc`c ldscrvmr jm amtrnz M cs snaîtrnhm plr jl tmetl sc prlhc`crí hle jm `nmilemjnzmhnle ∔=
t
lrtlilemj? @ 1 ( V ' ) M V ' @ 1 ( V ' ) M V '
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
48
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Híjhuj Hí jhujl l ` c mutlvmjlrcs M ∔ N 1 6 ?
3∔ 3 4 4 6 6 4 4 ∔0 ∔ 6 6 1 6 4 4 ∔0 4 6 6 4
4
4
4 ∔
∔0 1 6 4 ∔
∔0
6 4 + 4 ( 3 ∔ ) 4 ∔ (=/ = / 4 ) ( 4 ∔ ) ( 3 ∔ )
6
7∔
∔7 + 4 ∔
∔0
4
∔ ( ∔ 7)
4 4 ∔ =
= 4
( ∔ 4))(( ∔ 3 ) ∔ 7
∔ (= / 4 )( ) ( 3 ∔ ) g 8 + g 4 ' ∔ g8 + g 4 '
16 4
=
∔4 ( ∔ 7 ) 4 ∔ ( ∔ 4 )( ) ( ∔ 3 ) 4
∔ ( ∔ 7)
16
∔ 7
= 1 6 ∔ ( ∔ 7) 4 ( ∔ 7) ∔ 4 ∔ ( ∔ 4 ) ( ∔ 3) 1 6 4 4
( ∔ 7 ) ( 4 ∔ 8 ∔ =2) 1 6 ( ∔ 7 ) ( + 8) 1 6 = 1 8 , =,4 1 7
Híjhuj Hí jhujl l ` c mutlvchtlrcs M ∔ n N R 1 ? Vmrm = 1 ∔8 ?
∔= x= 1 4 4 7? Vmrm 4,8 1
3 4 4 7 6 6 x 6 ∔= 4 4 x 6 y 4 4 0 6 7 6 6 4 0 0 ∔ ∔ 1 ∔ ∔ y 1 6 rcsljvnce`l? 4 ∔0 4 6 6 7 z 6 4 ∔0 ∔0 z 6 ∔= 4 4
4
4 6
∔= 4 4 6
4 g= + g 4 6 ∔0 ∔0 6 6 ∔0 ∔0 6 4 g= + g 8
6 6 6 ∔ x + 4 y + 4 z 1 6 x 1 4 y + 4 z 6
66
x 4 y + 4 z 1 Tccapjmzme`l ce y y z z
4 4 4 4 y = + z 6 x 4 1 = ; x8 1 6 6 = 6 = x 4
x8
Lrtlelramjnzmhnle `c jls mutlvchtlrcs?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
40
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Vmrm x= ?
x= 1
x= x=
x= 1
( ∔=,4,4) 4
( ∔=) + 44 + 44
= x= 1 ( ∔=,4,4 ) 8
Vmrm x4 y x8 ?
x4 1
x8 1
x 4 x 4
x4 1
( 4,=,6) 44 + =4 + 64
x 8 ∔ x 8 , x4 x4 x 8 ∔ x 8 , x4
= x4 1 ( 4,=,6) 3
6,=) ∔ ( 4, 6, 6,=) , ( 4, 6,
=
=
(4,=, 6)
3 x8 1 = x4 6,=) ∔ ( 4, 6, 6,=) , ( 4, 6, (4,=, 6)
3 =
3
3
( 4,=, 6) ( 4,=, 6)
= x8 1 ( 4, ∔0,3 )
03
Hmjhujl `c jm amtrnz `c pmsl lrtlelramjnzm`m?
∔= 4 8 3 = 4 V 1 x= x4 x8 V 1 8 3 4 6 8
03 0 t ∔ 1 V ( ) 03 3 03 4
∔=
4 8 =
8 4 3 4
∔
03
3 0 03
4 8
6
3 03
t
Vmrm `nmilemjnzmr rccapjmzmals ce @ 1 ( V ' ) M V ' ?
