1 45

16 3 4 1

1 45

I 6 1 6 8.1........................... 8............................ 8 3 10 3.1 (a )............... 10 3. (a ). 11 4 1 4.1....................... 1 4............................ 13 4.3............................... 14 5 14 6 15 6.1............................ 15 6............................ 16 7 17 8 18 8.1............................ 18 8.1.1........................ 18 8.1........................ 19 8.......................... 1 9 3 10 7 11 9 1 35 3

13 38 13.1......................... 38 13........................ 39 13..1.............. 39 13........................ 40 13.3............................ 40 13.3.1..................... 41 14 4 15 45 15.1....................... 45 15.......................... 45 16 47 17 49 17.1......................... 49 17.................... 49 17..1................... 49 17......................... 49 17.3...................... 50 18 51 19 58 19.1................................ 58 19.1.1........................... 58 19.1........................ 58 19........................... 58 19..1..................... 58 19........................ 58 0 60 II 63 1 63 1.1................................ 63 4

69.1.............................. 69................................. 73 5

I 1 dq = η T θ s (1) l T t θ η l s dq = cmdt () c m (1) () η T θ s = cmdt l dt T θ = ηs (3) lcm T 0 (3) dt T θ = ηs lcm log T θ = ηs lcm t + c 6

( T θ = ± exp ηs ) ( lcm t + c = ± exp(c) exp ηs ) lcm t A = ± exp(c) ( T θ = A exp ηs ) lcm t t=0 T = T 0 T 0 θ = A ( T = θ + (T 0 θ) exp ηs ) lcm t (4) (4) cm(t θ D ) = c 1 m 1 (θ D T 1 ) (5) θ D t T 1 c 1 m 1 (5) θ D = c 1 m 1 cm T 1 + T (6) cm + c 1 m 1 cm + c 1 m 1 (4) (6) c 1 m 1 cm [ ( θ D = T 1 + θ + (T 0 θ) exp ηs )] cm + c 1 m 1 cm + c 1 m 1 lcm t (7) cm(t 0 θ 0 ) = c 1 m 1 (θ 0 T 1 ) (8) θ 0 (8) θ 0 θ 0 = c 1 m 1 cm T 1 + T 0 cm + c 1 m 1 cm + c 1 m 1 (4) cm cm + c 1 m 1 θ T [ ] c 1 m 1 cm θ T = θ + T 1 + T 0 θ cm + c 1 m 1 cm + c 1 m 1 [ exp ] ηs l(cm + c 1 m 1 ) t (9) 7

(7) (9) c 1 = 3.9 10 3 J/kg K c = 4.1 10 3 J/kg K η = 0.6V/m K m 1 = 10 kg m = 8 10 kg T 1 = 0 C θ = 0 C T 0 = 80 C s = 11 10 3 m l = 10 3.1 10cm 0cm 160 C 30 C 95 C. k 1.7 10 4 1m 4 T x Q = kf (x) dx = const (10) F (x) k Q F (x) = πxl l (cm) (10) Q = kf (x) dx 8

T = T = dt/dx dt = = kf (x)t Q kπxl Q kπl 1 x dx T = Q log x + C kπl T = 160 C x = 10cm,T = 30 C x = 10cm { 160 = Q kπl log 10 + C 30 = Q kπl log 0 + C 130 = Q log 10 + Q log 0 kπl kπl = Q log 10 + Q log 0 kπl Q = 130kπl log C = 160 + Q log 10 kπl 130kπl log = 160 + log 10 kπl log 10 = 160 + 130 log 130kπl log log 10 T = log x + 160 + 130 kπl log = 130 log x 10 + 160 + 130log log log log 10 log x = 160 + 130 log log 10 log x T = 160 + 130 log 1m 4 4 60 60Q = 7685J 9

3 dx = k(x a) (11) x t a k (11) a a 3.1 (a ) (11) t = 0 x(0) = x 0 dx x a = k dx x a = k log x a = kt + c x a = ± exp(c) exp( kt) A = ± exp(c) A = x 0 a x(t) = (x 0 a) exp( kt) + a (1) x 0 t = 0 x 31 C 9 C 10

37 C a = 1 C t = 0 (1) 9 = (31 1) exp( k) + 1 exp( k) = k = log 4 5 9 1 31 1... k = log 5 = log 1.5 = 0.314 (13) 4 k (13) x = 37 (1) 37 = (31 1) exp( 0.314t) + 1 37 1 exp( 0.314t) = 31 1 0.314t = log 1.6 t = 1 log 1.6 =.10630 0.314 3. (a ) dx + kx = ka(t) (14) a(t) t (14) x(t) = B(t) exp( kt) x (t) = B (t) exp( kt) kb(t) exp( kt) B (t) exp( kt) kb(t) exp( kt) + kb(t) exp( kt) = ka(t) B (t) exp( kt) = ka(t) B (t) = ka(t) exp(kt) B(t) = k a(t) exp(kt) + C (15) 11

t = 0 a(t) = 1 dx + kx = kt (16) (15) (16) x = D(t) exp( kt) (15) D(t) = k t exp(kt) + C D(t) = t exp(kt) + 1 k exp(kt) + C d(t) = ( t + 1 k ) exp(kt) + C... x(t) = t + 1 + C exp( kt) (17) k (17) t = 0 x 0 = 30 C x = (30 1 k ) exp( kt) t + 1 k 4 4.1 t = 0 hm x S(x) s x v v = k gx g=9,8m/s :, k: v s dx - sv = S(x)dx sk gx = S(x)dx dx gx = sk S(x) (18) 1

