1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
1W II K200 60 60 74 75 89 90 email 3 30 3 10+8+7=25 A4 1-453 email (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA email appointment Cafe David K2-1W04-00
1W II K201-1 Landau Talor : October 6, 2004 Version : 1.1 1. f(x) =2x + x 5 + 125x 20 x 0.1 f(x) (1) x =1, 0.1, 0.01 2x, x 5, 125x 20 (2) x =0.1 F (x) =2x f(x) x =0.01 2. x 0 x =0.000000000000000000001 x 10,x, x, x 1 3,x 2,x 2,x 1 2,x π x 2, 100 Landau f(x) x x 5 f(x) =2x + O(x 5 ) x 20 f(x) =2x + x 5 + O(x 20 ) Landau x =0 e(x) α, M 0 e(x) Mx α M x α e(x) M x α e(x) =O(x α ) e(x) x α K2-1W04-01 : C
1W II K201-2 3. e(x) =O(x α )(α>0) x 0 e(x) 0 Talor x =0 f(x) x f(x) =f(0) + f (0)x + + f (n) (0) x n + O(x n+1 ) n! 4. = f(x)+o(x 3 ) (1) =sinx (2) = e x sin x (3) =1 cos x (4) = 1 x 2 Hint: (1 + x) α =1+αx + α(α 1) 2 x 2 + O(x 3 ). 5. Landau tan x sin x (1) x 3 (2) (1 cos x) 1 3 x 2 3 1. f(x) =4 x + x + x 2 (1) x =1, 0.01, 0.0001 4 x, x, x 2 (2) x =0.01 F (x) =4 x f(x) x =0.00001 2. (1) e 1 (x) =O(x), e 2 (x) =O(x 2 ) e 1 (x)+e 2 (x) =O(x 2 ). (2) e 1 (x) =O(x), e 2 (x) =O(x) e 1 (x)+e 2 (x) =O(x). (3) e 1 (x) =O(x 3 ) e 1 (x) =O(x). (4) e 1 (x) =O(x), e 2 (x) =O(x) e 1 (x) e 2 (x) =O(x 2 ). (5) e 1 (x) =O(x 2 ), e 2 (x) =O(x 1/2 ) e 1 (x)e 2 (x) =O(x 5/2 ). 3. Landau (1) ex sin x (x + x 2 ) x 3 (2) tan x x x 3 K2-1W04-01 : C
1W II K202-1 1 2 Talor : October 13, 2004 Version : 1.2 (Cafe David) ( ) David Hilbert Talor Talor x = a = f(x) x a f(x) =f(a)+f (a)(x a)+ + f (n) (a) (x a) n + O((x a) n+1 ) n! 1. (1) X = x a Y = f(a) Y = F (X) (2) f 1 n x = a x F 1 n X =0 X (3) Y = F (X) G(X) =f (a)x + + f (n) (a) X n n! G F 0 n X ( ) 1 Y = cx Y = cx + c X 2 1 2 K2-1W04-02 : C
1W II K202-2 2. f(x) = 1 x 2 ( 1 x 1) (1) x =0 2 Talor f(x) = + O(x 3 ) (2) (1 + X) α =1+αX + O(X 2 ) x =0 2 Talor (3) (1) (2) ( ) 2 Talor z = f(x, ) (x, ) =(a, b) Talor f(x, ) =f(a, b)+f x (a, b)(x a)+f (a, b)( b) + 1 2!