.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 =,
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1 [ ] 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 y, y d y d + a dy d + by = f = dy d y dy y d f f f.4 { + y = 5 y + y = 7 s6 3 s36
2 .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 =, y = = [m], y y [kg/m ] [kg] 3 s366. y + y y = y = 4, y = 3 s367. 9y dy d + 4 = dy d + y = sin 3 s y + z = + 3y z =.3 A =.4 A = [ ] A 3 s369 3 s36 3 s36.5 f P P {,, } T : f fα + β 3 s36
3 .6 f = e sin y grad f v = 3zi + yj yz k div v i, j, k -, y-, z-.7 d log + d 3.8 f, y = log + y f + f y 3 s363 e d 5 s56 5 s56.9 = r cos θ, y = r sin θ r, θ, y f, y f f r θ r, f θ 5 s563.,,,,, D y ddy D. y =, y = y + 3y + y = 5 s564 5 s565. 3, 4 + y 3 = 45.3 A = 5 s566 P AP = D P D 5 s567.4 d.,. h r.5. D 3 ddy D : 6 s66 a + y a, b > b 6 s
4 dy d 3 dy = y d 3y = 5 6 s663.7., y, y, y. y y = y + 3y + y = 6 s664.8 yz π : + y + 3z 5 = g :. π. g. 3 = y 5 = z + 3,.9 A = 3 π g s665, AX = O, Y A = O 3 3 X, Y,..3. K k s666 f k k=5, p = a + b. G = f k p k, a, b. k= 6 s y 4y 3y = y = 4y + 7 s76.3., a >. I = e a d.33 z = y z =..34 A = 5 3, A A det A. 7 s76 7 s763 7 s764 4
5 .35 S 3 + y + z =, S P,, 3 S. 7 s765.36,. 7 s r = i + yj + zk, r = r,. r = r r, = i + j y + k z, i, j, k. 7 s , = y = y. dy d = a by a, b 7 s e iπ + =., i =. 7 s769.4 f = a,, ε δ. 7 s76.4 f, g. fg, f g + fg.,,..4 A = 7 s76 R R A : A,. 4 ImA. KerA. 3 3 A =.,. 6 4 A =.,. 7 s = R =, A A. A. 3 P, P AP T Λ., T 4,. P Λ. y y = P., y, y 5
6 5 4,,. 7 s a = 6,,, b = 3, 4,, c =,,,. a b, c., a, b, a b, a b, a b.. 7 s A. A = 7 s y = sin n. [ y y, y,, n. ] [ sin = sin + π ] 7 s y = sin 3 cos. 7 s767.48,, A, B, C.,.,,,,., A, B, C,,,,., A, B C. B C A B, C. A C B,,, A. A, A,,.. B.,,, B. B, B,,..,, C,,,, A, B..49 A = a a a a a 7 s768 A A = 6
7 8 s86.5 i i a + bi a, b 8 s86.5 dy d + e = d + + sin 7 cos d 8 s863 8 s f, y = + y e y f + f y 8 s y = + 9 D + yddy D 8 s A = a,, B = b, 3 A, B 3 6 b > a > a, b A 3 B 8 s y 4 = ABCD AB = a S D y C S a A O a B < a a.57 p p = + a < a a + a a a < a > pd 8 s868 pd 7
8 8 s A A = A A A A 3 + y + 5z = 3 A A 7 + 3y + 4z = 3 + y + z = i4 i =.6 = y = dy d = y 3 8 s86 8 s86 8 s86.6 A = 3, B = det A, det B, det C 3 3, C = A 3 C rank C 8 s863.6, y f, y D f + f y = f, y D D = R f, y = 3 ay a f, y g, y = y f + f y g, y f, y 8 s cosπd cos πd.64 y = + sin, 8 s s s867 8
9 .66 a =,,, b =,, α α < 8 8 s y = m y = m m.68 d y d + 6 dy + 9y = 5e d s869 8 s86 8 s86.7, y r r θ = y = n θ = π/n θ θ =, = r, y = θ i+ = θ i + θ i =,,,, n i+ = y i+ =, y,, y,, n, y n r, θ i, θ i+ 8 s86.7 f, y = loge y + e y.,. f f,..7 f y 9 s y z =. u v 9 s96.73 n A A 3 = O., O. A =., A A. a b n =. A =, A a + da = O. c d 9
10 3 n = A = O. 9 s , D y,. + y ddy.75 dy = yy d D,. y = a., a. 9 s964 y, lim y. 9 s A,,, B 3,,, C,, 3. e 3.77 lim. [ : lim log e + =.].78 f = , 3.79, dz dt. fd. z = sin cosy, = e t, y = log e 3t , s966 9 s967 9 s968 9 s969 9 s , , A. A = 3 9 s96 A. A P,, P AP P.