@ 1
∔=
4
8 4
8 =
3 4
3 0
∔
03
= =4 @ 1 3 0 3
03
∔4 7
∔
3 2 3
∔= 4 8 3 8 3 4 4 = 4 6 4 4 ∔0 8 3 4 ∔0 4 3 4 6 8 03 4
∔= 4 ∔4 8 3 = 4 6 8 3 =6 4 6 8 3
03 0 ∔ 03 3 03 4
03 ∔8 6 6 0 ∔ @ 1 6 7 6 03 6 6 7 3 03 4
h) t
t
Vmrm Me g ( M) 1 V ' g ( @ ) ( V ') Me 1 V ' @ e ( V ' )
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
43
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Tccapjmzme`l?
∔= 4 8 3 = 4 e M 1 8 3 4 6 8
e 03 ( ∔8) 6 0 7e ∔ 6 03 6 6 3 4
03 03
( ∔8)e 4 7e ∔ 8 3 e 4 ( ∔8) 7 e e M 1 8 3 e 4 ( ∔8) 6 8
03 3 3 7e 4 03 03
>e e ∔4 ( ∔8) + 4 7 >
∔
e
8 =
8
+
0 ( ∔8 ) > e
∔4 ( ∔8) + 4 7e > e
0 ( ∔8 ) + 3 7 e e
>
6
3 0
3
∔ 03
4
4
8 =
8
6
3 0
7e 3
e
∔
03
3 03
4 7e
e
>
0 ( ∔8) ∔ 0 7
8 4
+
4 ( ∔8)
2 7e
+ 3 ∔ 03
> 0 ( ∔8)
>
4
03
4 ( ∔8)
e
4
03
∔
∔
∔=
3 4
4 7e ∔= 03 8 0 7 e 4
( ∔8)e 0 7e 0 7e > + 3 + 03 4 ( ∔8)e 4 7e 2 7e e M 1 ∔ + ∔ > 3 03 e 4 8 ∔ ( ) =6 7 e ∔ + > 03 ( ∔8)e + 2 7e > ∔4 ( ∔8)e + 4 7e e M 1
6 6 7e
∔
e
=6 7e
+ 03 e 0 ( ∔8 ) 46 7e ∔ > 03 e 0 ( ∔8 ) 43 7 e + > 03
=7 7e 03
46 7 e 03
>
∔4 ( ∔8) + 4 7e > e e 0 ( ∔8 ) + 0 7 e
>
e
0 ( ∔8 ) + 3 7 e >
Xnapjngnhme`l?
( ∔8)e + 2 7e = e e e M 1 ∔4 ( ∔8 ) + 4 7 > e e ∔4 ( ∔8) + 4 7
e
∔4 ( ∔8) + 4 7e e
0 ( ∔8) + 3 7 e e
0 ( ∔8 ) ∔ 0 7 e
∔4 ( ∔8) + 4 7e e e 0 ( ∔8 ) + 0 7 e 0 ( ∔8 ) + 3 7 e e
`) Vmrm cj híjhujl `c amtrnz nevcrsm M ∔= plr Bmanjtle Hmyjcy pmrtnals `cj pljnelanl hmrmhtcrástnhl?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
47
OLXSC VM^C HBNVMEM
V ( ) 1 ( ∔ 7 )
4
OLXC VM^C HBNVMEM
( + 8) 1 ( ∔ 7 ) ( 4 ∔ 8 ∔ =2 ) 1 8 ∔ > 4 + =62 1 6 M8 ∔ > M4 + =62N 1 6 / / M∔=
M8 ∔ > M4 +=62 N 1 / / M∔= M4 ∔ > M + =62 M∔= 1 M∔= 1
= > M ∔ M4 ) ( =62
3 4 4 3 4 4 4 4 4 4 = ∔= = ∔= 4 ∔= ∔0 1 ∔ ∔ ∔ M 1 M =2 4 ∔0 ∔= =62 > 4 4 0 4 4 0 4 ∔0 4 4 ∔0 4
VTLDJCAM 0 Xcm jm trmesglramhnøe jnecmj P ? V4 ↔ T
4 R 4
`c jm humj sc hlelhce quc jm rcprcscetmhnøe amtrnhnmj
= = = = = = = 6 , = 6 , 6 6 , 6 6 quc cs jm amtrnz = =
rcspchtl m jms dmscs D 1 =,= + t ,= + t + t 4 y H 1
= = 4 ∔= 8 ∔= . Xc pn`c bmjjmr? m) jm gøraujm `c trmesglramhnøe, d) hle myu`m `c jm amtrnz M bmjjc M 1 4 6 8 ∔ ∔ 4 = 4 4 jm namice `c = + 4t + 8t , h) vcrngnquc cj tclrcam `c jm `nacesnøe.
