1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
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1 . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (,
2 (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/
3 c, c + c R n n R v ( 3 4 w ( 5 3v + w v + w v w 3 n R n m v,..., v m c,..., c m c v + +c m v m v,..., v m c,..., c m R 0,,
4 , ( + + ( Ke Words: 4. A e, e, v, v 0 e, e 0, v, v e, e v, v B e + e v, v
5 5 5. A,, B a, b, c + + ( ( + a + b + c + 6. A 0 ( + + ( + z + (z B ω ω 3 a, b, c z + a z + b c ω z ω 7. A v v 0 B (,, 0, (, 0,, (0,, H (, a, b H 3. (, 0 θ cos θ 0 sin θ R θ 0 cos θ sin θ
6 6. π 360 π (0, θ 0 R θ sin θ cos θ. (, R θ (c v + c v c R θ (v + c R θ (v R θ (v + v R θ (v + R θ (v, R θ (cv cr θ (v R n v n R n T (v R n 3 T : R n R n v T (v 5. id id(v v 3 T v v
7 6. c 7 D c (v cv R n c D id 7. a R n T a (v v + a R n a 0 T 0 id R n R R n T T (c v + c v c T (v + c T (v T (v + v T (v + T (v, T (cv ct (v ( 0 R θ R θ R θ + R 0 θ cos θ sin θ + sin θ cos θ cos θ sin θ sin θ + cos θ 0 + v 0 v ( θ cos θ sin θ R θ (v sin θ cos θ (, z + i
8 8 θ w cos θ + i sin θ z + i z wz + i (, θ (, R θ wz wz (cos θ + i sin θ( + i ( cos θ sin θ + i( sin θ + cos θ θ cos θ sin θ R θ sin θ + cos θ T w cos θ + i sin θ T (z wz v, v z, z c v + c v c z + c z T (c z + c z w(c z + c z c wz + c wz c T (z + c T (z T Ke Words: 8. ( 3/, / π/3 9. (, (, π/4 0. v v +. v w θ (v, w + (v, w v w cos θ
9 9. A θ l l S θ 0 S θ, S 0 θ B S θ S θ S θ cos θ + sin θ sin θ cos θ 3. 4 d d f(, g( c, c d ( d c f( + c g( c d d f( + c d d g( d n d n ( n n! ( ( n+ ( + n+ 4. A a, b, c f( f(a, f(b 0, f(c 0 B a f( a f( a f( a f(a f(, g( c, c ( cf( + cg( c f( + c (g( a a a f( f( p, f( q, f(3 r 4. R n (0,..., 0 T f( + g( c f( d d
10 T (0+T (0 T (0 T (0 T (0 0. R n T R n R n l v, 0 w l {v + tw R n t R} t v + tw R n T T (l {T u u l} l T T l T (l u v + tw T (u T (v + tt (w T (w T (l T v R R T R T ( c R T : R R T ( T ( T ( c c T c R c R R n Ke Words: R n 5. R 6. a T a (v v + a 7. (, ( +,
11 5. θ cos θ sin θ sin θ + cos θ a b 4 a, b, c, d c d a, b, c, d ( a b ( c d a c b a (, d (, ( b(, c(, d a b M v Mv c d a b a + b Mv c d c + d Mv M, a b Mv + c d cos θ sin θ R θ (, sin θ cos θ cos θ sin θ R θ sin θ cos θ cos θ sin θ θ sin θ cos θ ( R T T T
12 T ( 0 T T + T 0 T 0 T T a, T c a + c b d 0 a b c d b d a b c d T ( a b a b a b α + β α + β c d c d c d a b. T c d R 4 a b T, c d ( 0 a b T, T 0 c d 8. R id E 0 0 v Ev v
13 9. T T 3 a, T c b d T T ( T T + 0 T + T a + b, c + d ( 0 T T T T a b c d a + b a b T c + d c d T T + a + c b d a + b a b c + d c d 0. D c R θ S θ c θ t Z AFTER BEFORE X P Q t t.5 t C t 0.5 t t0 S R t0.5 A B t t.5 D C t Y Figure.