( fxx (a, b)(x a) 2 +2f x (a, b)(x a)( b)+f (a, b)( b) 2) + 3. (1) X = x a Y = b, Z = z f(a, b) Z = F (X, Y ) (2) f (x, ) =(a, b) x, n F (X, Y )=(0, 0) X, Y n (3) G(X) =F X (0, 0)X + F Y (0, 0)Y + 1 ( FXX (0, 0)X 2 +2F XY (0, 0)XY + F YY (0, 0)Y 2) 2! G F 0 2 X, Y 4. sin t = t t 3 /6+O(t 5 )(t 0) Talor (1) f(x, ) =sin(2x + ) (x, ) =(0, 0) 2 Talor (2) g(x, ) =sin(x 2 + 2 ) (x, ) =(0, 0) 2 Talor (3) xz- 1km z = f(x, ), z= g(x, ) (x,, z) =(0, 0, 0) K2-1W04-02 : C
1W II K202-3 1. f(x) =35x +2x 2 + x 4 (1) x =1, 10, 100 35x, 2x 2,x 4 (2) x =10 F (x) =x 4 f(x) x = 100 10 9.967 (10 9.967)/10 = 0.0033, 0.33 x f(x)/f (x) 1 2. x>0 x = 1000000000000000 x 10,x, 1 x,,x 2,x 2,x 1 2,x π x 2, x 100, 1 x, 1 x 2 3. (1 + t) α =1+αt + O(t 2 )(t 0) Talor (1) f(x, ) = 1 (1+x 2 + 2 ) 2 (x, ) =(0, 0) 2 Talor (2) g(x, ) = 1+x + 2 (x, ) =(0, 0) 2 Talor (3) xz- 1km z = f(x, ), z= g(x, ) (x,, z) =(0, 0, 1) A f(x) x = c 2 Talor f(x) =a 0 + a 1 (x c)+a 2 (x c) 2 + e(x) e(x) =O((x c) 3 ) B f(x) x = c 2 f(x) =b 0 + b 1 (x c)+b 2 (x c) 2 + E(x) E(x) =O((x c) 3 ) a i = b i i =0, 1, 2 es Hint E(x) e(x) =O((x c) 3 ) K2-1W04-02 : C
1W II K203-1 2 : October 20, 2004 Version : 1.2 10/27 30 11/10 1 = f(x) x = p f (p) =0 f (p) < 0 f x = p f (p) =0 f (p) > 0 f x = p Talor f(x) =f(p)+f (p)(x p) 2 /2! + = C + a(x p) 2 1. f(x) =x 3 (x 1) f(x) x =0, 1 Talor f (p) =f (p) =0 2 2 1 2 z = f(x, ) (x, ) =(p, q) Talor f(x, ) =f(p, q)+a(x p)+b( q) + 1 2!( a(x p) 2 +2b(x p)( q)+c( q) 2) + A = f x (p, q), B = f (p, q), a = f xx (p, q), b = f x (p, q), c = f (p, q) A = B =0 f(x, ) z = f(p, q) 1 K2-1W04-03 : C
1W II K203-2 f(x, ) A = B =0( f x (p, q) =f (p, q) =0) f(x, ) =f(p, q)+ 1 2!( a(x p) 2 +2b(x p)( q)+c( q) 2) + ac b 2 > 0 a<0 f (x, ) =(p, q) ac b 2 > 0 a>0 f (x, ) =(p, q) ac b 2 < 0 ac b 2 =0 ac b 2 ( ) a b = b c ( fxx (p, q) ) f x (p, q) f x (p, q) f (p, q) 3 2. A = B =0 (1) X = x p Y = q, Z = z f(p, q) Z = F (X, Y ) (2) 2 G(X, Y )=ax 2 +2bXY + cy 2 (a 0) (X, Y ) (0, 0) G(X, Y ) > 0 a, b, c G(X, Y ) < 0 ac b 2 0 (1) z = x (2) z = x 2 (3) z = 2 + x 4 3. (0, 0) (1) z = f(x, ) = x (2) z = f(x, ) =x 2 x + 2 (3) z = f(x, ) = 1+x 2 + 2 (4) z = f(x, ) =cos(x + 2 ) K2-1W04-03 : C
