( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
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1 6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = a 3 b 2 3 c 2 () ( 2 ) 2 = 2 2 ( ) 2 ( 2 ) = ( )( + 2 ) 2 ( 2 ) = ( + ) 2 c (2) lim sin 0 5 = sin 5 lim 0 = lim ( ) 3 sin 5 = 3 5
2 2 y = ( > 0) log y = 3 log 4 dy d = 5 dy d = y 9 y a y 2 b y 2 c y d y e y 2 f y log 2 log log log 2 8 log log 2 6 log 2 y = log y = log = log 3 4 y dy d = 2 log + ( )2 = log 2 4, 5 c 6 2
3 3 I = I = 6 tan d I = tan d 7 d = 6 8 ( ) tan 4 tan tan 6 cos 2 7 cos cos cos 2 8 cos tan 3 log + 4 log 5 log + 6 log 7 log( + 2 ) 8 2 log( + 2 ) 9 2 log( + 2 ) 3
4 tan d = tan d = tan (tan ) d = tan + 2 d = tan 2 log( + 2 ) 6, 7,
5 4 yz u = 2 + y 2 + z 2 ( (, y, z) (0, 0, 0) ) r = 2 + y 2 + z 2 r = 9 u = r u = du dr 2 u 2 = r = 0 2 u y 2, 2 u z 2 2 u u y u z 2 = r r 2 r 2 r 3 r 3 7 r 8 r a b 2 r 2 r 3 r 3 0 r 3 r r 4 r r 5 r 32 4 r 6 4 r 3 5 r r 4 6 r r 5 7 r r 6 5
6 r 3 r r 3 5 r 6 3 r r 3 3 r r = 2 + y 2 + z 2 = 2 + y 2 + z 2 = r 9 2 u = r u = du dr r = r 2 r = r 3 0 a 2 u 2 = r r r 4 = r r u y 2 = r 3 + 3y2 r 5, 2 u z 2 = r 3 + 3z2 r 5 2 u u y u z 2 = 3 r 3 + 3(2 + y 2 + z 2 ) r = 3 r 3 + 3r2 r 5 = 0 6
7 5 y D D = {(, y) 2 + y 2 9, + y 0} I = 2 + y ddy D = r cos θ, y = r sin θ D (r, θ) E = {(r, θ) 0 r 3, π } 4 θ 4 3 I = 0 4 π 4 5 dθ dr 4 π 4 5 dθ = 6 I = dr = π π 7 π π 9 π 2 a π 5 0 r cos θ r r sin θ r r2 cos θ r r2 sin θ r
8 r r r 2 r r 2 2 (r2 + 9) 4 r 2 (r2 + 9) 5 2 r r r 2 6 r r 2 2 (r2 + 9) 8 r 2 (r2 + 9) ( 2 + π ) ( 2 π ) ( 2 + π ) ( 2 π ) ( 2 + π ) ( 2 π ) 2 D y D O = r cos θ, y = r sin θ D { E = (r, θ) 0 r 3, π 4 θ 3π } 4 3 3, 4 6 ddy rdθdr 3 3π 4 r 2 cos θ I = r dθ dr 0 π 4 8
9 5 2. 3π 4 π 4 r 2 cos θ r dθ = r2 r π 4 [sin θ] π 4 = 2r 2 r r 2 I = 0 r dr = 3 { 2 9 } 0 r 2 dr + 9 = [ 2 r 3 tan r ] = 2 { 3 3 tan } = ( π ) 4 9
10 6 t y C t : y = t 2 + t {C t } t>0 φ(t), ψ(t) C : = φ(t) y = ψ(t) (t > 0) C {C t } t>0 (i) (ii) t C P(φ(t), ψ(t)) C t t C C t P(φ(t), ψ(t)) (i) ( ) ψ(t) = tφ(t) 2 + t ( ) t ( ) ψ (t) = 8 t 2 (ii) P(φ(t), ψ(t)) C C t ( ) ψ (t) φ (t) = 9 ( ) ( ) φ(t) 2 = 20 ( ) = φ(t) y = ψ(t) y = 2 ( 0) tφ(t) φ(t) + 2tφ(t) 2 φ(t) 2 + 2tφ(t) 3 2tφ (t) 4 φ(t) + 2tφ (t) 5 φ(t) 2 + 2tφ (t) 6 2φ(t)φ (t) 7 φ(t) + 2φ(t)φ (t) 8 φ(t) 2 + 2φ(t)φ (t) 9 2tφ(t)φ (t) a φ(t) + 2tφ(t)φ (t) b φ(t) 2 + 2tφ(t)φ (t) 0
