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 )

Similar documents
応力とひずみ.ppt

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

II 2 II


.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 =,

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

B line of mgnetic induction AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B = B ds 2π A B P P O s s Q PQ R QP AB θ 0 <θ<π

高等学校学習指導要領

高等学校学習指導要領

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 +

PSCHG000.PS


Gmech08.dvi

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


1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π

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

50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq

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,

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)) ( )

() 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)

Gmech08.dvi

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

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

( : 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)

pdf

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {

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

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :

70 : 20 : A B (20 ) (30 ) 50 1

genron-3

mugensho.dvi

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

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

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

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

untitled

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

18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (

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 ) = (,

8.3 ( ) Intrinsic ( ) (1 ) V v i V {e 1,..., e n } V v V v = v 1 e v n e n = v i e i V V V V w i V {f 1,..., f n } V w 1

さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n

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

m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a

KENZOU

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =


(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

difgeo1.dvi


Quz Quz

, 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

(1) 1 y = 2 = = b (2) 2 y = 2 = 2 = 2 + h B h h h< h 2 h

0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()]

2 p T, Q

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

Untitled


( ) 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) (

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

.1 z = e x +xy y z y 1 1 x 0 1 z x y α β γ z = αx + βy + γ (.1) ax + by + cz = d (.1') a, b, c, d x-y-z (a, b, c). x-y-z 3 (0,

(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

( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

) 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)

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

Gmech08.dvi

(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

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト

all.dvi

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

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, 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 =

取扱説明書[N-02B]

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

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

高等学校学習指導要領解説　数学編

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4

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

B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:

数学Ⅲ立体アプローチ.pdf

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

曲面のパラメタ表示と接線ベクトル


notekiso1_09.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 +

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

, 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,,,,

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

24.15章.微分方程式

[ ] Table

2 Chapter 4 (f4a). 2. (f4cone) ( θ) () g M. 2. (f4b) T M L P a θ (f4eki) ρ H A a g. v ( ) 2. H(t) ( )

( )

A

36 3 D f(z) D z f(z) z Taylor z D C f(z) z C C f (z) C f(z) f (z) f(z) D C D D z C C 3.: f(z) 3. f (z) f 2 (z) D D D D D f (z) f 2 (z) D D f (z) f 2 (

Transcription:

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 1 P Q PQ PQ A B A B cos A B A B A B A B B A A( B C) A B + AC A A A A A, B A, B

A B 0 A, B A, A, A B, B, B x y z x y z A B Ax Bx AyBy Az Bz A B A B A B B A cos A B C A A A B C C A B sin A B A B A B AB A( B + C) A B + AC A B = B A 3

A B A B = 0 A, B A, A, A B, B, B A B A B A B i y z z y A B A B z x x z A B A B x y y x j k x y z x y z A, B A, A, A B, B, B x y z x y z A B i j k A A A x y z B B B x y z A B C A B C A B C A( BC) A B C 4

A ( B C ) A( B C) ( AC) B ( A B) C ( A B) C ( AC) B ( B C) A A( B C) ( A B) C 5

t f ( t) t f ( t) df f lim dt t0 t f f t t f t f t s( t) s s( t) s( t1) t t t 1 t ds s lim t dt t0 t f ( t) df / dt t f ( t) Taylo t df t f ( t t) f t t dt A( t) d dt A A A A t t A t lim t0 t A( t) ( x, y, z) A, A, A x y z 6

da da da da dt dt dt dt x y z i j k ( x, y) t ( t) xi yj t t ( t t) ( x x) i ( y y) j u d u dt lim t0 t t t t / t x y s t t t t s s x y ( t) x( t) i y( t) j z( t) k u( t) dx t dy t dz t u( t) i j k dt dt dt 7

dx dy dz u u dt dt dt du u α lim dt t0 t u ut t u t du m F dt [1] d( fa) df da A f dt dt dt (, ) ( t) t e e e cosi sin j 8

e sini cos j e, e de d de d e, e dt dt dt dt i, j e e u d d u e e dt dt d u d d 1 d d a e e dt dt dt dt dt d d m F dt dt 1 d d dt dt m F F, F [] d( A B) da db B A dt dt dt 9

u( t) u du dt d ( uu ) 0 dt [3] d( A B) da db B A dt dt dt L p p p mu d L dt du N N m F dt // F dl dt 0 L m k 10