11 3. d dt t t = A t t, =, =.83 log = a, Taylor., a. 9 s96 9 s lim sin =. 9 s logsin t + t s965 e e + e d. 9 s966.87,. dy + y =, = y =. d 9 s F, P Q, W, W = F P Q. F = 4i 3j + k, P 3,,, Q,, 4 W., i =,,, j =,,, k =,,. 9 s e, =,., e 3,. 9 s969.9 φ, y, z, A, y, z = A i + A y j + A z k,.. rot grad φ = :. div rot A =, i, j, k 3, y, z 9 s96.9., C. sin sin 3d sin sin 3, cos,.. 9 s96
12 .9. 9 s d.94 y dy d = e.95 A = 3.96 A A = A λ. e log d 9 s963 y =.. 4 λ X. 9 s964 9 s965,. s6.97 f, y f =, f, y. f, y. = / + / y. f = + y.98.,, y, f = e sin y y = y + y 3 y + 6y + y = s6.99 A = 3.. s63 λ λ. A. s64. + y = + z =. s65. a, a, a 3. a =, a =, a 3 = s66
13 . A 3. A. A = A. 3 A P P AP..3 lim + log e =. s67 log e d. s68.4 X = { : π, R}, Y = {y : y, y R}. f,., R. f : X Y, sin. f. f..5 A = 3. s69 A. A. 3 A. 4 A..6, 7. s6 d dt = a d. y =, y t d dt + b dt. 3. dy, dt = dy d d, y 4 dt. 5.,, ν[m/s] 37.5 ν + 5 [m/s ]., ν = [m/s].75[m/s ],. 75[m/s]= 7[km/h] 6 [s], 7 [m]..7 f. f = e + f = + sin 3 s6 3
14 s6.8,., y, z, u. y, = y =,. u, y y z,., y, z, u. u, y, z.,,,,. s63.9. π 4 cos d y e y dy d s64. a,. + y + z + w = + y + 4z + w = y + 3z + w = y + w = a s65. y,. y +y = y 7y +y = y 4y +4y = cos s66.. A = s67.3. s68.4 -y. f = 3 + 3y, OABC, O,, A,, B,, C,. OABC 4, OAB 3 f f ds OAB f ds OABC,. ds. s6 4
15 .5 A =, P a AP = P 5 3 b a, b. s6.6 n. a, b, c, d ad bc, c, f = a + b c + d n. D = {, y y 4, },. + y ddy D s63.7 3y 5z = a y 4z = b 3 + y + 3z =, a, b., a, b,. s A,, 3,., E 3. 3A E A A s65.9 y.,. dy + y cos = sin cos d y =. y π.. n,. n d π 6. + y 3z =, 3 y z = 4. s66 s67 s68. 3 CPU. CPU.8,.,. 5
16 3 CPU CPU,., 3 CPU,. 3, CPU CPU,, a 5 a 3 a. 3 a A = a c b d s69 s6 A a + da + ad bce =.. E. A = A A. 3 A =, A n. 4 s m n d π cos n d 3 e d s6.6,, α α. t ft., f = β. s63.7. tan θ = /3. sin θ + cos θ. + tan α/ tan α = + 3, cos α. s64.8 y,, 3,., y + =, + y 4 =, 3 + 3y + 3 =. 6
17 ,, 3 y.,, 3 y. 3 A, 3 B, 3 C, A, B, C. 4 A, B, C ABC. s65.9. y + y = y 6y + 8y = s66.3 y = + sin π V. y O π.3 A = s67 det A. A. 3 A y z = 3. s68.3 f = e, 4 fd. -y, = at sin t, y = a cos t, a >, t π. sin.33 lim A = a b c. bc ca ab s69 s6 s6 7
18 .35 f = log >,., log. f = f. y = f. 3 f log. 4 y = f y = log S..36 A ranka. A = 3 4 s6 a,. a a y a a z = 3 4 w a.37 A =. s6 B = A 4A 4E, BA., E 3. 3 yz P =, y, z, XY Z Q = X, Y, Z T A. T : X Y = y Z yz P, T XY Z H Q. H. z s63.38 yz C l,., P, Q, P Q. = cos θ C : y = sin θ θ π l : z = cos θ + sin θ = t y = 3t z = t P C, Q l, P Q θ t. < t < + P C, Q l, P Q,, P Q..39 y, A,. s64 y tan y = e sin, π/ < < π/ A 8
19 y; y y = y A, y; y., y. y; y, lim y; y y. π/, y lim y; y.,, π/. s65.4 a. tan = a +,., tan, tan =. 3 s36.4 I = log + y ddy, D :, + y 4 D., log. 3 s36.4 R 3 a =, a = 3, a 3 =,. a, a, a 3 3 y z = α.,. a, a, a 3. a 3 a a..43 A = 3 a a b = a + ya a, y. 3 3 b = A,. b A. 3 A 5 b α 3 s363,. 3 s364 9
20 .44 f, y. f,, f y,. f h, f, h lim h h 3 s F, F = f. I = b a f d F., a, b., yf + y J = + y ddy, D :, y, p + y q D F., p, q..46 y 3 s366 y + y + ay =., a. a =,. a >, y lim y =. 3 s f = e e e + e. f = Taylor 3. 4 s46.48 a, b. I. ab I = a b b a y ddy, D : a + y b < D 4 s46.49 D. D = s463.5 A,. A = s464
21 .5 Γt.., Γt = e t d, t >. t > lim e t =. t > Γt + = tγt. n Γn + = n!. 3 Γ, e d e y y dy, y. 4, y r, θ o, y r, θ. o { = r cos θ, y = r sin θ Γ.5 A. 4 4 A = P AP P. 3 {a n } a =, a = 5, a 3 =,. a n+3 = a n+ + 4a n+ 4a n n a n+3 a n+ = B. a n+ a n+ 6 s66 3 B. 4 a n. a n+ a n 6 s66.53 m.. Im = m + d I, I. r, N,. N r + = n r n + N+ r N+ r + n=
22 3. n n= n = s76.54 α. A = α α. A A. A α. 3 A α., A..55 a, b >. y 7 s76 y a + = C. b,, y C A.. = a cos 4 t. y = b sin 4 t., y a + b., d t dt. C y = y, y = dy d. 3 A. 4 A..56 k = = 3k + E 4 +k = k 4 =.. E. E k.., lim a y 3 E k,. lim + y 8 s86 8 s86 a a a 3.57 yz R 3 f : R 3 R 3 A = a a a 3., A f = A, a 3 a 3 a 33 R 3..
23 f, A. f, A. 3, + y + z = 5 y + 7z = f A.58 y = y, y = dy. d E y + y cos = sin cos.. y + y cos =. E. 8 s863 3 E, y =. y 4 3 y, lim. 8 s864 3
[ ] 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](/thumbs/94/118579915.jpg "[ ] 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 =
() 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 ()
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 ),
( ) 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) (
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 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