Xljuhnøe? m) Jm glraujm `c trmesglramhnle jnecmj scrí bmjjm`m ac`nmetc? (=) ( m + dt + ht 4 ) 1 (=) + (= + t ) + (= + t + t 4 ) Mblrm hmjhujmals jls cshmjmrcs? = = = m = = = m
∔ g 4 + g=°
6 = = 1 d 6 = = d 6 6 = h 6 6 = h 1 m ∔ d, 1 d ∔ h, 1 h Tccapjmzme`l ce (=)?
= 6 6 m ∔d
6 = = d 6 6 =
∔ g 8 + g 4° 6 = 6 d ∔ h 6 6 =
h
( m + dt + ht ) 1 ( m ∔ d ) (=) + (d ∔ h ) (= + t ) + h (= + t + t ) / / P ( ) P ( m + dt + ht ) 1 ( m ∔ d ) P (=) + ( d ∔ h ) P (= + t ) + h P (= + t + t ) 4
= 6 6 m ∔d
h
4
4
4
(4)
(
)
Xc puc`c ldscrvmr quc els bmhce gmjtm jms naíicecs `c P (=) , P (= + t ) , P = + t + t 4 .Vmrm pl`cr bmjjmr `nhbms naíicecs trmdmomrcals hle jm rcprcscetmhnøe amtrnhnmj rcspchtl m jms dmscs D y H? = = 4 = = =
∔= 8 ∔= 1 4 M 1 Amt (P ) 1 4 6 8 8 ∔4 ∔= 4 0 H D
= = = = = P (=) 1 = + 4 + 8 = = = 6 6
4
4
8
8
0
0
=
= 6 + 0 1 = (=,=,=,=) + 4 (=,=,=, 6 ) + 8 (=,=, 6, 6 ) + 0 (=, 6 , 6, 6 ) 6 6 6
6 4 P (=) 1 (=) (=,=,=,=) + ( ∔=) (=,=,=, 6) + ( 4 ) (=,=, 6, 6, 6) + ( ∔4 ) (=, 6, 6, 6, 6, 6) 1 ( 6, 4, 4, 6, 6,=) P (=) 1 6 =
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
49
OLXSC VM^C HBNVMEM
= = = = = P (= + t ) 1 = + + 4 8 = = = 6 6
OLXC VM^C HBNVMEM
=
= 6 + 0 1 = (=,=,=,=) + 4 (=,=,=, 6 ) + 8 (=,=, 6, 6 ) + 0 (=, 6, 6, 6 ) 6 6 6 8 0 P (= + t ) 1 (=) (=,=,=,=) + ( 8) (=,=,=, 6) + ( 6 ) (=,=, 6, 6) + ( ∔=) (=, 6, 6, 6) 1 (8, 0, 0, 0, 0,=) P (= + t ) 1 0 = = = = = = = = 6 P (= + t + t 4 ) 1 = + + + 4 = 6 8 6 6 0 6 6 1 = (=,=,=,=) + 4 (=,=,=, 6 ) + 8 (=,=, 6, 6 ) + 0 (=, 6 , 6, 6 ) = =
P (= + t + t 4 ) 1 ( 4 ) (=,=,=,=) + ( ∔=) (=,=,=, 6 ) + ( 8) (=,=, 6, 6 ) + ( 4 ) (=, 6, 6, 6 ) 1 ( 7, 0,=, 4 ) P (= + t + t 4 ) 1 7 0 = 4 Hlal ym sc tncece jms naíicecs `cscm`ms jms rccapjmzmals ce (4)? 6 4 8 0 7 0 4 + ∔ + P ( m + dt + ht ) 1 ( m ∔ d ) d h h ( ) 0 = = 4 6 = Gnemjacetc jm glraujm `c trmesglramhnle scrí?