Quiz T ( x, y, z) u( x, y, z) T ( x, y, z, t) S W h S f e h W f S e h S n 11

S S Sn S n h S W f h S W f ( hn 0 ) ( hn 0 ) f kg/(m s) f f v v 1

f ( x, y, z) v( x, y, z) f ( x, y, z) M f f S f S M f 13

Quiz ( x, y) f ( x, y) x f / x f x,, f x x y f x y lim x 0 x ( x, y) f ( x, y) y f / y f y,, f x y y f x y lim y 0 y f ( x, y) df x, y dx, dy f ( x, y) df f ( x dx, y dy) f ( x, y) f f df dx dy x y i + j k x y z ( x, y, z) ( x, y, z) i + j k x y z 14

x y z 1 1 xi yj zk A( x, y, z) A A Az x y z x y A + h 6 hi Si i1 ( h) V hi S i 6 i1 15

Q Q h V t ( x, y, z, t) v( x, y, z, t) ( x, y, z) t v SM S M A( x, y, z) ( x, y, z) A ( ) A v v 3 16

0 3 + x y z A( x, y, z) A A A A A A y z z x x y z y x z y x A i j k i j k A x y z A A A x y z ω v ( x, y) z ω k 17

v( x, y) vxi vy j, vx y vy x v k v v A( x, y, z) ( x, y, z) A ( A) A+ A 0 ( ) 0 0 1 0 A A 18

A ( ) 0 F U F U E E B A B A A B ( A ) 0 B B A B A 19

Quiz C A A C d Ad C P Q A d Q Q d d Axdx Aydy Az dz C A A P P P Q (P) (Q) Q P d Q P C P Q C C A d Ad C C R P R C 1 R P C A d Ad A d C C1 C 0

a C (,, z) d ade e e sini + cos j A S A d S S ds ds ds n ds n ds h S h W f h d S s 1

f S f f ds v ds M f s S, v : ds (,, z) ds nds n k ds dd k : ds (,, z) ds nds n ds ddz a : ds (,, ) ds nds n a ds a sindd a

/ 3 E( x, y, z) ( x, y, z) K E ( x, y, z), xi yj zk x y z K const. a E( x, y, z) E d S 4 K S S V A S A ds diva dv V A S d S h h ( divh ) h S W 3

S S hds hdv V f f ( divf ) f S M S S f ds fdv V S S W ( ) 0 S 0 4

S ( ) W 0 0 ( ) 0 ( ) dv 1 V 0 V ( ) dv 0 V 0 V / 3 E( x, y, z) ( x, y, z) 1 E K xi yj zk, x y z K const. S E E d S 4 K S A C A C d A A C A A C d 5

A A( ot A) C S A ds C A d d C A S S z ω k ( x, y, z) v v v v ω v =, v, ω 6

E E E 0 C E d 0 C B A B A B 0 S A S S d 0 7

Quiz ( u 1, u, u 3 ) u1( x, y, z) C1, u( x, y, z) C, u3( x, y, z) C3 ( C1, C, C3 const.) ( x, y) x y C1 1 y ( x, y) tan C x z z C3 ( u 1, u, u 3 ) e 1, e e 3 e e i j 1i j 0i j e1, e e 3 f ( u, u, u ) f e f e f e 1 3 1 1 3 3 f 1, f f 3 f f e ( j 1,,3) e 1, e e 3 u 1, u h1, h h 3 j j u3 8

e 1 h1 u1, hu e, e 3 h3u3 du 1, du, du 3 ds 1, ds, ds 3 h 1, h h 3 ds h du ds h du ds 3 h 3 du 3 1 1 1 ds 1, ds, ds 3 du 1, du, du 3 h 1, h h 3 dv ds1ds ds3 h1 hh3 du1dudu3 h 1, h, h 1 dv z dddz h 1, h, h sin dv sinddd e 1, e e 3 h 1, h h 3 f ( u 1, u, u 3 ) 9

1 f 1 f 1 f f e e e h u h u h u 1 3 1 1 3 3 f 1 f f f e e e z z f 1 f 1 f f e e e sin f ( u 1, u, u 3 ) f 1, f, f 3 1 f h h h u u u h h f h h f h h f 3 1 3 1 1 3 1 3 1 3 1 1 f f f = f z z 1 1 1 f f ( f ) (sin ) f sin sin f ( u 1, u, u 3 ) f 1, f, f 3 30

h f h f e h h u u 1 3 3 f 3 3 h f h f e h h u u 1 1 3 3 3 1 3 1 h f h f e 3 h h u u 1 1 1 1 1 f z f f f z 1 f f f e e e z z z sin f f e f sin e f sin e sin f f f f ( u 1, u, u 3 ) 1 hh3 f h3h1 f h1h f f h1h h3 u1 h1 u1 u h u u3 h3 u3 1 f 1 f f f z 1 1 1 f f sin f f sin sin 31

2024 © docsplayer.net プライバシー | 利用規約 | フィードバック