11 20 0 t t 2 2 t 3 3 t 4 t 2 5 t 3 6 t 7 t 2 8 t 3 9 t a t 2 b t 3 P = (φ(t), ψ(t)) C t : y = t 2 + t ψ(t) = tφ(t) 2 + t. t ( ) ψ (t) = φ(t) 2 + 2tφ(t)φ (t) t 2. 8 b. C : = φ(t), y = ψ(t) P = (φ(t), ψ(t)) dy dy d = dt d dt = ψ (t) φ (t). C t : y = t 2 + dy = 2t P = (φ(t), ψ(t)) t d 2tφ(t). ψ (t) φ (t) = 2tφ(t) ψ (t) = 2tφ(t)φ (t). ( ) φ(t) 2 = t , y = 2 C t (t = /4, /2,, 3/2, 2).
12 y 2
13 ( ) A rank A A () u = 0, v = 0 u cu + v c c = 2 u cu + v e, e 2 e = 2 u, e 2 =
14 (2) () u cu + v u (cu + v) = 0 c. u = 0, cu + v = c u (cu + v) = c ( ) ( c ) = 2c + = 0 c = c = u + v = = 2 2 2, ( ) ( ) 2 2 = (2) A = λ 2 A λi = 4 3 λ = ( λ)(3 λ) 2 4 = λ 2 4λ 5 = (λ 5)(λ + ) = 0 4
15 , λ = 5, = (A 5E) = 0 y 4 2 = y 0 y = 2 = c y = 2 = 2c = c
16 2 3 R a =, a 2 = 4, a 3 = c, c 2, c 3 c a + c 2 a 2 + c 3 a 3 = 0 c + 3c 2 + 5c 3 = 0 c + 4c 2 + 7c 3 = 0 c + 5c 2 + 9c 3 = 0 c c 2 = t c 3 t. t = a + 25 a a 3 = 0, a, a 2, a a
17 3 5 3 a =, a 2 = 4, a 3 = 7 3 c, c 2, c c a + c 2 a 2 + c 3 a 3 = 0, (c, c 2, c 3 ) (0, 0, 0), a, a 2, a 3. c a + c 2 a 2 + c 3 a 3 = 0 (c, c 2, c 3 ) = (0, 0, 0),. a, a 2, a 3 c a + c 2 a 2 + c 3 a 3 = 0. c + 3c 2 + 5c 3 = 0 c + 4c 2 + 7c 3 = 0 c + 5c 2 + 9c 3 = c c 3 = 0, c 2 + 2c 3 = 0. c = t (t ) c c 2 = t 2 c
18 t = c =, c 2 = 2, c 3 = a 2a 2 + a 3 = 0., a, a 2, a
19 0 3 X = 2 X. 2 0 X = E + A (E 3 ) A 2 = A3 = O (O 3 ) (E + A)(E A + A 2 ) = 28 X = 29 = O E 2 E + A 3 E A 4 E + A + A 2 5 E A + A 2 6 E A A n M M MM = M M = E n (E n n ) n M () M n E n (M E n ) (2) (E n N) (3) N M 9
20 X X = E + A 0 0 A = X E = 2 A 2, A 3 A 2 = 0 0 0, A3 = O E 2 = E, AE = EA = A (E + A)(E A + A 2 ) = E 2 EA + A 2 + AE A 2 + A 3 = E + A 3 = E. 28 X = E + A X = E A + A X 2 0 X = m M m = O n (O n n ) n M ( ) M E n + M E n M + M 2 + ( M) m m = 3 20
21 4 2y + 2z = 3 y + cz = cy z = c A = B = c 2 2 c 2 2 c 2 c () c = 3 A B 32 (2) c = 33 A B c B A t (3 2 ), B. B (T) c 2 0 c (T2) c (c )(c + 3) c 2
22 (T) 2 (T2) (c + 4) B B 0. B A B A ranka rankb B A, B c B 0 c A B 2 3 () ranka rankb ranka = 2 rankb = 3 (c )(c + 3) = 0 c 0, c = y + 2z = 3 y 5z = 0 = (2) ranka = rankb = 2 (c )(c + 3) = 0 c = 0 c = 33 2y + 2z = 3 y z = 0 = 0 =, y = z y z