8d + 8h
4m + 4d
0d ∔ 8h
m+h
P ( m + dt + ht 4 ) 1
d) Vmrm hmjhujmr P (= + 4t + 8t 4 ) trmdmomals hle? 4
( (
Hmjhujmals = + 4t + 8t
4
)
D
P (= + 4t + 8t
4
))
H
(
1 M (= + 4t + 8t
quc cstm `m`l plr = + 4t + 8t 4
)
D
) D
(8)
= 1 4 . 8
= + 4t + 8t 4 1 = (=) + 4 (= + t ) + 8 (= + t + t 4 ) Mblrm hmjhujmals jls cshmjmrcs?
∔ g 4 + g =°
= = = = = = = = = 6 = = 4 1 4 6 = = 4 6 6 = 8 8 6 6 = 8 1 ∔=, 1 ∔=, 1 8 =
4
= 6 6 ∔=
6 = = 4 6 6 = 8
= 6 6 ∔=
∔ g8 + g 4° 6 = 6 ∔=
6 6 = 8
8
= ∔= 4 4 (= + 4t + 8t ) D 1 4 (= + 4t + 8t )D 1 ∔= 8 8 Tccapjmzme`l ce (8)?
(P (= + 4t + 8t )) 1 M (= + 4t + 8t ) 4
4
D
H
= = 4 ∔= 8 ∔= ∔= ∔= (P (= + 4t + 8t 4 ) ) 1 H 4 6 8 8 4 = 4 ∔ ∔
0 ∔3 (P (= + 4t + 8t
4
))
H
1 9 >
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
42
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
Vlr ujtnal usmals?
P (= + 4t + 8t
4
) 1 Amt (P (= + 4t + 8t ) ) 4
H
H
(0)
@c jm dmsc H `c `mtl?
= = = = = = = 6 = = = = = = = 6 H 1 , , , Amt H 1 = = 6 6 = = = 6 6 6 6 6 = 6 6 6
Tccapjmzme`l ce (0)?
= = 4 4 4 P (= + 4t + 8t ) 1 AmtH (P (= + 4t + 8t ) ) P (= + 4t + 8t ) 1 = H =
= = = 0
=3 = = 6 ∔3 7 1 = 6 6 9 ∔= 6 6 6 > 0
Gnemjacetc pl`cals meltmr cstm namice hlal?
=3 7 4 P (= + 4t + 8t ) 1 ∔= 0
h) Vmrm cj eÿhjcl? P ( x ) 1 6 4 4 E (P ) 1 ( m + dt + ht ) V4 / P ( mt + dt + h ) 1 Jjcvmals jm glraujm `c trmesglramhnle m su glram amtrnhnmj? 8d + 8h 6 m m 8d + 8h 4m + 4d 4m + 4d 4 4 P m + dt + ht 1 P d 1 0d ∔ 8h P d 1 6 0 8 d h m h ∔ + h m +h h =
(
)
6 m 4 P d 1 6 h = 6 8
6 4 6 0 = 6
6 8 8 6 6 8 8 6 m 4 6 d 1 4 4 6 1 6 4 4 6 6 6 0 ∔8 6 6 0 ∔8 6 0 ∔8 = 6 = 6 = 6 = 6 6 = h 8 6 ∔ ( 8 / 4 ) g 4 + g = ' 6 6 7 6 ∔7 g8 + g = ' 6 6 g8 + g 4 ' 6 = ∔4 6 6 = ∔= 6 (= / 4 ) g 4 ' 6 6 = 6 6 6 ∔8 6 ∔4 g 4 + g 8 ' ∔ g 8 + g 0 ' = 6 = 6 = 6 = 6 8
8
8
m 6 d 0 ∔8 h 6 = 4
8
∔4 g 0 + g 4 '
66
m 1 6 d 1 6 =6 h 1 6 66
66
Vmrm icecrmr jms dmscs `cj eÿhjcl rccapjmzmals jms hle`nhnlecs ce m + dt + ht 1 6m + 6t + 6t 1 6 4
4
D E (P ) 1
Jm `nacesnøe `cj eÿhjcl cs? @na E (P ) 1 6 P
Vmrm bmjjmr cj rmeil prnacrl hmjhujmals M `c jm glraujm `c trmesglramhnle?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
4>
OLXSC VM^C HBNVMEM
6 m 4 P d 1 h 6 =
8
OLXC VM^C HBNVMEM
8
m 6 4 6 = 6 d Mt 1 8 4 0 6 0 ∔8 ∔ h 8 6 8 = 6 = 4
M t
Cshmjleme`l M ?