1W II K203-3 4. z = f(x, ) =x 2 + x + 2 4x 2 +1 1. z = f(x, ) =x 3 3 3x +12 2 x + + z =1 x + z + zx 2 2 G(X, Y )=ax 2 +2bXY + cy 2 a 0 (X, Y ) (0, 0) G(X, Y ) > 0 a, b, c G(X, Y ) < 0 G(X, Y ) 3 f(x, ) =sin(x 2 + 2 ) (x, ) =(0, 0) f(x, ) 0.5 x, 4 K2-1W04-03 : C
1W II K204-5 : November 10, 2004 Version : 1.2 1 (1) A B 1 0 0 a b c a b c BA = 0 1 0 d e f = d e f 0 0 3 g h i 3g 3h 3i 3 3 (2) A B a b c 1 0 0 a b 3c AB = d e f 0 1 0 = d e 3f g h i 0 0 3 g h 3i 3 3 I A 2 100 1 0 0 0 100 0 0 0 1 1 2 2 0 0 0 1 0 0 0 1 ( )A( ) (3) B (x,, z) x 1 0 0 x x G : 0 1 0 = z 0 0 3 z 3z B z 3 Q 3 vol G(Q) =3 vol Q =3 (4) det B =3 vol G(Q) = det B vol Q K2-1W04-04
1W II K204-6 2 (1) A B 0 1 0 a b c d e f BA = 1 0 0 d e f = a b c 0 0 1 g h i g h i 1 2 (2) A B a b c 0 1 0 b a c AB = d e f 1 0 0 = e d f g h i 0 0 1 h g i 1 2 II 2 II (3) (x,, z) x 0 1 0 x G : 1 0 0 = x z 0 0 1 z z B x = x vol G(Q) =volq =1 (4) det B = 1 vol G(Q) = det B vol Q ( 1) II det BA = det B det A = det A 3 (1) A B 1 0 2 a b c a +2g b+2h c+2i BA = 0 1 0 d e f = d e f 0 0 1 g h i g h i 1 3 2 K2-1W04-04
1W II K204-7 (2) A B a b c 1 0 2 a b c+2a AB = d e f 0 1 0 = d e f +2d g h i 0 0 1 g h i+2g 3 1 2 III A 2 1 10 1 10 0 0 1 0 0 0 1 2 1 ( 1 ) 2 1 0 0 1/2 1 0 0 0 1 (3) (x,, z) B (1) x z 2 (2) z x 1 0 2 x G : 0 1 0 = z 0 0 1 z x +2z z Q vol G(Q) =volq =1 x z =0 0 x, 1 Q z B Q 0 1 0 0 1 1 0 1 0, 0, 1, 0, 1, 0, 1, 1 0 0 0 1 0 1 1 1 K2-1W04-04
1W II K204-8 z z 1 1 x 1 1 2 3 x 1: 0 1 0 2 1 3 2 3 0, 0, 1, 0, 1, 0, 1, 1 0 0 0 1 0 1 1 1 G(Q) 1 III G(Q) (4) det B =1 vol G(Q) = det B vol Q III det BA = det B det A =deta (1) 0 1 0 1 0 2 1 0 0 1 0 0 A = 1 0 2 = 0 1 0 = 0 1 0 = 0 1 0 = I 0 0 3 0 0 3 0 0 3 0 0 1 1 2 3 1 ( 2) 3 1/3 K2-1W04-04
1W II K204-9 (2) I, II, III 0 1 0 L 1 = 1 0 0 0 0 1 1 0 2 R 1 = 0 1 0 0 0 1 1 0 0 L 2 = 0 1 0 0 0 1/3 1 0 0 0 1 0 1 0 2 0 1 0 1 0 0 A 0 1 0 = I. 0 0 1/3 0 0 1 0 0 1 (3) 0 1 0 1 0 2 L 1 1 = 1 0 0, R1 1 = 0 1 0, 0 0 1 0 0 1 L 1 2 = 1 0 0 0 1 0 0 0 3 0 1 0 1 0 0 1 0 2 A = L 1 1 L 1 2 R1 1 = 1 0 0 0 1 0 0 1 0. 0 0 1 0 0 3 0 0 1 (4) A = L 1 1 L 1 2 R 1 1 x R1 1 x L 1 2 (R 1 1 x) L 1 1 (L 1 2 R 1 1 x)=ax Q det R1 1 =1 det L 1 2 =3 det L 1 1 =1 vol F (Q) =1 3 1 vol Q =3 det A = 3 vol F (Q) = det A vol Q K2-1W04-04