14 4 C t v t v t t C t BEFORE AFTER t 0 D ABCD PQRS t 0 C t t ± XY SZ R Ke Words: R T T a, T b c d T 0. A 3 A v B t λ(t ( t t Av + v λ(tv v λ(t λ(t v t. A R T T + 3
15 T B R T 0 3 T, T 0 T 5. A π/3 B R P (, 0 π/3 (, P 3. (, (, (, (, 4. + (, 6. θ θ v v R θ (v v R θ (v R θ (R θ (v v v R θ R θ R θ R θ R θ R θ R θ +θ ai b A i i (i, T i T T c i d i ( a b a b T T T + T c d + T c d a a a T T A + b c b A c c a + d c d a b + b d A c b + d d A A A a b a b a a + b c a b + b d c d c d c a + d c c b + d d R θ R θ R θ R θ R θ +θ
16 6 S θ (cos θ sin θ cos θ sin θ cos (θ θ sin (θ θ sin θ cos θ sin θ cos θ sin (θ θ cos (θ θ S θ S θ R (θ θ S θ S θ R (θ θ (, ( +, T (, (, + T + ( + + T T T T T (, ( +, + T T T T T ( (, ( +, + T, T 0 T, T 0 0 T T 0, T T ( 0 0 A, B A B Ke Words: 5. X, Y X, Y 3 XY Y X 3 6. A R θ R θ R θ +θ
17 7 B R θ S 0 S θ 7. A θ l θ (r, ϕ l θ B l, l P l i (i, S i P θ l l S S P θ P 7. m n m n m n m n m n (m, n (i, j A (a ij i,...,m j,...,n A (a ij i,j,...,n n a a a n a a a n A... a m a m a mn m (i, j i a a j a n a a a n... a ij... a m a m a mn. ( i + j i, j,,3 ( (m, n A + B (a ij i,...,m j,...,n ca c(a ij i,...,m j,...,n + (b ij i,...,m j,...,n (ca ij i,...,m j,...,n (a ij + b ij i,...,m, j,...,n
18 8 (l, m A (m, n B C AB m c ij a ik b kj a i b j + a i b j + + a im b mj k AB (a ik i,...,l (b kj k,...,m k,...,m j,...,n AB (l, n 00 k k , ( m a ik b kj j i,...,l j,...,n m j 00 k k A, B A B AB Figure. 3 3 A B AB m v n t w v t w m n t wv m n
19 3. v, t w ( 3 v t w 9 ( 3 6, t wv 5 m n m n m n Ke Words: m n z + z + + z 30. A,B /3 A, B 0 /3 0 AA, AB, BA, BB 3. A X, Y, Z X ( i j i0,,, Y ( i j j0,, Z ( j i i0,, j0, i0,, j0, B A, B, C A ( i j i0,...,l, B (z j w k j0,...,m, C ( i w k i0,...,l j0,...,m k0,...,n k0,...,n AB (/zm /z C
20 0 8. A, B, C c (ABC A(BC, A ( ( 0, B 0 ( ( 0 (ABC 0 A(B + C AB + AC, (A + BC AC + BC, (cab A(cB c(ab, C ( ( ( A(BC ( ( (0 ( 0 A ( 0, B, C 0 A(B + C ( ( 0 + ( ( AB + AC ( + ( 0 ( 3 + ( 3 ( c 5, A ( 0, B 0 (cab ( 5 ( 0 ( ( A(cB ( ( 0 5 ( 0 5 ( ( ( 0 c(ab 5 5 ( ( 5 5 0
21 a ij b jk c kl a ij b jk c kl k j j k a ij b jk c kl a ij b jk c kl j k j k X jk j j k X jk k n A A k (k N k 0 A 0 E, A k+ AA k E n AE A, EB B 0 O AO O OB O O Ke Words: M a b c M a b c ( a b c za zb zc z
22 g ( a b c a + b + cz z M A α, β M(α, β αe + βa M(α, β M(γ, δ B A, A, A A, A 0, A N(α, β αa + βa N(α, βn(γ, δ N(γ, δn(α, β (αδ βγa A A, B A (, B (, A A, B B, AB BA O, A + B E B 3A + B n + 3 n 3 n 3 n + 3 n 35. M 3 N 3 MN
23 3 9. R n T S T T S S T id v T T v Sw S w R n 4. D c D c T a T a R R θ R θ R S θ n A B AB BA E B A A B B A B BE B(AB (BAB EB B B B A A A n A (a ij i,j,...,n i j a ij 0 A 0 0 a ii 0 n (b jk j,k,...,n (AB ii n a ik b kj a ii b ii 0 k AB E
24 4 k N k O N N E N E + N + N N k (E N(E + N + N + + N k (E + N + N + + N k (N + N + + N k E N k E n A (a ij i,j,...,n i > j a ij 0 A N 0 n N , N , N , N N 3. X E + X + X + X X E X n A, B B A AB (B A (AB E (AB (B A A(BB A AEA AA 3. A (a i,j i,j,...,n 0 i a i,i 0 i a i,i 0 A D N E D A N n 0 A D(E N
25 i n 5 { 0 j < i a jj 0 j i n v Av 0 A 0 A 0 A Av Ev v A A B b C A 0 0 C O 0 C 3 B (i (i b (i B b v 0 Av : Ke Words: A A A, B a A 0 b, B a b
26 6 B + a a 0 AB a + b b 0 b 38. N 4 ( 0 + ( a b (. A A ad bc c d 4. a b A c d 0 A ad bc 0 A d b ad bc c a d b à c a a b d b ad + b( c a( b + ba Aà (ad bce c d c a cd + d( c c( b + da ÃA (ad bce A ad bc 0 A d b ad bcã ad bc c a
27 7 A A 0 AÃ O A AÃ O A AÃ O Ã O A O E A A O A. A. A u, w A u t w A ( A 0 a b A ad bc 0 c d 0 (c (a, b (0, 0 A d (a a 0 d bc/a A b c/a (a b 0 c ad/b A b d/b A v 0 Av 0 {Av v R } R v A(v A
28 8 A O A O A u t w (u, t w 0 (i 0 t w n 0 0 n t w ( 3 3 n An (u t wn u ( t w n 0 n (ii v R t w v Av (u t wv u ( t w v ( t w vu Av {tu t R} A 3 n An v Av { t t R} 3 6 ( ( ( 3 ( 3 ( r s + r, + 4 s
29 A, v 4 Av b 9, b r, s 4 A v A b 4r s r + s, 0 r, s A b a b r A, b c d s Av b A b A Av b v A b O b 0 v A u t w u w b {tu t R} b su, t wn 0 n t wa a {sa + tn t R} Ke Words:
30 A p, q, r, s A p (i A 0, q (ii v Av B { t r t R} s 40. A R T B R l (a, b (0, 0 a, b, c R a + b c T l R C T l 4. α, β A A α, A β 4. (, 3 6 s 3 s
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1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
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取扱説明書 -詳細版- 液晶プロジェクター CP-AW3019WNJ