23 5 A = , B = A, B A, B () A = 0 35 (2) B = 0 36 (3) rank A = 2 B = , 6 2, 2 7, 0 8, 2 9 2,, 2 a, 0, 2 b,, 2 c 0,, 2 () A 2 ( + ) (T) + 2 A = 2 = ( + ) ( + )( ) 2 = ( + ){( + 2)( ) 2 } = ( + )( 2) A = 0 =, (2) B () 23
24 B B (T2) (T3) = = (2 ) (T4) = (2 ) 0 = (2 )( + )(2 + ) (T2) (T3) (2 ) (T4) ( 4) 4 B = 0 = 2,, (3) A 2 A = 0 rank A = 2 () 2 B = 0 (2) = 2,, 2 =
25 ( ) y, z, y, z, y dy d, dz d, d2 y d 2 () y = y 2 y = C log 2 + C 2 sin + C 3 sin + C 4 tan + C 5 tan + C C 7 + Ce 8 2 ( + Ce ) 9 + Ce a C + 2 C 2 c + log(2 + C) log(2 + C) (C ) d log(2 + C) + log(2 + C) b + Ce2 Ce 2 25
26 (2) y + 2y = 2 e 2 y = Ce e + C 2 Ce e 2 + C 4 Ce Ce C3 3 e a e 2 ( ) 3 c e C d e2 + C ( 3 3 e 2 + C 8 e 2 3 ( ) 3 C3 3 e2 3 + C b C3 3 e 2 e e 2 ( 3 ) 3 + C ) 3 + C f C3 3 e 2 (C ) () y 2 y y 2 =. y y 2 d = (y )(y + ) dy ( = 2 y ) dy y + = 2 log y y + + C, d = + C 2 (C, C 2 ) 26
27 2 log y y + = + C (C ). y = + Ce2 Ce 2 (C ) 38 b (2) e 2 (e 2 y) = e 2 y + 2e 2 y = 2. e 2 y = 2 d = C (C ) ( ) 3 y = e C. 39 e 27
28 2 ( ) e y + y y = 0, > 0. u() = y() y = 40 u() ( ) u = 4 ( ) u() = 42 ( ) y() = u u 2 u + u 3 u u 4 u 5 u 6 u + u 7 u u 8 u u 9 u + u a u u b u + u 4 0 u u 2 e u 3 e u 4 e u 5 e u 6 eu 7 e u 8 2 e u 9 2 e u a eu b e u
29 42 0 C + C 2 C 3 C 4 C + log 5 C log 6 C + log log 7 C log log 8 log(c + log ) 9 log(c + log ) a log(c log ) b log(c log ) ( c log C + ) ( f log C ) ( d log C ) ( e log C + ) (C ) u() = y() y = u, y = (u) = u + u ( ) e u + u (u + u ) = 0 u ( ) u = eu ( ) e u e u = e u du = e u u d = d = log + C (C ) u = log(c log ) (C ). ( ) y = u = log(c log ) 40, 4, 42 6, 6, b 29
30 3 ( ) y 4y = 8 4. () y 4y = 0 y() = C e 4 + C 2 C e 4 + C 2 2 C e 2 + C 2 3 C e 2 + C 2 4 C e 2 + C 2 e 2 5 (C + C 2 )e 2 6 C cos 2 + C 2 sin 2 (C, C 2 ) 30
31 (2) ( ) y p () = a + b, a = 44, b = a 2 (3) (), (2) ( ) y() = C e 4 + C 2 2 C e 4 + C C e 2 + C 2 e C e 2 + C 2 e (C + C 2 )e (C + C 2 )e C cos 2 + C 2 sin C cos 2 + C 2 sin (C, C 2 ) 3
32 () y 4y = 0 λ 2 4 = 0 λ = 2, 2 y() = C e 2 + C 2 e 2 (C, C 2 ) 43 4 (2) y p () = a + b y p () = a, y p () = 0 ( ) y p 4y p = 4(a + b) = 4a 4b = 8 4. a = 2, b = 44, 45 7, (3) ( ) y h ( ) y p y() = y h + y p = C e 2 + C 2 e (C, C 2 )