1W II K204-10 R 3 A f : x Ax R 3 A det A>0 det A<0 K2-1W04-04
1W II K204-1 : November 10, 2004 Version : 1.1 a b c A = d e f g h i det A Q = { (x,, z) R 3 :0 x,, z 1 } F : R 3 R 3 x F (x) =Ax E R 3 vol E vol F (E) = det A vol E vol F (Q) = det A vol Q (1) E (1) Jacobian 1. E Lebesgue E f(e) 1. (1)(2) (3)(4) 1 0 0 (1) B = 0 1 0 A B A 0 0 3 (2) (3) G : R 3 R 3 x Bx vol Q. K2-1W04-04 : B
1W II K204-2 (4) vol G(Q) = det B vol Q 0 1 0 2. B = 1 0 0 0 0 1 1 0 2 3. B = 0 1 0 0 0 1 3 (1) I 0 0 0 0 1 k 0 1 0 0 (2) II 0 1 0 1 0 0, 0 0 1 1 0 0 0 0 1, 0 1 0 0 0 1 0 1 0 1 0 0 (3) III 1 1 1 k 0 0 1 A I k k k A II 2 2 1 A III k k 1 K2-1W04-04 : B
1W II K204-3 I, II, III I, II, III I, II, III 1 0 2 B = 0 1 0 B 1 1 0 2 = 0 1 0 0 0 1 0 0 1 4. 0 1 0 A = 1 0 2 0 0 3 (1) A I (2) I, II, III L 1,L 2,...,L m,r 1,R 2,...,R n L 1 L 2 L m AR 1 R 2 R n = I (3) A L 1,L 2,...,L m,r 1,R 2,...,R n (4) F : R 3 R 3 x F (x) =Ax vol F (Q) = det A vol Q 4 I, II, III L 1,L 2,...,L m,r 1,R 2,...,R n L 1 L 2 L m AR 1 R 2 R n = I A = L 1 m L 1 2 L 1 1 R 1 n R 1 2 R 1 1 A I, II, III K2-1W04-04 : B
1W II K204-4 1. 1 3 0 A = 0 5 1 2 1 4 (1) A I (2) I, II, III L 1,L 2,...,L m,r 1,R 2,...,R n L 1 L 2 L m AR 1 R 2 R n = I (3) A L 1,L 2,...,L m,r 1,R 2,...,R n (4) F : R 3 R 3 x F (x) =Ax vol F (Q) = det A vol Q 4 3 I, II, III 3 3 K2-1W04-04 : B
1W II K205-1 : November 17, 2004 Version : 1.1 1. 1 2 1 3 A = 2 1 5 12 1 4 1 1 x 0 A z = 0 0 w (x,, z, w) R 4 A rank A R V W a, b W a + b W a W α R αa W 2 W V 2. W R 4 3. R 4 x x 3 (1) A z = 7 0 (2) A z = 2 7 1 w w 4. (1) W c R 4 W = W + c := { a + c R 4 : a W } K2-1W04-05 : B
1W II K205-2 U R u 1,...,u n U a U (a 1,...,a n ) a = a 1 u 1 + + a n u n =(u 1,...,u n ). u 1,...,u n V U n a 1 a n U U R n U a a 1. a n R n 5. (1) R 4 1 (2) W 1 R 5 x 1 1 2 3 4 5 x 1 0 1 3 4 9 5 x 2 0 (1) 3 9 1 5 7. = 5 4 3 2 1 x 2 0 (2) 1 2 2 1 1. = 0 0 2 6 0 2 6 0 3 1 1 3 3 0 x 5 x 5 5 R U A u 1,...,u n B v 1,...,v m m = n K2-1W04-05 : B