6 4 6 = M 1 8 4 0 6 8 6 ∔8 = t
6 4 6 = ∔ g8 + g 4 ' M 1 6 4 9 ∔= 8 6 ∔8 = t
,6,=) , ( 6,6,9, 6,6,9, ∔4 ) ( 8,6, ∔8.= ) D N (P ) D N (P ) 1 ( 6, 4,6,=
6 4 6 = ∔ g= + g 4 ' M 1 6 6 9 ∔4 8 6 ∔8 = 6 4 6 6 8 6 1 , , 6 = 9 ∔4 ∔8 = t
Jm `nacesnøe `c jm namice l rmeil scrí?
@naN (P ) 1 ( Mt ) @naN (P ) 1 8
Cj tclrcam `c jm `nacesnøe ne`nhm quc? @naV4 1 @na E(P ) + @na N (P ) 8 1 6+8 8 1 8, cetlehcs quc`m hlaprldm`l cj tclrcam `c `nacesnøe.
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
86
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
CRMACE? NN-46=9 VTLDJCAM = @m`ls jls mutlvmjlrcs ∔8,8,8 y mutlvchtlrcs v , (=,=, 6 ) , ( ∔k , 6 6,, k ) `c uem amtrnz M, `le`c v hlrrcsple`c mj mutl vmjlr el rcpctn`l, sc pn`c? m) Bmjjmr cj vmjlr `c k y cj vchtlr v pmrm quc M scm snaîtrnhm. d) Bmjjc jm amtrnz M. h) Bmjjc jm amtrnz nevcrsm `c `c M plr Hmyjcy-Bmanjtle. Hmyjcy-Bmanjtle. Xljuhnøe? m) Híjhujl `cj vmjlr `c k ? Cj mutlvmjlr 1 1 8 icecrm jls mutlvchtlrcs v= 1 (=,=, 6 ) , v4 1 ( ∔k , 6, k ) , cetlehcs sc huapjc quc?
x = ∔k x = ∔= hlapmrme`l hle v4 1 ( ∔k , 6, k ) y 1 = = + = 6 y 1 = = + = ( k ) 6 v4 1 ( ∔=, 6,=) z 6 k z 6 = v=
v4
v=
v4
1 = Xc ldtncec k 1 Híjhujl `cj vmjlr `cj mutlvchtlr v ?
