1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1

Size: px
Start display at page:

Download "1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1"

Transcription

1 1 Ricci V, V i, W f : V W f f(v = Imf W ( f : V 1 V k W 1

2 {f(v 1,, v k v i V i } W < Imf > < > f W V, V i, W f : U V L(U; V f : V 1 V r W L(V 1,, V r ; W L(V 1,, V r ; W (f + g(v 1,, v r = f(v 1,, v r + g(v 1,, v r (kf(v 1,, v r = kf(v 1,, v r U, V W : U V W W =< Im >=< (U V > X f : U V X g : W X 2

3 f = g W U V U V W = U V U V W (= U V g f X (u, v = u v, (u U, v V (2 g g 1, g 2 : W X g 1 = g 2 u U, v V g 1 (u v = f(u, v, g 2 (u v = f(u, v W u v g 1, g 2 u, v g 1 = g 2 g 3

4 U V W L(U, V ; W W U, V U, V U u, V v u v L(U, V ; R α U, β V u v(α, β = u(αv(β u(α, v(β : U V L(U, V ; R (u 1 + u 2 v(α, β = (u 1 + u 2 (αv(β = u 1 (αv(β + u 2 (αv(β = u 1 v(α, β + u 2 v(α, β u v (u U, v V L(U, V ; R W W L(U, V ; R W {e i }, {f i } U, V U, V {e i }, {f i } {e i }, {f i } u v = u(e i e i v(f j f j = u(e i v(f j e i f j e i f j W g L(U, V ; R g(e i, f j = g ij e i e j g = g ij e i e j g(α, β = g(α i e i, β j e j = α i β j g ij = g ij e i (αe j (β = g ij e i e j (α, β 4

5 L(U, V ; R W W = L(U, V ; R W f : U V X f u U, v V g : W X g(u v = f(u, v f = g g W = U V U, V u U, v V u, v u v (u 1 + u 2 v = u 1 v + u 2 v u (v 1 + v 2 = u v 1 + u v 2 (ku v = u (kv = k(u v u v U V : U V W 5

6 : U V W f : W W g : W W f =, g = U V W f g W W, W, f g, g f (W,, (W, U V (= W U V L(U, V ; R u v(α, β = α(uβ(v, α U, β V u, v U V (1, (2 ( (1, (2 6

7 U V W (= U V g f X f L(U, V : X g L(U V ; X Φ : L(U, V ; X L(u V ; X Φ L(U, V ; X = L(U V ; X X R L(U, V ; R = L(U V ; R = (U V (U V = U V U V = L(U, V ; R U V = L(U, V ; R = (U V L(U, V ; W = L(U V ; W = (U V W = U V W V 1 V 2 V k V 1,, V k W 7

8 : V 1 V k W W =< Im >=< (V 1 V k > X f : V 1 V k X g : W X f = g W W = V 1 V k W = L(V1,, V ; R k V R V V, (1 x, y V V x + y (2 x V, a R V ax x, y, z V, a, b R a(x + y = ax + ay, (a + bx = ax + bx (abx = a(bx 8

9 V (32 V Intrinsic Intrinsic Intrinsic Intrinsic V {e 1, e 2,, e n } V x x = x 1 e 1 + x 2 e x n e n {e 1, e 2,, e n } x = (x 1, x 2,, x n V e i V v = v 1 e 1 + v n e n = v i e i 9

10 i V V V e i V w = w 1 e 1 + w n e n = w i e i V {e 1,, e n } {e 1,, e n } V V V V {e 1,, e n } v V v = x i e i n {x i } v = (x 1,, x n V {f 1,, f n } {e i } {f i } f i = a j i e j 10

11 (f 1 f n = a (e 1 1 a 1 n 1 e n ( a 1 1 a 1 n A = a n 1 a n n a n 1 a n n v = x i e i = y i f i (f 1 f n y 1 y n ( (e 1 e n A (e 1 e n A y 1 y n = x 1 x n ( = y 1 y n ( = e 1 e n x 1 x n e 1 e n x 1 x n V {e 1,, e n }, {f 1,, f n } 11

12 (f 1 f n = (e 1 e n A v V v = x i e i = y i f i A y 1 y n = x 1 x n a j i yi = x j 1 2 {e i } {f i } V 3 4 {e i } {f i } (f 1 f n = (e 1 e n A, f 1 e 1 = At f n e n 12

