1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D

Save this PDF as:
Size: px
Start display at page:

Download "1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D"

Transcription

1 1W II K200 : October 6, 2004 Version : 1.2, TA M1, TA Talor Jacobian K2-1W04-00

2 1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe David K2-1W04-00

3 1W II K201-1 Landau Talor : October 6, 2004 Version : f(x) =2x + x x 20 x 0.1 f(x) (1) x =1, 0.1, x, x 5, 125x 20 (2) x =0.1 F (x) =2x f(x) x = x 0 x = 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

4 1W II K 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 f(x) =4 x + x + x 2 (1) x =1, 0.01, x, x, x 2 (2) x =0.01 F (x) =4 x f(x) x = (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

5 1W II K 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 K2-1W04-02 : C

6 1W II K 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 ) (x, ) =(0, 0) 2 Talor (3) xz- 1km z = f(x, ), z= g(x, ) (x,, z) =(0, 0, 0) K2-1W04-02 : C

7 1W II K f(x) =35x +2x 2 + x 4 (1) x =1, 10, x, 2x 2,x 4 (2) x =10 F (x) =x 4 f(x) x = ( )/10 = , 0.33 x f(x)/f (x) 1 2. x>0 x = 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 (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

8 1W II K : October 20, 2004 Version : / /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) = 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

9 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 (4) z = f(x, ) =cos(x + 2 ) K2-1W04-03 : C

10 1W II K z = f(x, ) =x 2 + x + 2 4x z = f(x, ) =x 3 3 3x 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 ) (x, ) =(0, 0) f(x, ) 0.5 x, 4 K2-1W04-03 : C

11 1W II K204-5 : November 10, 2004 Version : (1) A B a b c a b c BA = d e f = d e f g h i 3g 3h 3i 3 3 (2) A B a b c a b 3c AB = d e f = d e 3f g h i g h 3i 3 3 I A ( )A( ) (3) B (x,, z) x x x G : = z 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

12 1W II K (1) A B a b c d e f BA = d e f = a b c g h i g h i 1 2 (2) A B a b c b a c AB = d e f = e d f g h i h g i 1 2 II 2 II (3) (x,, z) x x G : = x z 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 a b c a +2g b+2h c+2i BA = d e f = d e f g h i g h i K2-1W04-04

13 1W II K204-7 (2) A B a b c a b c+2a AB = d e f = d e f +2d g h i g h i+2g III A ( 1 ) / (3) (x,, z) B (1) x z 2 (2) z x x G : = z z x +2z z Q vol G(Q) =volq =1 x z =0 0 x, 1 Q z B Q , 0, 1, 0, 1, 0, 1, K2-1W04-04

14 1W II K204-8 z z 1 1 x x 1: , 0, 1, 0, 1, 0, 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) A = = = = = I ( 2) 3 1/3 K2-1W04-04

15 1W II K204-9 (2) I, II, III L 1 = R 1 = L 2 = / A = I / (3) L 1 1 = 1 0 0, R1 1 = 0 1 0, L 1 2 = A = L 1 1 L 1 2 R1 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

16 1W II K R 3 A f : x Ax R 3 A det A>0 det A<0 K2-1W04-04

17 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) B = A B A (2) (3) G : R 3 R 3 x Bx vol Q. K2-1W04-04 : B

18 1W II K204-2 (4) vol G(Q) = det B vol Q B = B = (1) I k (2) II , , (3) III k A I k k k A II A III k k 1 K2-1W04-04 : B

19 1W II K204-3 I, II, III I, II, III I, II, III B = B = A = (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

20 1W II K A = (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

21 1W II K205-1 : November 17, 2004 Version : A = 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 = w w 4. (1) W c R 4 W = W + c := { a + c R 4 : a W } K2-1W04-05 : B

22 1W II K205-2 U R u 1,...,u n U a U (a 1,...,a n ) a = a 1 u 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 x x 2 0 (1) = x 2 0 (2) = x 5 x 5 5 R U A u 1,...,u n B v 1,...,v m m = n K2-1W04-05 : B