=, 6,=) v 1 ( m, d, h ) ; v= 1 (=,=, 6 ) , v4 1 (∔=, 6,=) Jls mutlvchlrcs scríe v , (=,=, 6 ) , ( ∔=, Vlr ltrm pmrtc pmrm quc M scm snaîtrnhm jls mutlvchtlcs `cdce cstmr lrtlelramjnzm`ls. ( m, d, h ) v v'1 v 1 ( m, d, h ) v ' 1 4 4 4 v m +d +h
v= 1 (=,=, 6 ) v= ' 1
v=
,=, 6 ) (=,=,6
1
= += + 6
v=
v4 1 ( ∔=, 6,=) v4 ' 1
4
4
=
v= ' 1
4
4
(=,=, 6 )
( ∔=, 6,=) ∔ ( ∔=, 66,,=) ,
v4 ∔ v4 , v= ' v= '
1
v4 ∔ v4 , v= ' v= '
( ∔=, 6,=) ∔ ( ∔=, 66,,=) ,
=
(=,=, 6)
=
4 =
4 =
4
4
(=,=, 6)
(=,=, 6 ) v4 ' 1
(=,=, 6 )
= 7
( ∔=,=, 4 )
Hlal M cs snaîtrnhm v ' cs lrtlilemj m v= ' y v4 ' ?
v ', v= ' 1 6 v ', v4 ' 1 6
( m, d, h )
4 4 4 m +d +h
( m , d, h )
m +d +h 4
4
4 =
,
4
v ', v ' 1 v ' v ' hls ( 6 ) 1 v '
=
,
4
7
=,=, 6 ) 1 6 m + d 1 6 d 1 ∔m (=,
(=)
( ∔=,=, 4 ) 1 6 ∔m + d + 4h 1 6
(4)
m +d +h 4
4
1
(m
4
+ d4 + h4 )
4
4
4
4
4
4
4
4
,
( m, d, h ) m +d +h 4
4
1 4
( m , d, h )
m +d +h
4
4
4
4
4
m 4 + d 4 + h 4 1 =
+ + + + m d h m (4) ce (=)? ∔m ∔ m + 4h 1d6 hh 1 m 4
m +d +h 4
= 4
( m, d, h )
(8)
(0)
(=) y (0) ce (8)? m 4 + ( ∔m ) + m 4 1 = m 1
= 8
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
8=
OLXSC VM^C HBNVMEM
=
8
Ce (=)? d 1 ∔ Ce (0)? h 1
= 8
OLXC VM^C HBNVMEM
Tccapjmzme`l ce v 1 ( m, d, h ) v 1
= 8
=
8
,∔
,
=
=
(=, ∔=,=) v 1 8 8
= Gnemjacetc v 1 (=, ∔=,=) v ' 1 (=, ∔=,=) 8
d) ∔=
t
Vmrm bmjjmr M trmdmomals hle M 1 V ' @ ( V ' ) M 1 V ' @ ( V ' ) t
Híjhujl `c V ' y ( V ' ) ?
= 8 = 1 V 1 v ' v= ' v4 ' V ∔ 8 =
8
= 4 = 4 6
∔
=
= 7 8 = t = V ( ) 1 4 7 4 = ∔ 7 7
∔
= 8 = 4 = 7
=
8
6
4 7
Híjhujl `c amtrnz `nmilemj?
∔8 6 6 Hlelhnce`l ∔8,8,8 tcecals @ 1 6 8 6 6 6 8 t
Tccapjmzme`l ce M 1 V ' @ ( V ') ?
= 8 = M 1 ∔ 8 = 8
= 4 = 4 6
=
= ∔ 7 8 ∔8 6 6 = = 6 8 6 7 6 6 8 4 4 = ∔ 7 7
=
∔ 8 = 4 = 7
= 8
6 4 7
Aujtnpjnhme`l?
= 4 ∔4 M 1 4 = 4 ∔4 4 = h) Híjhujl `c jm amtrnz nevcrsm nevcrsm M ∔= plr Bmanjtle Hmyjcy? Vmrtnals `cj pljnelanl hmrmhtcrástnhl?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
84
OLXSC VM^C HBNVMEM
V ( ) 1 ( ∔ 8 )
4
OLXC VM^C HBNVMEM
( + 8) 1 8 ∔ 8 4 ∔ > + 4 9 1 6 M8 ∔ 8 M4 ∔ > M + 49 N 1 6 / / M∔=
M4 ∔ 8 M ∔ > N + 49 M∔= 1 6 M∔= 1
=
(∔ M 49
4
+ 8 M + > N )
Tccapjmzme`l? 4
∔=
M 1
6 = = 4 ∔4 =4 4= ∔44 + > =6 6 4 = 4 8 = 6 ∔ + 49 ∔4 4 = ∔4 4 = 6 6 =
Lpcrme`l?