4. 6m m 1/1m t s = 1/144π S(x) = 4π k=0,6 (18) dx 144 = π 0, 6 9, 8 x 4π 17, 15 = x 1 dx/ = x x = x 17, 15 x 1 = x 17, 15 dx 1 = x 17, 15 x 1/ = 1 17, 15 + C x t 1 x(t) = 17, 15 t + C t = 0 x(t) = 6 6 = C 1 x = 17, 15 t + 6 1 x = 434, 304 t + 6 ( 1 x = 434, 304 t + 6 x = 0 ) t = 434, 304 6 = 1038, 31 18 13

4.3 (18) dx/ = a a = sk gx S(x) S(x) = sk gx a πr = sk gx a x = aπr sk g (19) a (19) x = Cr 4 (0) (0) x 5 B t N x B B N/γ dx = kx(n x) (1) t = 0 x = N/γ (1) k (1) dx x(n x) = k 14

( 1 1 N x 1 ) dx = k N x A = ± exp(nc) x = 1 (log x log N x ) = kt + C N log x N x = Nkt + NC x = ± exp(nc) exp(nkt) N x x N x = A exp(nkt) x = (N x)a exp(nkt) x = NA exp(nkt) A exp(nkt)x [1 + A exp(nkt)]x = NA exp(nkt) NA exp(nkt) 1 + A exp(nkt) = N 1 + exp( Nkt)/A x(0) = N/γ N γ = N 1 + 1/A x =... 1 A = γ 1 N 1 + (γ 1) exp( Nkt) 6 6.1 15

6. p p 1kg 1 t 1kg p(t) q = 4p p + 39 s = 44p + p 1 p = dp/ 4p p + 39 = 44p + p 1 40p 4p + 40 = 0 10p + p 10 = 0 10 dp = p + 10 1 10 dp = p 10 1 10 p 10 dp = 10 log p 10 = t + C log p 10 = t 10 C ( 10 p 10 = Cexp t ) 10 ( p = Cexp t ) + 10 10 16

t = 0 p = 1 1 = C + 10 C = 9 ( p = 9exp t ) + 10 10 7 A + B 3C A B C A B 10 0 C 6 0 t C t() C ( ) x A B x/3 x/3 A 10 x/3 B 0 x/3 dx = K dx ( 10 x 3 ) ( 0 x ) 3 = k(15 x)(60 x) () k (k = k/9) t = 0 x = 0 dx = k(15 x)(60 x) x(0) = 0 x(1/3) = 6 x(0) = 0 () 1 (15 x)(60 x) = k ( 1 1 45 60 x 1 ) dx = k 15 x 17

α = ±e C log 60 x log 15 x = 45kt + C 60 x 15 x = ±ec e 45kt 60 0 15 0 = α... α = 4 60 x = 4 exp(45kt) (3) 15 x t = 1/3 x = 6 (3) exp(15k) = 3/ 60 x 15 x = 4[exp(15k)]3t = 4(3/) 3t 1 (3/) 3t x = 15 (1/4) (3/) 3t t C A B C x t t (/3) 3t = 4 x x t t(t 0) 8 8.1 8.1.1 x(t) t A B 18

x dx = A B A B x A = ax, B = bx a b dx = ax bx = (a b)x t = t 0 x = x 0 dx (a b)x = dx (a b)x = 1 log x a b = t + C log x = (a b)(t C) x = ±exp(a b)(t C) x = Cexp(a b)t x 0 C = exp(a b)t 0 x = x 0 exp(a b)t exp(a b)t 0 x = x 0 exp(a b)(t t 0 ) (4) a > b x t a < b t x 0 8.1. (4) dx/ = f(x) f(x) f(x) = ax bx a > 0, b > 0 19