1W II K206-1 : November 24, 2004 Version : 1.2 2 35 42 C C O O 2 u 1, u 2 1. Q (x, ) OQ = xu 1 + u 2 2. V u 1, u 2 = {αu 1 + βu 2 : α, β R 2 } V = u 1, u 2 OQ = xu 1 + u 2 OQ = xu 1 + u 2 =(u 1, u 2 ) Q (x, ) x x C 2 C ( ) x ax 2 + bx + c 2 + dx + e + f =0 C 6 x a,b,...,f 2x 2 2x +5 2 9=0 K2-1W04-06 : B
1W II K206-2 C x 3. C V = u 1, u 2 C O O 2 v 1, v 2 4. Q (s, t) OQ = sv 1 + tv 2 5. v 1, v 2 = {αv 1 + βv 2 : α, β R 2 } V = v 1, v 2 OQ = sv 1 + tv 2 OQ = sv 1 + tv 2 =(v 1, v 2 ) Q (s, t) st C s, t 2 ( ) s t as 2 + bst + ct 2 + ds + et + f =0 C 6 st a,b,...,f s 2 + t 2 1=0 s 2 + t 2 =1 6. C V = v 1, v 2 K2-1W04-06 : B
1W II K206-3 7. ( ) v 1, v 2 Q s OQ = sv 1 + tv 2 =(v 1, v 2 ) t s, t (1) S 1 = {sv 1 + tv 2 : s =2} (2) S 2 = {sv 1 + tv 2 : s 0, t 0, s+ t 1} (3) S 3 = {sv 1 + tv 2 :2s +2t =1} (4) S 4 = {sv 1 + tv 2 : s 2 + t 2 =1} (5) S 5 = {sv 1 + tv 2 : s + t =2} 8. C ( a 2 P = c ( a (u 1, u 2 )P =(u 1, u 2 ) c ) b d V Q ( ) ( OQ = (u 1, u 2 ) =(u 1, u 2 )PP 1 x ) b =(au 1 + cu 2,bu 1 + du 2 )=(v 1, v 2 ) d x ) =(v 1, v 2 )P 1 ( ) x ( ) s OQ = (v 1, v 2 ) t ( ) ( ) s = P 1 x t K2-1W04-06 : B
1W II K206-4 n V u 1,...,u n v 1,...,v n n P (u 1,...,u n )P =(v 1,...,v n ). x V u 1,...,u n (a 1,...,a n ) v 1,...,v n (b 1,...,b n ) a 1 x =(u 1,...,u n ). =(v 1,...,v n ).. b 1 a n b n b 1 a 1. = P 1.. b n a n P = P 1 (P ) 1 = P P =( ab cd (u 1, u 2 )P =(v 1, v 2 ) ) a, b, c, d v 1 = u 1 u 2 a =1,c= 1 (1) C 2 (v 1, v 2 )=(u 1 u 2,bu 1 + du 2 ) b, d P (2) P b 0 x v 1, v 2 x (3) st u 1, u 2 st (4) x C (5) st C (6) x { xu1 + u 2 : x 2 4x +4 2 3x 3 =0 } Hint st 5 1 u 1, u 2 v 1, v 2 P K2-1W04-06 : B
1W II K207-1 : December 1, 2004 Version : 1.2 2 mm, cm, m, km L L = 2cm = 20mm (cm mm) (2 20) cm 1 =mm 2 10 = 20 10 1 10 10 n V u 1,...,u n v 1,...,v n n n =3 V x V (a 1,...,a n ) x = a 1 u 1 + + a n u n =(u 1,...,u n ). (b 1,...,b n ) a 1 a n b 1 b n x = b 1 v 1 + + b n v n =(v 1,...,v n ). x a 1 b 1 x =(u 1,...,u n ). =(v 1,...,v n ). =(,..., ). a n n P (u 1,...,u n )P =(v 1,...,v n ), P 1. b n a 1 = b 1.. a n b n K2-1W04-07 : B
