1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1

Similar documents
(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 {

= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k

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,

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

() 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)

( ) 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) (

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

Gmech08.dvi

Chap11.dvi


lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d

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)) ( )

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

Gmech08.dvi

( ) 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 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

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

notekiso1_09.dvi

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


( : 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)

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 =

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

[ ] 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

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

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

高等学校学習指導要領

高等学校学習指導要領


1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h


i

() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y

n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x

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

2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP

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 (

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

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

Chap10.dvi

Z: Q: R: C: sin 6 5 ζ a, b

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

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +


(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z

1 1 x y = y(x) y, y,..., y (n) : n y F (x, y, y,..., y (n) ) = 0 n F (x, y, y ) = 0 1 y(x) y y = G(x, y) y, y y + p(x)y = q(x) 1 p(x) q(

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

Untitled

2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30

r III... IV.. grad, div, rot. grad, div, rot 3., B grad, div, rot I, II ɛ-δ web page (

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.

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

.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 =,

08-Note2-web

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

i

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 =

知能科学:ニューラルネットワーク

知能科学:ニューラルネットワーク

1

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

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

TOP URL 1

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

2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30

, 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

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

Acrobat Distiller, Job 128

KENZOU

di-problem.dvi

Chap9.dvi

mugensho.dvi

201711grade1ouyou.pdf

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

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

I 1

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

KENZOU

. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,

85 4

重力方向に基づくコントローラの向き決定方法

66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

pdf

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

DVIOUT

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

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10

A

Part () () Γ Part ,

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0


2 2 ( Riemann ( 2 ( ( 2 ( (.8.4 (PDF 2

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.

meiji_resume_1.PDF

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

Transcription:

. ( + + 5 d ( + + 5 ( + + + 5 + + + 5 + + 5 y + + 5 dy ( + + dy + + 5 y log y + C log( + + 5 + C. ++5 (+ +4 y (+/ + + 5 (y + 4 4(y + dy + + 5 dy Arctany+C Arctan + y ( + +C. + + 5 ( + log( + + 5 Arctan + C. ( sin sin cos + cos cos + sin + C. (3 6 9 6 9 3 + 3 log 3 + 3 + C.

(4 4 + (5 4 ( + + 3 + Arctan + C. + 3 + y + y + ydy. ( + dy (y y dy log y y + y + + C log + + C. + + (6 + 5 t + + 5 (t + 5 t 5 t t 5 t. ( + 5 dt t + 5 t t + 5 t + 5 t t + 5 ( dt log t +C log + t + 5 t + 5 +C. (7 ( + log y log dy ( + log dy log + y + C log + log + C. + y

(8 sin cos 4 I k 3 cos k I k sin cos k + (k sin cos k sin cos k + (k ( cos cos k sin cos k + (k (I k I k k sin cosk + k k I k. sin cos 4 I I 3 sin cos 4 6 sin cos5 + 6 I 6 sin cos5 + 6 4 sin cos3 + 3 6 4 I. + cos I sin cos + + C + sin 4 + C 6 sin cos5 + 4 sin cos3 +frac6 sin cos + 6 + C. (9 sin 3 cos 4 cos t sin t sin dt sin 3 cos 4 ( 4 dt cos7 cos5 + C. 7 5

4 ( sin cos 3 sin cos 3 (sin 5 sin sin cos 3 ( cos 5 cos + C. 5 ( t dt ( ( 4 /3 ( dt t log + C. t ( 4 /3 t 3 4 t 3 + 4 3t dt ( 4 /3 (t 3 + 4 t 3t dt (3t 6 + t 3 dt (3 + 6 + 3( 47/3 7 + 3( 4 4/3 + C. t + 3 + + 6 + t t( + 3 + ( + 3 + 6 +, t( + 3 t. + 3 t t, t + dt. t + 6 + t ( + 3 t + t

5 + 6 + t dt log t +C log +3+ + 6 + +C. (4 7 + 3 ( (3 + ( 3 7 + 3 ( (3 + ( 3 a + b 3 + + c 3 a(3 + ( 3 + b( ( 3 + c( (3 + (3a + b + 6c + ( 7a 7b + c + ( 6a + 3b c 3a + b + 6c 7, 7a 7b + c, 6a + 3b c 3. a, b, c 6 35 7 5 7 + 3 ( (3 + ( 3 (5 35 7 3 + + 6 5 3 log 7 log 3 + + 6 log 3 + C. 5 + 3 4 + 3 4 a + 4 + b a( + b( + 4 (a + b + 4b a a + b, 4b a a 3, b + 3 4 3 + 4 3 log +4 log +C.