23 1W II K206-1 : November 24, 2004 Version : 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 =0 K2-1W04-06 : B

24 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

25 1W II K ( ) 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

26 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 x 3 =0 } Hint st 5 1 u 1, u 2 v 1, v 2 P K2-1W04-06 : B

27 1W II K207-1 : December 1, 2004 Version : mm, cm, m, km L L = 2cm = 20mm (cm mm) (2 20) cm 1 =mm 2 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 a n u n =(u 1,...,u n ). (b 1,...,b n ) a 1 a n b 1 b n x = b 1 v 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

28 1W II K R e 1 = 0, e 2 = 1, e 3 = 0, u 1 = 0, u 2 = 1, u 3 = 1, v 1 = 0, v 2 = 1, w 1 = 0, w 2 = 1, w 3 = 1, w 4 = (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 = V v 1,...,v N a 1,...,a N a 1 v 1 + a 2 v 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

29 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 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

30 1W II K 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 =0 1 (1) x = a = b 1 (2) x = a = b 1 (3) x 0 1 Hint: (4) x 0 1 (5) x (6) x K2-1W04-07 : B

31 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

32 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 ( ) ( ) ( ) ( ) (e 1, e 2 ) =(u 1, u 2 ), (e 1, e 2 ) =(u 1, u 2 ) u 1, u 2 x (2) f : R 2 R 2 ( ) x f : x = ( ) x f :(e 1, e 2 ) (e 1, e 2 ) ( )( x ) ( )( 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

33 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 ( ) ( ) 2 A U X A V Y A ( (5) X, Y 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

34 1W II K209-1 : December 22, 2004 Version : 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

35 1W II K 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) = (5) dim U = 2005, dim(im f) = 2004, dim(ker f) =1. K2-1W04-09 : B

36 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) = R 3 e 1 = 0, e 2 = 1, e 3 = 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 x f : a = = f(a) z z x x f : a =(e 1, e 2, e 3 ) (e 1, e 2, e 3 ) = f(a) z 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

37 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 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) f : U V dim(im f) =dimv K2-1W04-09 : B

38 1W II K210-1 : Januar 12, 2005 Version : f : R 2 R 2 f : ( x ) ( 2x /2 ) (1) f x =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

39 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

40 1W II K210-3 ( ) ( ) f : R 2 R 2 f : x 2x 2 2x +2 (1) x =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 ( ) ( ) (1) (2) (3) (4) 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) ( ) 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

41 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 ) )( (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

42 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 } D dxd (1 + x ) 2 3 K2-1W04-11 : B

2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C

2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1  appointment Cafe David K2-2S04-00 : C 2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b) 2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+

2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+ R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x

More information

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i [ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk

More information

i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a = [ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =

More information

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (

II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x ( II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )

More information

数学Ⅱ演習(足助・09夏)

数学Ⅱ演習(足助・09夏) II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a 3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (

More information

f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y

f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y 017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n

More information

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) = 1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,

More information

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

6. Euler x

6. Euler x ...............................................................................3......................................... 4.4................................... 5.5......................................

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b) 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x

More information

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =

More information

A S- hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A %

A S-   hara/lectures/lectures-j.html r A = A 5 : 5 = max{ A, } A A A A B A, B A A A % A S- http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r A S- 3.4.5. 9 phone: 9-8-444, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office

More information

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 ( 1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +

More information

,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1)

,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1) ( ) 1., : ;, ;, ; =. ( ).,.,,,., 2.,.,,.,.,,., y = f(x), f ( ).,,.,.,., U R m, F : U R n, M, f : M R p M, p,, R m,,, R m. 2009 A tamaru math.sci.hiroshima-u.ac.jp 1 ,.,. 2, R 2, ( ).,. 2.1 2.1. I R. c

More information

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( ( (. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a + 6 ( ) 6 5 ( ) 4 I II III A B C ( ) ( ), 5 7 I II III A B C ( ) () x π y π sin x sin y =, cos x + cos y = () b c + b + c = + b + = b c c () 4 5 6 n ( ) ( ) ( ) n ( ) n m n + m = 555 n OAB P k m n k PO +

More information

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a 9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,

More information

( )

( ) 18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

( ) ( )

( ) ( ) 20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

More information

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6 1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67

More information

II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y

More information

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X 4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P 4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0

More information

21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........