1W II K207-2 1. R 3 1 0 0 1 1 1 e 1 = 0, e 2 = 1, e 3 = 0, u 1 = 0, u 2 = 1, u 3 = 1, 0 0 1 0 0 1 1 0 1 1 1 0 v 1 = 0, v 2 = 1, w 1 = 0, w 2 = 1, w 3 = 1, w 4 = 1 1 0 1 0 1 1 12 (1) R 3 9 {e 1, e 2, e 3 }, {u 1, u 2, u 3 }, {v 1, v 2 } 5 {w 1, w 2, w 3, w 4 } 1 (d 1,d 2,d 3,d 4 ) 12 9 = d 1 w 1 + d 2 w 2 + d 3 w 3 + d 4 w 4 5 (2) {e 1, e 2, e 3 }, {u 1, u 2, u 3 }, {v 1, v 2 } {w 1, w 2, w 3, w 4 } R 3 N N =2 N = 65535 V v 1,...,v N a 1,...,a N a 1 v 1 + a 2 v 2 + + a N v N = 0 a 1,...,a N a 1 = = a N =0 a 1,...,a N 2. (1) V u 1,...,u n K2-1W04-07 : B
1W II K207-3 (2) x V A u 1,...,u n (a 1,...,a n ) B u 1,...,u n (b 1,...,b n ) a i = b i i =1,...nu 1,...,u n 1 V v 1,...,v N a 1,...,a N 1 x = a 1 v 1 + a 2 v 2 + + a N v N v 1,...,v N 3. v 1,...,v N span{v 1,...,v N } (1) V u 1,...,u n V = u 1,...,u n (2) v 1,...,v N V V = v 1,...,v N V = v 1,...,v N v 1,...,v N V 4. (1) V = R 3 v 1, v 2 V = v 1, v 2 (2) V = R 3 w 1, w 2, w 3, w 4 V = w 1, w 2, w 3, w 4 K2-1W04-07 : B
1W II K207-4 5. V u 1,...,u n u 1,...,u n V = u 1,...,u n u 1,...,u n V 2 2 TA 1 x =0 =0 x =0 1 x =0 =0 x 2 + 2 =0 1 (1) x = a = b 1 (2) x = a = b 1 (3) x 0 1 Hint: (4) x 0 1 (5) x 0 0 1 (6) x 0 0 1 K2-1W04-07 : B
1W II K208-1 : December 15, 2004 Version : 1.2 R U, V f : U V a, b U f(a + b) =f(a)+f(b) a U α R f(αa) =αf(a) 1. ( ) ( ) ( ) (1) f : R 2 R 2 x x +, f : (2) f : R 2 x R, f : x + 1 (3) f : R R, f : x 2 x (4) f : R R, f : x sin x (5) f : R 2004 R, f : x 1. x 2004 x 32 (6) f : R 2 R 2, f : ( x ) ( a c )( b d x 2 2 ) 2. R 2 R f 1 (x, ( ) ) =sin(3x ( +2), ) f 2 (x, ) =sin(x +4) f : R 2 R 2 f : x f 1 (x, ) f 2 (x, ) (1) f (2) (x, ) =(0, 0) f 1 f 2 3 Talor ( ) (3) 0 = 0 2 A =( ab cd 0 ( ) ( )( ) f : R 2 R 2 F : x = x a b x Ax = c d A ( ) (4) 0 x, 0.01 x = x f(x) F (x) K2-1W04-08 : B
1W II K208-2 A f : R 2 R 2 Jacobi ( ) ( ) 3. R 2 e 1 = 1 0, e 2 = 0 1 Q(x, ) a ( ) a = xe 1 + e 2 =(e 1, e 2 ) x (1) R 2 u 1, u 2 ( ) ( ) ( ) ( ) 3 1 1 2 (e 1, e 2 ) =(u 1, u 2 ), (e 1, e 2 ) =(u 1, u 2 ) 2 2 1 1 u 1, u 2 x (2) f : R 2 R 2 ( ) x f : x = ( ) x f :(e 1, e 2 ) (e 1, e 2 ) ( )( 2 98 0 100 x ) ( )( 2 98 0 100 x ) ( ) ( s a f :(u 1, u 2 ) (u 1, u 2 ) t c )( ) b s d t B =( ab cd ) B f : R 2 R 2 u 1, u 2 K2-1W04-08 : B