(6 3 8 ( + 3( 4 + 5 3 8 ( + 3( 4 + 5 7 7 + 6 ( + 3( 4 + 5 a + 3 b + c 4 + 5 3 (a + b + (3b + c 4a + 5a + 3c a + b 7, 4a + 3b + c 7, 5a + 3c 6. a 5, b 4, c 4 3 3 3 6 ( 4 4 4 3 4 + 5 6 log 4+5 + 68 3 4 + 5. 4 + 5 ( + dt (t, dt + t Arctan( + C 3 8 5 4 log + 3 ( + 3( 4 + 5 3 6 log 4 + 5 68 Arctan( + C. 3 (7 e e e e e e + C. (8 sin sin cos cos + sin cos + +C. 4

7 (9 3 log 3 log 4 3 4 log 4 ( Arctan 4 4 Arctan Arctan ( cos sin log 4 6 + C. + Arctan + Arctan + C. cos sin ( 3 cos3 cos sin cos 3 3 cos3 +. 3 cos 3 ( sin cos cos 3 sin 3 sin3 + C. cos sin cos3 + sin 3 3 ( 5 3 + 4 sin3 9 + C. y 3 + 4 3 y 4, 3 dy 5 3 + 4 (y 4 y dy 3 ( 3 5 y5/ 8 3 y3/ + C ( (3 + 4 3 5 5 8 (3 + 4 3 3 + C.

8. ( + 4 + 4 ( [ ] + Arctan π + π 4 π 4. π ( cos 4 π cos 4 [ sin cos 3 ] π + 3 π 3 3 4 π π (cos cos 4 3 4 + cos 3π 8. cos sin π cos (3 4 log 4 [ log log ] 4 4 [ 8 log 4 4 ] 4 8 log 4 5 4. (4 (5 8 π/3 /3 cos 8 /3 [ ] 8 3 4 4/3 45 4. y / dy y [, π/6] π/3 cos π/6 cos ydy.

9 4 (6 + 3 4 4 + 3 4 ( ( + 3( 4 + 3 [ ] log 4 4 + 3 4 (log 3 7 + log 5 4 log 5 7. (7 Arctan (8 e Arctan [Arctan] + log (9 ( 3 e ( /3 3 ( /3 π 4 [ log( + ] e log [ log ] e. ( /3 t /3 dt 3 /3. t dt ( /3 t /3 dt [ ] 3 4 t/3 3 4. π 4 log.

( 3 9, t 9 dt 3 9 9 t 3. 3. ( ( 6 + 6 + 3 N N 6 + 3 [ ] N N N / N log e log < e log e N log e log e e N e e e N e. (3 N log log t et 3 dt, dt, et dt

N 3 log log N log 3 t e t et dt log N log 3 (4 3 log N 3 N log 3 log N log 3. [ ] N N. N log N 3 4. (a n [ cos cos 4 cos n ] cos k n n n n cos n n + cos cos 4 cos n cos n k k n+ (k n cos n+

(k cos cos n n+ [ ( (k cos + [ n cos (4k n+ + cos n+ ( (k + cos ] n ] n+ (4k 3 n+ k k n n n cos n k n cos n k (k n cos (k n+. n+ (b n [, ] n f(t cos t cos t t lim n n n k cos (k n+ (c cos t dt. lim cos n cos 4 cos sin n [ sin t cos t dt ] t t sin lim n cos cos 4 cos n sin.

3 (d π + (Viète + + + f( n f (n+ ( f(f n ( f n ( f(f( f(. }{{} n cos π n+ + cos π n cos cos +. f (cos π ( f (n cos π. n 4 (c 5. f( Laplace Lf(s f(e s Laplace (a α > α Laplace Γ(α + /s α+ s > α > α e (, L( α e s α. s > v s dv s L( α v α s α e v s dv Γ(α + s α.

4 (b f( e α Laplace /(s α s > α L(e α e s+α s > α e (s α. e (s α s α lim [ ] e (s α N N s α (c f( sin(α Laplace α/(s + α s > s > N e s sin α [ ] N s e s sin α + α N e s cos α s s e sn sin αn α s [ e s cos α ] N + α s N α s e s sin α [ s e sn sin αn + α s α s e sn cos αn N e s sin α α α + s. (d f( cos(α Laplace s ].