33 4 y(), z() y = 2y + z ( ) z = ay a () ( ) 2 z y 2 ( ) y + 2y + 47 y = 0. (2) ( ) λ 2 + 2λ + 47 = 0 a = a a a b 2a c 2a (3) a = 48 ( ) y = 49 y(0) = 2, y (0) = y = 50 z = C e + C 2 C e + C 2 2 C e + C 2 e 3 (C + C 2 )e 4 (C + C 2 )e 5 C cos + C 2 sin 6 (C cos + C 2 sin )e 7 (C cos + C 2 sin )e (C, C 2 ) 33
34 50 0 e + e (3e + e ) 3 2 (e + 3e ) 4 ( + 2)e 5 (3 + 2)e 6 ( + 2)e 7 (3 + 2)e 8 2 cos + sin 9 cos + 2 sin 5 0 3e + 2 3e (9e + e ) (e e ) 4 ( 3 + 5)e 5 9( + )e 6 ( + 3)e 7 (3 + 5)e 8 5 cos 9 3 sin (4) a < 48 y(0) = 2, y (0) = ( ) y()
35 () ( ) y = ( 2y + z) = 2y + z = 2y + ay, ( ) y + 2y ay = 0 47 a (2) ( ) λ 2 + 2λ a = 0 a = λ = 48 2 (3) a = y + 2y + = 0 (2) y = (C + C 2 )e (C, C 2 ) y = {(C + C 2 )e } = {C ( ) C 2 }e y(0) = C 2 = 2 y (0) = C C 2 = C = 3, C 2 = 2, y = (3 + 2)e z = 2y + y = 2(3 + 2)e + {3( ) 2}e = (3 + 5)e. 49, 50, 5 4, 7, 7 (4) ( ) λ 2 + 2λ a = 0 D = 2 2 4( a) = 4(a + ) a < λ = ± i (a + ) ( ) ( y() = e C cos (a + ) + C 2 sin ) (a + ) (C, C 2 ) y() 0 ( )
36 5 y 0 a ( ) = ay 2 y ( ) ( ) y = 53 ( ) ( ) a y = a ay 2 2a 3 2ay 4 a 5 y a 6 2a 7 y 2a 8 a 9 ay a 2a b 2ay 54 0 y y ( y ) ( ) 2 y y y 6 ( y ) ( ) y 8 y 2 9 y2 2 a y3 b y3 2 ( ) ( ) a = y 2 = (ay 2 ) = 2ayy ( ) y = 2ay y = ( ) 2 y 2 y = y 2 53, 54 b, 2 36
37 ( ) A P (A) A X E(X), V (X), D(X) X ( ) () X, Y X Y a a 6 6 a a = 55 E(X 2 ) = 56 Y 2X V (Y ) = a 6 b 24 37
38 (2) 2 A, B B A P (A B). P (B) = 8, P (A B) = P (A) = 4 A B 58 P (B A) = 59 P (A B) = a 3 4 () ( a = ) = 6 8 E(X 2 ) = ( 3) ( 2) = 4 55, 56 2, 7 E(X) E(X) = ( 3) 6 + ( 2) = 0 V (X) = E(X 2 ) {E(X)} 2 V (X) = E(X 2 ) = 4 Y Y 2X V (Y ) = V (2X) = 2 2 V (X) = 6 57 a (2) 2 2 A, B P (A B) = P (A)P (B) P (A) 0 38
39 A B P (B) = P (B A) P (A B) = P (A) A B P (B A) = P (B) = 8 P (A B) = P (A) + P (B) P (A B) A B P (A B) = P (A)P (B) = 4 8 = 32. P (A B) = = , 3, 7 39
40 2 X F () = P (X ) 0, <, F () = ( + ) 2, 0,, > 0 ( P X > ) = 6 f() 2 f() = 62, < < 0 E(X) = ( + )3 2 3 ( + )3 2 2( + ) a 3 4 P (X 2 ) = F ( 2 P ( X > 2 ) = ( 2 + ) 2 = 4 ) = 4 = 3 4 f() = F () = 2( + ), < < 0 40
41 E(X) = f() d 0 = 2( + ) d [ 2 = ] 0 = , 2, 7 4
42 3 X, Y, Z [ 2, 4] P ( X 3) = 64, E(X) = 65 X, Y, Z [, 3] N N 66 P (N = 2) = a b t U [a, b] P (c U d), a c < d b [a, b] [c, d] P ( X 3) = 3 4 ( 2) = 3 E(X) [ 2, 4] ( 2) + 4 = 2 X, Y, Z P ( Y 3) = P ( Z 3) = 3 N 3 3 B(3, 3 ) ( ) 2 ( 2 ) P (N = 2) = 3 C 2 = , a,, 2 42