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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 ( )
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0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
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HITACHI 液晶プロジェクター CP-AX3505J/CP-AW3005J 取扱説明書 -詳細版- 【技術情報編】
B A C E D 1 3 5 7 9 11 13 15 17 19 2 4 6 8 10 12 14 16 18 H G I F J M N L K Y CB/PB CR/PR COMPONENT VIDEO OUT RS-232C LAN RS-232C LAN LAN BE EF 03 06 00 2A D3 01 00 00 60 00 00 BE EF 03 06 00 BA D2 01
IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
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ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE
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A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
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29
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HITACHI 液晶プロジェクター CP-EX301NJ/CP-EW301NJ 取扱説明書 -詳細版- 【技術情報編】 日本語
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数学Ⅱ演習(足助・09夏)
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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
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振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
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x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
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,
[ ] 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 =
BD = a, EA = b, BH = a, BF = b 3 EF B, EOA, BOD EF B EOA BF : AO = BE : AE, b : = BE : b, AF = BF = b BE = bb. () EF = b AF = b b. (2) EF B BOD EF : B
2000 8 3.4 p q θ = 80 B E a H F b θ/2 O θ/2 D A B E BD = a, EA = b, BH = a, BF = b 3 EF B, EOA, BOD EF B EOA BF : AO = BE : AE, b : = BE : b, AF = BF = b BE = bb. () EF = b AF = b b. (2) EF B BOD EF :
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
) ] [ 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
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,
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) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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
5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i
i j ij i j ii,, i j ij ij ij (, P P P P θ N θ P P cosθ N F N P cosθ F Psinθ P P F P P θ N P cos θ cos θ cosθ F P sinθ cosθ sinθ cosθ sinθ 5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6
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( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a
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04 zz + iz z) + 5 = 0 + i z + i = z i z z z 970 0 y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory
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}
(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
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
(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
04年度LS民法Ⅰ教材改訂版.PDF
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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
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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
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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( 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
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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,
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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 =
(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n
3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n
Part2 47 Example 161 93 1 T a a 2 M 1 a 1 T a 2 a Point 1 T L L L T T L L T L L L T T L L T detm a 1 aa 2 a 1 2 + 1 > 0 11 T T x x M λ 12 y y x y λ 2 a + 1λ + a 2 2a + 2 0 13 D D a + 1 2 4a 2 2a + 2 a
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
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CV N-A (-' by T.Koyama ennard-jones fcc α, β, γ, δ β α γ δ = or α, β. γ, δ α β γ ( αβγ w = = k k k ( αβγ w = ( αβγ ( αβγ w = w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( βγδ w = = k k k ( αγδ
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