13 V V V {e i } {e i } v V v = x i e i n {x i } v = (x 1,, x n V V {e i } {f i } V (f 1 f n = (e 1 e n A e 1 f (e 1 1 e n = (f 1 f n = E e n f n E e 1 e n = A (f 1 f n = f 1 f n (f 1 f n e 1 e n (e 1 e n A = A (f 1 f n 13

14 A f 1 f n = v V e 1 e n v = x i e i = y i f i (x 1 x n e 1 e n (x 1 x n A ( = f 1 f n (x 1 x n A = x i a i j = y j ( = y 1 y n f 1 f n y 1 y n (y 1 y n f 1 f n V {e 1,, e n }, {f 1,, f n } (f 1 f n = (e 1 e n A {e 1,, e n }, {f 1,, f n } {e 1,, e n }, {f 1,, f n } f 1 A f n = e 1 e n v V 14

15 v = x i e i = y i f i (y 1 y n = (x 1 x n A y j = a i j x i V V V {e 1,, e n }, {f 1,, f n } (f 1 f n = (e 1 e n A A B = (b i j v V v = x i e i = y i f i y 1 A y n = y 1 y n = B x 1 x n x 1 x n v V v = x i e i = y i f i (1 (y 1 y n = (x 1 x n A (2 y i = x j a j i V 15

16 V V V f : V V V {e 1,, e n } f(e i = f j i e j n 2 {f j i } {d 1,, d n } (d 1 d n = (e 1 e n A d i = a j i e j f(d i = f j i d j { f j i } {f j i } f(d i = f(a j i e j = a j i f k j e k f(d i = f j i d j = f j i ak j e k f j i ak j = aj i f k j (a j i (bj i bl k k f j i ak j bl k = aj i f k j bl k f l i = f k j aj i bl k (3 {f j i } 16

17 V V n 2 {f j i } (3 f : V V F : V V R α V, v V F (α, v = α(f(v F V {e i } {e i } f(v = f(e i, ve i well defined 30 f j i f(e i = f j i e j f(e i = F (e j, e i e j f j i = F (ej, e i f j i F (α, v = α(f(v F : V V R 17

18 f j i = F (ej, e i f j i {e i }, {d i } d i = a j i e j d i = b i j ej f j i = F (dj, d i = F (b j k ek, a l i e l = b j k al i f k l 1 V n V r + s r s {}}{{}}{ f : V V V V R r,s (r,s V {e i } {e i } n r+s f i 1i r y1 j s = f(e i 1,, e i r, e j1,, e jr f {e i } f i 1i r y1 j s 18

19 ( 2 V n V n r+s n r+s r,s {e i } n r+s {f i 1i r y1 j s } {d i } n r+s f i 1i r j1 j s d i = a j i e j f k 1k rl1 l s = b k 1 i 1 b k r i r a j 1 l1 a j s ls f i 1i r j1 j s (b j i (aj i 1 2 d i = a j i e j d i = b i j ej f k 1k rl1 l s = f(d k i,, d k r, d l1,, d ls = b k 1 i 1 b k r i r a j 1 l1 a j s ls f i 1i r j1 j s 2 1 (2, 2 V {e i } n 2+2 f i 1i 2j1 j 2 19

20 f k 1k 2l1 l 2 = b k 1 i 1 b k 2 i 2 a j 1 l1 a j 2 l2 f i 1i 2 j1 j 2 u, v V, α, β V u = u i e i, v = v i e i, α = α i e i, β = β i e i, f(α, β, u, v f(α, β, u, v = α i1 β i2 u j 1 v j 2 f i 1i 2 j1 j 2 V d i = a j i e i v i e i = v i d i v i = b i j vj α i e i = ᾱ i d i ᾱ i = a j i α j ᾱ i1 βi2 ū j1 v j 2 i f 1 i 2 j1 j 2 = (a k 1 i 1 α k1 (a k 2 i 2 β k2 (b j 1 l1 u l 1 (b j2 l 2 v l 2 f i 1i 2 j1 j 2 = α k1 β k2 u l 1 v l 2 f k 1k 2l1 l 2 20