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 =
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
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 =
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.
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.
( 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
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
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
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f
208 3 28. f fd f Df 関数 接線 D f f 0 f f f 2 f f f f f 3 f lim f f df 0 d 4 f df d 3 f d f df d 5 d c 208 2 f f t t f df d 6 d t dt 7 f df df d d df dt lim f 0 t df d d dt d t 8 dt 9.2 f,, f 0 f 0 lim 0 lim
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.................................
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
all.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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
9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
s s U s L e A = P A l l + dl dε = dl l l
P (ε) A o B s= P A s B o Y l o s Y l e = l l 0.% o 0. s e s B 1 s (e) s Y s s U s L e A = P A l l + dl dε = dl l l ε = dε = l dl o + l lo l = log l o + l =log(1+ e) l o Β F Α E YA C Ο D ε YF B YA A YA
r III... IV.. grad, div, rot. grad, div, rot 3., B grad, div, rot I, II ɛ-δ web page (
r 8.4.8. 3-3 phone: 9-76-4774, e-mail: hara@math.kyushu-u.ac.jp http://www.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html Office hours: 4/8 I.. ɛ-n. ɛ-δ 3. 4. II... 3. 4. 5.. r III... IV.. grad, div,
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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,
Gmech08.dvi
63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
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
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)
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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
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 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)
(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)
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
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() P. () AB () AB ³ ³, BA, BA ³ ³ P. A B B A IA (B B)A B (BA) B A ³, A ³ ³ B ³ ³ x z ³ A AA w ³ AA ³ x z ³ x + z +w ³ w x + z +w ½ x + ½ z +w x + z +w x,,z,w ³ A ³ AA I x,, z, w ³ A ³ ³ + + A ³ A A P.
ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y
01 4 17 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy + r (, y) z = p + qy + r 1 y = + + 1 y = y = + 1 6 + + 1 ( = + 1 ) + 7 4 16 y y y + = O O O y = y
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](/thumbs/95/125540821.jpg "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 =
Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) r x i y j zk P r r r r r r x y z P ( x 1, y 1, z 1 )
Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) x i y j zk P x y z P ( x 1, y 1, z 1 ) Q ( x, y, z ) 1 OP x1i y1 j z1k OQ x i y j z k 1 P Q PQ 1 PQ x x y y z z 1 1
2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l
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
l µ l µ l 0 (1, x r, y r, z r ) 1 r (1, x r, y r, z r ) l µ g µν η µν 2ml µ l ν 1 2m r 2mx r 2 2my r 2 2mz r 2 2mx r 2 1 2mx2 2mxy 2mxz 2my r 2mz 2 r
2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
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TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................
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) = (, )
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
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, 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. +
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( ) x y f(x, y) = ax
013 4 16 5 54 (03-5465-7040) nkiyono@mail.ecc.u-okyo.ac.jp hp://lecure.ecc.u-okyo.ac.jp/~nkiyono/inde.hml 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy
( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i
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5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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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)
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z