∔4 M 1 4 = 4 > ∔4 4 = =
=
4
VTLDJCAM 4 8
8
Xcm uem trmesglramhnøe jnecmj P ? T ↔ T `cgnen`m hlal jm prlychhnøe `c ue vchtlr sldrc cj pjmel x ∔ y + 8 z 1 6 . Xc pn`c bmjjmr? m) jm gøraujm `c trmesglramhnøe jnecmj, d) vcrngnhmr cj tclrcam `c jm `nacesnøe, h) jm ,=, ∔=) , (=, =, ∔=, 4 ) , (=, =, 6,= 6,=) , `) hle myu`m amtrnz `c jm trmesglramhnøe hle rcspchtl m jm dmsc D 1 (=,=, `c jm metcrnlr amtrnz jm namice `cj vchtlr ( 8,0, ∔=) .
Xljuhnøe? m) Jm gøraujm `c trmesglramhnøe cstmrí `m`m plr uem prlychhnøe lrtlilemj, snce`l jm ansam? P ( x, y, z ) 1 Vr ly D ' v 1 v, u '= u '= + v, u '4 u '4 `le`c? v 1 ( x, y , z ) cs cj vchtlr quc sc prlychtmrm sldrc cj pjmel.
D ' 1 u '= , u '4 cs uem dmsc lrtlelramjnzm`m `cj pjmel. Híjhujl `c jm dmsc D `cj pjmel? @cj pjmel x ∔ y + 8 z 1 6 x 1 y ∔ 8 z rccapjmzmals ce ( x, y , z ) 1 ( y ∔ 8z , y , z ) 1 y (=,=, 6 ) + z ( ∔8, 6,=) u=
u4
ldtcence`l jm dmsc D 1 u= , u4 D 1 (=,=, 6 ) , ( ∔8, 6,=) . Híjhujl `c jm dmsc lrtlelramjnzm`m D ' `cj pjmel? Cstm dmsc scrí D ' 1 u '= , u '4 Lrtl elramjnzmals jls vchtlrcs?
u= 1
u= u=
u= 1
∔ u4 1
,=,6 6) (=,=, =4 + =4 + 64
= ,=,6 6) u= 1 (=,=, 4
∔
∔
∔
=
6,=) ( 8, 6,=) , 4 (=,=, 6) u4 1 ( 8, 6, = u 4 ∔ u 4 , u= u= 6,=) ∔ ( ∔8, 6,=) , ( ∔8, 6, (=,=, 6) u4
u4 , u = u =
=
4
4 (=,=, 6) u 1 = ∔8,8,4 4 ( ) = 44 (=,=, 6) 4
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
88
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
=
Jm dmsc lrtlelramjnzm`m scrí? D ' 1 u '= , u '4 D ' 1
4
=
(=,=, 6 ) ,
44
( ∔8, 8, 4 )
Híjhujl `c jm gøraujm `c ` c trmesglramhnøe?
P ( x, y, z ) 1 Vr ly D ' v 1 v, u '= u '= + v, u '4 u '4
P ( x, y , z ) 1 ( x, y , z ) ,
= 4
(=,=, 6 )
= 4
(=,=, 6 ) + ( x, y, z ) ,
= 44
( ∔8, 8, 4 )
= 44
( ∔8, 8, 4 )
Tcmjnzme`l lpcrmhnlecs sc ldtncec?