5 N e s cos α [ ] N s e s cos α α N e s sin α s s ( e sn cos αn + α s [ e s sin α ] N N α s e s cos α s s + α ( e sn cos αn α s e sn sin αn. s s +α s > s (N kπ, kπ + π, k

6 6. a > f(, ydy ( (a f(, y dy (b f(, ydy {(, y ;, y } (, y y y y y. ( f(, ydy f(, y dy. ( f(, y dy f(, ydy {(, y;, y } (, y y y y y y y y y

7 (c π 4 ( a cos θ f(, ydy f(r, θ dr dθ ( y f(, y f(r, θdrdθ { (r, θ; θ π 4, r a } cos θ dy. (r, θ a a a r r cos θ r a r a/ cos θ θ π/4 a r a r θ θ π a cos θ > r 4 cos θ cos θ a r, θ Arccosa r. (d a f(r, θdrdθ ( π/4 f(r, θdθ dr + ( F (, ydy a F (, ydy a ( π/4 f(r, θdθ dr. Arccos(a/r { (, y;, y }. y y y, y y y

8 { (, y ; y, y y }. (e ( 6 y y F (, ydy F (, y dy ( y F (, y dy. y F (, ydy { (, y ; y, y 6 y } y 6 y y y 6 y y 6 y 6, y min{, 6, } 6 (3 + ( 4 6 min{, 6, }. 4 6 6 6 min{, 6, } 6. F (, ydy ( 4 F (, ydy + 6 ( 6 4 F (, ydy.

9 (f ( 3y F (, y dy F (, ydy y { (, y ; y, y } 3y y y 6 y 3y y 3y, y 3 y y (, y y 3 6 y 3 F (, ydy ( 6 ( F (, ydy + F (, y dy. /3 7. (a ( ye y dy { +y, y 3} /3 u + y, v y (u + v/, y (u v/ (, y (u, v (b ( ue v dudv { u, v 3} 3 ( udu zdydz { +y +z, z } e v dv 4 (e3.

r cos θ, y r sin θ, z z r, θ, z z + r, z, θ π (, y, z (r, θ, z cos θ r sin θ sin θ r cos θ r zrdrdθdz {z +r, z, θ π} 4π zrdzdr. {z +r, z } (c (d z ρ cos ϕ, r ρ sin ϕ π ϕ π, ρ 4π zrdzdr 4π {z +r, r,z } π π/ π/ ρ 3 cos ϕ sin ϕdρdϕ sin ϕ dϕ π. (3 y dy, {, y } ( (3 ydy ( 3 6 5. ( + y dy, {, y }

(e ( ( + ydy [y + y ( 3 + 4 4 + 7. ] ye y3 dy {, y } (f ( ye y3 dy {, y } { y, y} ( y y e y3 dy ye y3 dy ye y3 dy ye y3 dy y dy {, y, + y 4} [ ] 3 ey3 e 3. r cos θ, y r sin θ {(r, θ ; r, θ π} r π/ ( y dy r 3 sin θdr dθ {,y, +y 4} 4 4 π/ cos θ dθ [ θ sin θ 4 ] π/ π.

8. a > (a V {(, y, z ;, y, z + y } ( ( +y V dz dy ( [ y + y3 3 ( + y dy ] ( 3 + 83 3 7 6. (b V {(, y, z ; + y a, z y} z V { +y a, z y} { +y a, y } π a dydz ydy r sin θdrdθ a3 3 [ cos θ]π a3 3. (c V {(, y, z ; + y a, + y + z a } ( (r, θ, z (r, θ, z r { π θ π, r a cos θ, a r z a r }

3 V π/ ( a cos θ π/ π/ π/ r a r dr dθ [ 3 ] a cos θ (a r 3/ dθ 3 (a3 a 3 sin 3 θdθ ( π 3 8 a 3. 9 (d z y, + y, z V. {(, y, z ;, y, + y, z y} V {, y, +y } ( ( y ydy ( [ ] 3 3 4 4 4. dz dy ( (e z 4a + y a V. { (, y, z ; + y a,, a z a } ( V {, +y a} 4 ady. {, +y a} dz { a z a} dy

4 r, { (r θ ; r a cos θ, π θ π } 4 ar cos θ 4 ady {, +y a} π/ ( a cos θ 4 4 8a3 5 π/ π/ π/ 3a3 5. 5 a3 cos 3 θdθ r 3/ a cos θdr dθ ( t dt (t sin θ