1W II K208-3 z = x + i R 2 R 2 ( ) ( ) ( ) x Re z Re z 2 f : = Im z Im z 2 ( = u v ) ( ) 1 (1) u, v x, f 1 (2) f : R 2 R 2 (3) X := x 1, Y := 1, U := u, V := v 2 U, V X, Y (4) X, Y ( ) ( 0.0001 ) 2 A U X A V Y A ( (5) X, Y 0.0001 U V ) ( A X Y ) R R f : R R f : R R f f(x) =Kx K Hint: 2 f : R 2 R 2 K2-1W04-08 : B
1W II K209-1 : December 22, 2004 Version : 1.1 2 U, V f : U V f(u) Im f U f (image) Im f V Im f f rank f rank f := dim(im f). {0} dim({0}) = 0 V dim V =0 V = {0 V } 1. f : U V Im f V ( ) x 2. f : R 2 R 3 x f : x Im f x + rank f =dim(imf) 3. f : U V u 1,...,u n U U n n (1) f(u 1 ),...,f(u n ) =Imf. (2) f(u 1 ),...,f(u n ) Im f (3) f(u 1 ),...,f(u n ) V f : U V U V f : U V f(a) =0 (= 0 V (kernel) Ker f V ) a U Ker f := {a U : f(a) =0} = f 1 ({0}). Ker f U K2-1W04-09 : B
1W II K209-2 4. f : U V Ker f U ( ) x + 5. f : R 2 R 3 x f : 0 Im f R 3 0 Ker f R 2 f : U V f, V dim(im f)+dim(kerf) =dimu. 6. f : U V (1) dim U =3, dim(im f) =1, dim(ker f) =2. (2) dim U =3, dim(im f) =2, dim(ker f) =1. (3) dim U =5, dim(im f) =5, dim(ker f) =0. (4) dim U = 2004, dim(im f) =0, dim(ker f) = 2004. (5) dim U = 2005, dim(im f) = 2004, dim(ker f) =1. K2-1W04-09 : B
1W II K209-3 f : U V (1) dim U =3, dim(im f) =1, dim(ker f) =2 (2) dim U =3, dim(im f) =2, dim(ker f) =1 (3) dim U =5, dim(im f) =5, dim(ker f) =0 (4) dim U =5, dim(im f) =0, dim(ker f) =5 1 0 0 2 R 3 e 1 = 0, e 2 = 1, e 3 = 0 0 0 1 Q(x,, z) a x a = xe 1 + e 2 + ze 3 =(e 1, e 2, e 3 ) z (1) u 1 = e 1 + e 2, u 2 = e 2 + e 3, u 3 = e 3 + e 1 R 3 (2) f : R 3 R 3 x 1 1 0 x f : a = 0 2 0 = f(a) z 1 0 1 z x 1 1 0 x f : a =(e 1, e 2, e 3 ) (e 1, e 2, e 3 ) 0 2 0 = f(a) z 1 0 1 z u 1, u 2, u 3 f s s f : a =(u 1, u 2, u 3 ) t (u 1, u 2, u 3 ) B t = f(a) u u 3 B K2-1W04-09 : B
1W II K209-4 dim(ker f) =0 Ker f = {0 U } 9-1 f : U V (1) Ker f = {0 U } (2) f f(a) =f(a ) V = a = a U. (3) f 2 1 9-2 u 1,...,u n U Ker f = {0 U } f(u 1 ),...,f(u n ) Im f = f(u 1 ),...,f(u n ) f : U V Im f = f(u) =V f (surjective) b V a U f(a) =b f : U V (bijective) (isomorphism) 2 9-3 f : U V dim(im f) =dimv K2-1W04-09 : B