43 4 B 350 g g µ g g 3% H 0 : µ = 350, H : µ X, X 2,..., X 36 X = X i i= H 0 Z = X P ( Z 2.7) X = g H 0 3% µ µ µ µ µ t 43
44 a 8 b N(m, σ 2 ) X, X 2,..., X n X = n n X k N ( m, n σ2) k= X E(X) = µ, V (X) = 36 D(X) D(X) = 68 70, 8, 3 6 N(m, σ 2 ) X X m σ N(0, ) Z P ( Z 2.7) = P ( X ) = P ( X ) = 0.4 > 0.36 H 0 3%, 7, 72 5, 0 44
() 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 ()
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
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 ),
Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
![c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l](/thumbs/85/91372642.jpg "c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l")
c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
7 4.., ], ], ydy, ], 3], y + y dy 3, ], ], + y + ydy 4, ], ], y ydy ydy y y ] 3 3 ] 3 y + y dy y + 3 y3 5 + 9 3 ] 3 + y + ydy 5 6 3 + 9 ] 3 73 6 y + y + y ] 3 + 3 + 3 3 + 3 + 3 ] 4 y y dy y ] 3 y3 83 3
1 2 1 No p. 111 p , 4, 2, f (x, y) = x2 y x 4 + y. 2 (1) y = mx (x, y) (0, 0) f (x, y). m. (2) y = ax 2 (x, y) (0, 0) f (x,
No... p. p. 3, 4,, 5.... f (, y) y 4 + y. () y m (, y) (, ) f (, y). m. () y a (, y) (, ) f (, y). a. (3) lim f (, y). (,y) (,)... (, y) (, ). () f (, y) a + by, a, b. + y () f (, y) 4 + y + y 3 + y..3.
, 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)
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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
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[ ] 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 =
) ] [ 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
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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
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08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
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II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
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)
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
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