21 r s {}}{{}}{ f : V V V V R r s {}}{{}}{ L( V,, V, V,, V ; R Vs r r s f, g V r s, k R (f + g(v = f(v + g(v (kf(v = kf(v f + g, kf V r s r s {}}{{}}{ Vs r = L( V,, V, V,, V ; R r = 2, s = 2 V 2 2 = L(V, V, V, V ; R V {e i } V {e i } e i e j e k e l V 2 2 α = α i e i, β = β i e i, u = u i e i, v = v i e i, e i e j e k e l (α, β, u, v = α(e i β(e j e k (ue l (v = α i β j u k v l 21

22 f V 2 2 e i e j e k e l f(α, β, u, v = f(α i e i, β j e j, u k e k, v l e l = f ij kl α iβ j u k v l f(e i, e j e k, e l = f ij kl = f ij kl e i e j e k e l (α, β, u, v α, β, u, v f = f ij kl e i e j e k e l f V 2 2 n4 {e i e j e k e l } V 2 2 {e i e j e k e l } f ij kl e i e j e k e l = 0 f ij kl = f(ei, e j, e k, e l = 0 {e i e j e k e l } V 2 2 dim V 2 2 = n 4 L(V, V, V, V = V V V V ( u v α β(u, v V, α, β V V V V V ( Vs r e i1 e ir e j 1 e j s f Vs r 22

23 f i 1i r j 1 j s e i1 e ir e j 1 e j s dim V r s = n r+s V {e i }, {f i } {e i }, {f i } f j = a i j e i A A = (a i j A B B = (b i j A i j a i j b i j ai j V ( x i e i = y i f j y i = b i j xj V ( x i e i = y i f i 23

24 y i = a j i x j (2,2 T V {e i }, {f i } T ij pq kl, T rs T pq rs = b p i bq j ak ra l st ij kl (b i j y i = b i j xj a j i yi = x j T rs pq = b p i bq j ak ra l st ij kl a i pa j pq q T rs = a k ra l st ij kl 24

25 (u, v = g(u, v (u, v V g : V V R, (u, v g(u, v (1 g g (2 g g(u, v = g(v, u, u, v V (3 g g(u, u 0, u V u = 0 g g g V g V {e 1, e n } {ω 1,, ω n } g g = g ij ω i ω j g ij = g(e i e j u = u i e i, v = v i e i 25

26 g(u, v = g(u i e i, v j e j = u i v j g(e i, e j = g ij u i v j = g ij ω i (uω j (v = g ij ω i ω j (u, v g {e i } n 2 {g ij } g ē i ω i ē i = a j i e j (A g = ḡ ij ω i ω j ḡ ij = g(ē i, ē j = g(a k i e k, a l j e l = a k i al j g kl (B g {e i } n 2 {g ij }, {ē i } n 2 {ḡ ij } (A g ij ḡ ij (B {e i } n 2 {g ij } {ē i } n 2 {ḡ ij } (B V g : V V R V {e 1,, e n }, {ē 1,, ē n } ē i = a j i e j u, v V u = u i e i = ū i ē i, v = v i e i = v i ē i u i = a i jūj, v i = a i j vj 26

27 g(u, v = g ij u i v j = g ij a i kūk a j l ūl = ḡ kl ū k v l g(u, v g {e i } {ω i } g = g ij ω i ω j g ij ω i ω j (u k e k, v l e l = g ij ω i (u k e k ω j (v l e l = g ij u i u j {ē i } ω i g = g ij ω i ω j = ḡ kl ω k ω l g ij ω i ω j = g ij a i k aj l ωk ω l ḡ kl = g ij a i k aj l g ij, ḡ ij {g ij } {g ij } {g ij } V {e i } n 2 {ē i } ē i = e j a j i g ij = g(e i, e j, ḡ ij = g(ē i, ē j {g ij }, {ḡ ij } {g ij }, {ḡ ij } {a i j } {bi j } a i b i i a i b i = a j b j = a p b p 27