1W II K210-1 : Januar 12, 2005 Version : 1.1 1. f : R 2 R 2 f : ( x ) ( 2x /2 ) (1) f x 2 + 2 =1 f R 2 (2) f e 1, e 2 A ( ) ( ) 1 1 (3) u 1 = u 2 = f B 1 1 f A f B ( ) ( ) 2. F : R 2 R 2 x 3x +2 F : u 1, u 2 x +4 F u 1, u 2 B ( λ 1 ) 0 0 λ 2 ( ) ( )( ) s λ 1 0 s F : x =(u 1, u 2 ) (u 1, u 2 ) = f(x) t 0 λ 2 t (1) F e 1, e 2 A Au 1 = λ 1 u 1,Au 2 = λ 2 u 2 (2) A Au = λu u 0 λ 0 det(λe A) =0 E (3) A det(λe A) =0 (4) λ Au = λu u 1 K2-1W04-10 : B
1W II K210-2 (5) F 3. R 2 ( ) ( ) x 3x +2 1 x +4 ( ) 0 P 1 0 n P n A det(λe A) =0 λ A Au = λu u 0 λ 4. n λ 1,...,λ n u 1,...,u n P =(u 1,...,u n ) λ 1 O AP = P... O P =(u 1,...,u n ) P 1 u 1,...,u n C n λ 1 O P 1 AP =... O A λ n f e 1,...,e n A A u 1,...,u n f P 1 AP = λ 1 O... λ n O λ n K2-1W04-10 : B
1W II K210-3 ( ) ( ) f : R 2 R 2 f : x 2x 2 2x +2 (1) x 2 + 2 =1 f (2) det(λe A) =0 λ (3) λ Au = λu u 0 1 ( ) ( ) (4) f : C 2 C 2 z 2z 2w f : C 2 w 2z +2w f ( ) ( ) 0 1 1 2 2 1 1 2 0 1 (1) (2) (3) 2 3 2 (4) 6 5 2 1 0 1 0 0 0 1 2 1 1 R 2 R f 1 (x, ) =sin(3x +2), ( ) f 2 (x, ( ) =sin(x ) +4) f : R 2 R 2 f : x f 1 (x, ) f 2 (x, ) ( f : x =(e 1, e 2 ) x ) ( (e 1, e 2 ) f 1 (x, ) f 2 (x, ) ) = f(x) ( 32 14 ) u 1, u 2 ( ) ( ) s F 1 (s, t) f : x =(u 1, u 2 ) (u 1, u 2 ) = f(x) t F 2 (s, t) F 1 (s, t), F 2 (s, t) K2-1W04-10 : B
1W II K211-1 : Januar 12, 2005 Version : 1.1 = f(x) x = x(t) df dt = df dx dx dt df dx x x f(x) t t dx dt x t dx dt 2 Jacobian 1. x, u, v (x, ) =(x(u, v),(u, v)) f(x, ) ( ) ( ) ( 2. x = ( (1) x u u x v v 2 1 3 4 ) )( (f u,f v )=(f x,f ) u v ) x u u ( u x v x u v ) x v v (2) Q u 1 x P (3) P x 1 uv Q (4) Q uv 1 D x P K2-1W04-11 : B
1W II K211-2 x, u, v (x, ) =(x(u, v),(u, v)) f(x, ) ( ) (f u,f v )=(f x,f ) Jacobi (u, v) =(u(x, ),v(x, )) ( ) ( ) 1 u x u x u x v = v x v u x =1/x u uv x u x v Jacobian Jacobi u v u x u u v x v v 3. (x, ) =(r cos θ, r sin θ) ( ) (1) Jacobi x r r (2) (1) x θ θ ( r x θ x r θ ) (3) rθ r, θ R (r 0,θ 0 ) x D Area(D) (4) r, θ Jacobi Area(D) 4. D = {(x, ) R 2 :1 x 2 + 2 4} D dxd (1 + x 2 + 2 ) 2 3 K2-1W04-11 : B