=6 x + y ∔ 8 z x + =6 y + 8 z ∔8 x + 8 y + 4 z , , P ( x, y, z ) 1 == == ==
d) Jjcvmals jm gøraujm m jm glram amtrnhnmj? x =6 = x y z x y z x y z = 6 8 = 6 8 8 8 4 = + ∔ + + ∔ + + P ( x, y , z ) 1 , , P y 1 = =6 == == == z == ∔8 8
∔8 x 8 y 4 z
M
( )
Vmrm cj eÿhjcl? P x 1 6
E(P ) 1 ( x, y, z ) T 8 / P ( x, y, z ) 1 x =6 = ∔8 x 6 = = =6 8 y 1 6 P y 1 z == ∔8 8 4 z 6
∔8 6 ∔=6 g 4 + g '=
=6
=
=
=6
8 6
∔8
8
4 6
6 6
6
6
∔>> ∔88 6
=
=6
8 6
8 g 4 + g '8
6
88 88
== 6
z
z
8
8
6
8 g8 + g '=
6
6
66
= =6 8 6
(= / ==) g '8
6
8
∔ (=6 / 8) g8 + g '4
=6
= 6 ∔= / 8 6 x 1 , y 1 ∔ 6
8
=
6
z z z , ∔ , z (= , ∔=, 8) D E (P ) 1 (=, ∔=, 8) 8 8 8
Tccapjmzme`l ce ( x, y , z )
Jm `nacesnøe `cj eÿhjcl cs? @na E (P ) 1 = Vmrm bmjjmr jm dmsc `c jm namice hmjhujmals MP . P Hlal jm amtrnz M cs snaîtrnhm cetlehcs jm amtrnz trmespucstm scrí jm ansam M 1 M ?
=6 = ∔8 = = =6 8 P M 1 == ∔8 8 4 Vmrm jm dmsc `c jm namice dmstmrm hle cshmjlemr jm amtrnz MP .
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
80
OLXSC VM^C HBNVMEM
OLXC VM^C HBNVMEM
∔8 == g '= =6 = ∔8 P 8 = = g '4 M 1 = = 6 8 4 == g '8 ∔8 8 4 6 6 6 P ,8,=) Cshmjleme`l M 1 8 6 ∔= D N (P ) 1 ( 8,6, ∔=) , ( 6,8,= 6 8 = =6 = = = =6 P M 1 == ∔8 8
Jm `nacesnøe `c jm namice cs? @na N (P ) 1 4
Cj tclrcam `c jm `nacesnøe ne`nhm quc? @na T 8 1 @na E (P ) + @na N (P ) 8 1 =+ 4 8 1 8, cetlehcs sc hlaprucdm cj tclrcam `c `nacesnøe.
h) Jm amtrnz Amt D (P ) cstí `m`m plr?
= D Amt D (P ) 1 4 8
=
=
4
4
8
8
Hmjhujmals D =,= ,=,, =jls , =, =cshmjmrcs? , =, 4 , =, =, 6,= 6,=
1 (
∔ ) ( ∔
)
)(
=0 2 ∔4 , , 1 = (=,=, ∔=) + 4 (=, ∔=, 4 ) + 8 (=, 6,=) == == == 8 ∔8 ∔4 P (=, ∔=, 4 ) 1 , , 1 = (=,=, ∔=) + 4 (=, ∔=, 4 ) + 8 (=, 6,=) == == == 9 0 ∔= P (=, 6,=) 1 , , 1 = (=,=, ∔=) + 4 (=, ∔=, 4 ) + 8 (=, 6,=) == == == P (=,=, ∔=) 1
Icecrmals cj sniuncetc snstcam? = = = = = = =0 / ==
=
∔= 6 4 ∔= 4 = 8
4
4 1
8
8
8 / ==
9 / ==
2 / == ∔8 / == 0 / == ∔4 / == ∔4 / == ∔= / ==
Tcsljvnce`l? = = =
4 8
4 8
/ == 2 //= == 0 / == 2 /= 4 1 6 = 6 8 7 / == == ∔=7 / == == 8 / == ==
Cetlehcs?
= D Amt D (P ) 1 4 8
= 4 8
2 = D 4 Amt D (P ) 1 6 == 7 8 =
2 ==
∔=7
0
6 8
`) Vmrm hmjhujmr P ( 8 ,0, ∔=) trmdmomals hle?
MJICDTM JNECMJ ^ PCLTNM PCLTNM AMPTNHNMJ - HL@CR
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