28 ḡ kl = g ij a i k aj l b k i b k i ḡkl = b k i g sta s k at l = δs i at l g st = a j l g ij ḡ lr g si b k i ḡklḡ lr g si = b k i δr k gsi = b r i gsi a j l g ijḡ lr g si = a j l δs j ḡlr = a s l ḡlr b r i gsi = a s l ḡlr a k r g sk = a s l ak rḡ lr {g ij }, {g ij } g ij = a i k aj l ḡkl (1 ḡ ij = a k i aj l g kl (2 {g ij } (1 V G : V V R {e i } {ω i },{ē i } { ω i } ē i = a j i e j ω i = a i j ωj α, β V α = α i ω i = ᾱ i ω i, β = β i ω i = β i ω i ᾱ i = α j a j i, β i = β j a j i G(α, β = g ij α i β j = a i k aj l ḡkl α i β j = ḡ kl ᾱ k βl G 28

29 G = g ij e i e j {g ij } {g ij } {g ij } {g ij } V f : V V V {e i } f(e i = f j i e j n 2 {f j i } V {ē i } ē i = e j a j i f(ē i = f j i ēj f(ē i = f(a k i e k = a k i f j k e j f k i ēk = f k i aj k e j a k i f j k = aj k f k i (1 {f j i }, { f j i } (1 V v = v i e i = v i ē i f(v = v i f j i e j = a i k vk f j i e j = f l k aj l vk e j = f l k vk ē l (1 29

30 V (1,1 F : V V R {f j i }, { f j i } a k i f j k = aj f k i k (1 f : V V F : V V R, L(V ; V L(V, V ; R v V, α V,V {e i } {ω i } Φ : L(V ; V L(V, V ; R, f F F (α, v = α(f(v Ψ : L(V, V ; R L(V ; V, F f f(v = F (ω i, ve i Φ Ψ F (ω i, ve i = F (a i j ωj, ve i = F ( ω j, va j i e i = F ( ω j, vē j F Ψ Φ (Ψ Φ(f(v = Φ(f(ω i, ve i = ω i (f(ve i = f(v Ψ Φ Φ U, V 30

31 f : U V, u v u v Φ : L(V ; V L(V, V ; R, f F f F V V a V V f f(a R V R( a V V a V V V V V V V {e 1,, e n } {ω 1,, ω n } e i ω i f : V V, a i e i a i ω i {ē i } { ω i }, ē i = a j i e j f : ē i = a j i e j a j i ωj ω i = b i j ωj ē i ω i v V v f : V W w W f(v = w v v 1 v 2 f(v 1 f(v 2 31

32 V, W V, {e i } V v = v i e i v = (v 1,, v n V n {e i } n {v i } {ē i } n v i ē i = a j i e j v i = a i j vj V v i e i = a i j vj e i = v j ē j (, V V, V (Intrinsic f : V V 32

33 x V f(x(y = (x, y y V f(x V Intrinsic f(x = 0 x = 0 V, V f : V V V {e 1,, e n }, {ω 1,, ω n } (e i, e j = g ij f(e i = a ij ω j a ij = (a ik ω k (e j = f(e i (e j = (e i, e j = g ij f(e i = g ij ω j V e i V g ij ω j e i = g ij ω j e i, ω j e 1 g 11 g 1n ω 1 = e n g n1 g nn (g ij (g ij ω 1 g 11 g 1n e 1 = ω n g n1 g nn e i, ω i ω n e n 33

34 e i = g ij ω j, ω i = g ij e j V x i e i V x i g ij ω j V y i ω i V y i g ij e j V x i e i x i ω i V x i = g ij x j, x i = g ij x j f = f ijk ω i ω j ω k = f ijk (g ih e h ω j ω k = f hjk (g hi e i ω j ω k = f i jk e i ω j ω k f i jk = f hjkg hi V (, f = f i jk ei ω j k, h = h i jk ei ω j k (f, g (f, h = f i 1 j 1 k 1 h i 2 j2 k 2 (e i1, e i2 (ω j 1, ω j 2 (ω k 1, ω k 2 = f i 1 j 1 k 1 h i 2 j2 k 2 g i1 i 2 g j 1j 2 g k 1k 2 f 2 = (f, f = f i 1 j 1 k 1 f i 2 j 2 k 2 g i1 i 2 g j 1j 2 g k 1k 2 34 = f ijk f ijk

35 (V, g V V V {e 1,, e n } V R n, x i e i (x 1,, x n (V, g V v V α α(v V {e i } {e i } v V, α V α(v v = v i e i, α = α i e i α(v = α i e i (v j e j = α i v i n α i v i i=1 v V, α V v α v α = v i α j e i e j fj i = vi α j 35