More information

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f

, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f ,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)

More information

i 6 3 ii 3 7 8 9 3 6 iii 5 8 5 3 7 8 v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35

More information

1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1

1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1 1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +

More information

(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )

More information

2011de.dvi

2011de.dvi 211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37

More information

untitled

untitled yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/

More information

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2 θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =

More information

20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................

More information

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

More information

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

mugensho.dvi

mugensho.dvi 1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

II

II II 16 16.0 2 1 15 x α 16 x n 1 17 (x α) 2 16.1 16.1.1 2 x P (x) P (x) = 3x 3 4x + 4 369 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = 2 3 7 2 + 4 = 8 14 +

More information

40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,

40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x, 9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,

More information

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

More information

2014 S hara/lectures/lectures-j.html r 1 S phone: ,

2014 S hara/lectures/lectures-j.html r 1 S phone: , 14 S1-1+13 http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 1 S1-1+13 14.4.11. 19 phone: 9-8-4441, e-mail: hara@math.kyushu-u.ac.jp Office hours: 1 4/11 web download. I. 1. ϵ-δ 1. 3.1, 3..

More information

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ

t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ 4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ

More information

K E N Z OU

K E N Z OU K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................

More information

DVIOUT

DVIOUT A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)

More information

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th 1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2

More information

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ 1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c

More information

70 : 20 : A B (20 ) (30 ) 50 1

70 : 20 : A B (20 ) (30 ) 50 1 70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................

More information

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta 009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n

More information

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 : 9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log

More information

II 2 II

II 2 II II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................

More information

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 - M3............................................................................................ 3.3................................................... 3 6........................................... 6..........................................

More information

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y

III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +

More information

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx

More information

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B 1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト 名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim

More information

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information

( 12 ( ( ( ( Levi-Civita grad div rot ( ( = 4 : 6 3 1 1.1 f(x n f (n (x, d n f(x (1.1 dxn f (2 (x f (x 1.1 f(x = e x f (n (x = e x d dx (fg = f g + fg (1.2 d dx d 2 dx (fg = f g + 2f g + fg 2... d n n

More information

ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2

More information

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y 5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x

More information

A

A A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................

More information

I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x

I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x 11 11.1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a,

More information

dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp

dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp + P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy

More information

CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)

CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) CALCULUS II (Hiroshi SUZUKI ) 16 1 1 1.1 1.1 f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) lim f(x, y) = lim f(x, y) = lim f(x, y) = c. x a, y b

More information

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc + .1 n.1 1 A T ra A A a b c d A 2 a b a b c d c d a 2 + bc ab + bd ac + cd bc + d 2 a 2 + bc ba + d ca + d bc + d 2 A a + d b c T ra A T ra A 2 A 2 A A 2 A 2 A n A A n cos 2π sin 2π n n A k sin 2π cos 2π

More information

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,

0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (, [ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =

More information

v er.1/ c /(21)

v er.1/ c /(21) 12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,

More information

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

,2,4

,2,4 2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................

More information

1

1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

More information

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2 7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)

More information

( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a,

( z = x 3 y + y ( z = cos(x y ( 8 ( s8.7 y = xe x ( 8 ( s83.8 ( ( + xdx ( cos 3 xdx t = sin x ( 8 ( s84 ( 8 ( s85. C : y = x + 4, l : y = x + a, [ ] 8 IC. y d y dx = ( dy dx ( p = dy p y dx ( ( ( 8 ( s8. 3 A A = ( A ( A (3 A P A P AP.3 π y(x = { ( 8 ( s8 x ( π < x x ( < x π y(x π π O π x ( 8 ( s83.4 f (x, y, z grad(f ( ( ( f f f grad(f = i + j

More information

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4

ax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4 20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,

17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1, 17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ

More information