36 v α = f i j e i e j α(v n α(v = v i α i = v α(e i, e i i=1 v α(e i, e i u, v V, α, β V u v α β V 2 2 u α α(uv β V 1 1 u = u i e i, v = v i e i, α = α i e i, β = β i e i u v α β = u i v j α k β l e i e j e k e l = f ij kl e i e j e k e l α(uv β = u i α i v j β l e j e l = f ij il e j e l (2,2 f ij kl T 2 2 (1,1 f ij il e j e l f V 2 2 g V 1 1 g(α, v = f(e i, α, e i, v 36

37 V {e i } {e i } p, q 1 p r, 1 q s (r,s f (r-1,s-1 g g(α 1, α r 1, v 1, v s 1 = f(α 1,, α p 1, e i α p,, α r 1, v 1,, v q 1, e i, e q,, v s 1 g f p q g g = contr (p,q (f f i p j q contr (ij,pq (f (i, p g = contr(f V {e i } f i 1i r j 1 j s (r,s f p q (r-1,s-1 g g i 1i p 1 i p+1 i r j 1 j q 1 i q+1 j r = f i 1i p 1 ki p+1 i r j 1 j q 1 ki q+1 j r g f p q g g = contr (p,q (f 37

38 f (2,2 u = u i e i, v = v i e i, α = α i e i, β = β i e i f(α, β, u, v = f ij kl α iβ j u k v l, (4, 4 f α β u v contr (1234,3412 f α β u v V f f : V V (1,1 F : V V R F (α, v = α(f(v V {e i } F contr(f = F (e i, e i = e i f(e i = fi i f f 38

39 ( 39

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

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

January 27, 2015

January 27, 2015 e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6

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

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

2016

2016 2016 1 G x x G d G (x) 1 ( ) G d G (x) = 2 E(G). x V (G) 2 ( ) 1.1 1: n m on-off ( 1 ) off on 1: on-off ( on ) G v v N(v) on-off G S V (G) N(v) S { 3 G v S v S G G = 1 OK ( ) G 2 3.1 u S u u u 1 G u S

More information

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev

More information

AI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id

AI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id 1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi

More information

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x

More information

2012 A, N, Z, Q, R, C

2012 A, N, Z, Q, R, C 2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)

More information

1/68 A. 電気所 ( 発電所, 変電所, 配電塔 ) における変圧器の空き容量一覧 平成 31 年 3 月 6 日現在 < 留意事項 > (1) 空容量は目安であり 系統接続の前には 接続検討のお申込みによる詳細検討が必要となります その結果 空容量が変更となる場合があります (2) 特に記載

1/68 A. 電気所 ( 発電所, 変電所, 配電塔 ) における変圧器の空き容量一覧 平成 31 年 3 月 6 日現在 < 留意事項 > (1) 空容量は目安であり 系統接続の前には 接続検討のお申込みによる詳細検討が必要となります その結果 空容量が変更となる場合があります (2) 特に記載 1/68 A. 電気所 ( 発電所, 変電所, 配電塔 ) における変圧器の空き容量一覧 平成 31 年 3 月 6 日現在 < 留意事項 > (1) 空容量は目安であり 系統接続の前には 接続検討のお申込みによる詳細検討が必要となります その結果 空容量が変更となる場合があります (2) 特に記載のない限り 熱容量を考慮した空き容量を記載しております その他の要因 ( 電圧や系統安定度など ) で連系制約が発生する場合があります

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1

u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1 5 K K Q R C 5.1 5.1.1 V V K K- 1) u, v V u + v V (a) u, v V u + v = v + u (b) u, v, w V (u + v)+w = u +(v + w) (c) u V u + o = u o V (d) u V u + u = o u V 2) α K u V u α αv V (a) α, β K u V (αβ)u = α(βv)

More information

LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University

LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y

More information

ver Web

ver Web ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

More information

II Time-stamp: <05/09/30 17:14:06 waki> ii

II Time-stamp: <05/09/30 17:14:06 waki> ii II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................

More information

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >

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

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij

More information

/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx,

/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx, 1 1.1 R n 1.1.1 3 xyz xyz 3 x, y, z R 3 := x y : x, y, z R z 1 3. n n x 1,..., x n x 1. x n x 1 x n 1 / 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point 1.1.2 R n set

More information

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s [ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =

More information

数値計算:有限要素法

数値計算:有限要素法 ( ) 1 / 61 1 2 3 4 ( ) 2 / 61 ( ) 3 / 61 P(0) P(x) u(x) P(L) f P(0) P(x) P(L) ( ) 4 / 61 L P(x) E(x) A(x) x P(x) P(x) u(x) P(x) u(x) (0 x L) ( ) 5 / 61 u(x) 0 L x ( ) 6 / 61 P(0) P(L) f d dx ( EA du dx

More information

linearal1.dvi

linearal1.dvi 19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352

More information

B ver B

B ver B B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................

More information

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}

More information

?

? 240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................

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

koji07-01.dvi

koji07-01.dvi 2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?

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

difgeo1.dvi

difgeo1.dvi 1 http://matlab0.hwe.oita-u.ac.jp/ matsuo/difgeo.pdf ver.1 8//001 1 1.1 a A. O 1 e 1 ; e ; e e 1 ; e ; e x 1 ;x ;x e 1 ; e ; e X x x x 1 ;x ;x X (x 1 ;x ;x ) 1 1 x x X e e 1 O e x x 1 x x = x 1 e 1 + x

More information

I , : ~/math/functional-analysis/functional-analysis-1.tex

I , : ~/math/functional-analysis/functional-analysis-1.tex I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................

More information

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n

. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n 003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........

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

さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a

さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a φ + 5 2 φ : φ [ ] a [ ] a : b a b b(a + b) b a 2 a 2 b(a + b). b 2 ( a b ) 2 a b + a/b X 2 X 0 a/b > 0 2 a b + 5 2 φ φ : 2 5 5 [ ] [ ] x x x : x : x x : x x : x x 2 x 2 x 0 x ± 5 2 x x φ : φ 2 : φ ( )

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

1 Euclid Euclid Euclid

1 Euclid Euclid Euclid II 2000 1 Euclid 1 1.1..................................... 1 1.2..................................... 8 1.3 Euclid............. 19 1.4 3 Euclid............................ 22 2 28 2.1 Lie Lie..................................

More information

: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i =

: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i = 1 1980 1) 1 2 3 19721960 1965 2) 1999 1 69 1980 1972: 55 1999: 179 2041999: 210 211 1999: 211 3 2003 1987 92 97 3) 1960 1965 1970 1985 1990 1995 4) 1. d ij f i e i x i v j m a ij m f ij n x i = n d ij

More information

1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1

1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1 1 1.1......... 1............. 1.3... 1.4......... 1.5.............. 1.6................ Bownian Motion.1.......... Einstein.............. 3.3 Einstein........ 3.4..... 3.5 Langevin Eq.... 3.6................

More information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information

December 28, 2018

December 28, 2018 e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................

More information

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4 [2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =

More information

+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm.....

+   1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm..... + http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm (1) n m () (n, m) ( ) n m B = ( ) 3 2 4 1 (2) 2 2 ( ) (2, 2) ( ) C = ( 46

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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................

More information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

I

I I 2008 I i 1 1 1.1.............................. 1 1.2................................. 7 1.3......................... 13 2 23 2.1......................... 23 2.2............................... 31 3 37

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

Jacobson Prime Avoidance

Jacobson Prime Avoidance 2016 2017 2 22 1 1 3 2 4 2.1 Jacobson................. 4 2.2.................... 5 3 6 3.1 Prime Avoidance....................... 7 3.2............................. 8 3.3..............................

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

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

A

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

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

16 B

16 B 16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..

More information

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d ) 23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ

More information

直交座標系の回転

直交座標系の回転 b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx

More information

量子力学 問題

量子力学 問題 3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,

More information

chap10.dvi

chap10.dvi . q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l

More information

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, 01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,

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

VI VI.21 W 1,..., W r V W 1,..., W r W W r = {v v r v i W i (1 i r)} V = W W r V W 1,..., W r V W 1,..., W r V = W 1 W

VI VI.21 W 1,..., W r V W 1,..., W r W W r = {v v r v i W i (1 i r)} V = W W r V W 1,..., W r V W 1,..., W r V = W 1 W 3 30 5 VI VI. W,..., W r V W,..., W r W + + W r = {v + + v r v W ( r)} V = W + + W r V W,..., W r V W,..., W r V = W W r () V = W W r () W (W + + W + W + + W r ) = {0} () dm V = dm W + + dm W r VI. f n

More information

°ÌÁê¿ô³ØII

°ÌÁê¿ô³ØII July 14, 2007 Brouwer f f(x) = x x f(z) = 0 2 f : S 2 R 2 f(x) = f( x) x S 2 3 3 2 - - - 1. X x X U(x) U(x) x U = {U(x) x X} X 1. U(x) A U(x) x 2. A U(x), A B B U(x) 3. A, B U(x) A B U(x) 4. A U(x),

More information

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2 1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac

More information

行列代数2010A

行列代数2010A a ij i j 1) i +j i, j) ij ij 1 j a i1 a ij a i a 1 a j a ij 1) i +j 1,j 1,j +1 a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a, ij i j 1,j 1,j +1 ij 1) i +j a

More information

(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y

(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y (2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b

More information

高校生の就職への数学II

高校生の就職への数学II II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................

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

A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa

A A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa 1 2 21 2 2 [ ] a 11 a 12 A = a 21 a 22 (1) A = a 11 a 22 a 12 a 21 (2) 3 3 n n A A = n ( 1) i+j a ij M ij i =1 n (3) j=1 M ij A i j (n 1) (n 1) 2-1 3 3 A A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33

More information

ii

ii ii iii 1 1 1.1..................................... 1 1.2................................... 3 1.3........................... 4 2 9 2.1.................................. 9 2.2...............................

More information

/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ

/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ 4 4.1 1 2 1 4 2 1 / 2 4.1.1 n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ λ U λ (local chart, local coordinate)

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

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b 23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b

More information

Jacobi, Stieltjes, Gauss : :

Jacobi, Stieltjes, Gauss : : Jacobi, Stieltjes, Gauss : : 28 2 0 894 T. J. Stieltjes [St94a] Recherches sur les fractions continues Stieltjes 0 f(u)du, z + u f(u) > 0, z C z + + a a 2 z + a 3 +..., a p > 0 (a) Vitali (a) Stieltjes

More information

Catalog No.AR006-e DIN EN ISO 9001 JIS Z 9901 Certificate: 販売終了

Catalog No.AR006-e DIN EN ISO 9001 JIS Z 9901 Certificate: 販売終了 Catalog No.AR006-e DIN EN ISO 9001 JIS Z 9901 Certificate:09 100 5919 DJ!0 DF DF @3 q w e 130 230 TR RA 0H R 130 230 RA TR R R RA 0.02MPa RA 130 230 130 230 R 0.06MPa RA 0.15MPa q R #1 TR #6 I N D E X

More information

1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915

More information

2

2 p1 i 2 = 1 i 2 x, y x + iy 2 (x + iy) + (γ + iδ) = (x + γ) + i(y + δ) (x + iy)(γ + iδ) = (xγ yδ) + i(xδ + yγ) i 2 = 1 γ + iδ 0 x + iy γ + iδ xγ + yδ xδ = γ 2 + iyγ + δ2 γ 2 + δ 2 p7 = x 2 +y 2 z z p13

More information

K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 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

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)

x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

More information

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V

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

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

ISTC 3

ISTC 3 B- I n t e r n a t i o n a l S t a n d a r s f o r Tu b e r c u l o s i s C a r (ÏS r c ) E d is i k e - 3 ) a =1 / < ' 3 I n t e r n a t i o n a l s t a n d a r d s f o r T B C a r e e «l i s i k e 3

More information

2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m

2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m 2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2

More information

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n . X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n

More information

Report98.dvi

Report98.dvi 1 4 1.1.......................... 4 1.1.1.......................... 7 1.1..................... 14 1.1.................. 1 1.1.4........................... 8 1.1.5........................... 6 1.1.6 n...........................

More information

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >

More information

等質空間の幾何学入門

等質空間の幾何学入門 2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................

More information

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan

More information

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P 9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)

More information

(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y

(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y (2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c 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

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

untitled

untitled 1 ( 12 11 44 7 20 10 10 1 1 ( ( 2 10 46 11 10 10 5 8 3 2 6 9 47 2 3 48 4 2 2 ( 97 12 ) 97 12 -Spencer modulus moduli (modulus of elasticity) modulus (le) module modulus module 4 b θ a q φ p 1: 3 (le) module

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