1 a dx = x(a bx) dx = x(a bx) ( ) 1 ax + b dx = a(a bx) 1 x dx + 1 b a a bx dx = t + C 1 a log x 1 log a bx = t + C a log x log a bx = a(t + C) log x a bx = a(t + C) x a bx = Cexp(at) C = x 0 (a bx 0 )exp(at 0 ) x a bx = x 0 (a bx 0 )exp(at 0 ) exp(at) = x 0 a bx 0 exp[(t t 0 )a] x = (a bx) exp{(t t 0 )a} a bx 0 ax 0 x = exp{(t t 0 )a} bxx 0 exp{(t t 0 )a} a bx 0 a bx ( 0 x 1 + bx ) 0 ax 0 exp{a(t t 0 )} = exp{a(t t 0 )} a bx 0 a bx 0 x(t) = x = x = x = x = x 0 ax 0 a bx 0 exp{a(t t 0 )} 1 + bx0 a bx 0 exp{a(t t 0 )} ax 0 exp{a(t t 0 )} a bx 0 + bx 0 exp{a(t t 0 )} ax 0 (a bx 0 )exp{ a(t t 0 )} + bx 0 x 0 a/b x 0 + (a/b x 0 )exp{ a(t t 0 )} x 0 a/b x 0 + (a/b x 0 )exp{ a(t t 0 )} (5) t x(t) a/b a/b > x 0 a/b < x 0 (5) 0

8. dx i = f i(x 1,, x n ) i = 1,,, n Vito Volterra,1860-1940 x y a b c d dx = ax + bxy (6) dy = cx dxy (7) (6) bxy (7) dxy u(τ) = d c x(t), v(τ) = b a y(t), τ = ct, α = a c (6) (7) du dτ = αu(v 1), dv dτ = v(1 u) (8) 1

du(τ) dτ dv(τ) dτ τ = τ 0 = d(d/cx(t)) = d(d/c)x(t) dτ = d c dx(t) d(τ/c) dτ = d c dx(t) = d c [ ax(t) + bx(t)y(t)] = a c d c x(t) + a c d c x(t) b a y(t) = αu(v 1) = d(b/ay(t)) dτ = d(b/ay(t)) dτ = b a dy(t) d(τ/c) dτ = b ac dy(t) = b a [cx(t) dx(t) y(t)] = b a x(t) a c x(t) b a y(t) = v(τ) u(τ)v(τ) = v(τ)(1 u(τ)) u(τ 0 ) = u 0, v(τ 0 ) = v 0 (9) u v (8) du du dv = dτ dv dτ = αu(v 1) v(1 u) 1 u du α(v 1) = u dv v 1 u α(v 1) u du = dv v ( ) 1 ( u 1 du = α α ) dv v log u u = αv α log v + C αv + u α log v log u = C αv + u log v α u = C αv 0 + u 0 log v α 0 u 0 = C αv + u log v α u = αv 0 + u 0 log v α 0 u 0 = H

H (9) α u v u 0 v 0 u 0 > 1, v 0 < 1 (8) u v v 1 v τ v v = 1 u = 0 u u v u v 3 1 3 9 N t t S(t) t I(t) t R(t) S(t) + I(t) + R(t) = N (30) I I(t) I { ds = αs (I(t) > I ) 0 (I(t) I ) (31) 3

{ di = αs βi (I(t) > I ) βi (I(t) I ) α,β dr/ = βi (3) t = 0 R(0) = 0 I(0) S(t) = S(0) = N I(0) α = β I(0) I ds/ = 0 R(0) = 0 (3) di = αi(t) I(t) = I(0) exp( αt) I(t) = R(t) R(t) = N S(t) I(t) = I(0)[1 exp( αt)] I(0) exp( αt) = I(0)[1 exp( αt)] = log t = log α. I(0) > I 0 t < T t I(t) > I I(t) 4

[0, T ] t 0 t < T S(t) = S(0) exp( αt) di(t) + αi(t) = αs(0) exp( αt) (33) I(t) exp(αt) d I(t) exp(αt) = αs(0) I(t) exp(αt) = αs(0)t + C (33) I(t) = C exp( αt) + αs(0)t exp( αt) (34) t = 0 C = I(0) (34) I(t) = [I(0) + αs(0)t] exp( αt) 0 t < T (35) R(t) = N S(t) I(t) R(t) = S(0)[1 exp( αt) αt exp( αt)] + I(0)[1 exp( αt)] T t m ax T (35) t = T I I(T ) = I I = [I(0) + αs(0)t ] exp( αt ) (36) S(t) = lim t S(t) = S( ) S(T ) = S( ) = S(0) exp( αt ) exp( αt ) = S( ) S(0) 5

αt = log S( ) S(0) T = 1 α log S(0) S( ) (37) S( ) (37) (37) T (36) [ I = I(0) + S(0) log S(0) ] S( ) S( ) S(0) I S( ) = I(0) S(0) + log S(0) S( ) I I(0) + log S( ) = + log S(0) (38) S( ) S(0) (38) I S( ) (35) (35) t di = αs(0) exp( αt) αi(0) exp( αt) α S(0)t exp( αt) = α exp( αt)[s(0) I(0) αs(0)t] = 0 S(0) I(0) αs(0)t = 0 αs(0)t = I(0) S(0) S(0) I(0) t = αs(0) = 1 [ 1 I(0) ] α S(0) t I (35) I max = S(0) exp [1 I(0)/S(0)] = S(t max ) t max t > T I(t) = I exp[ α(t T )] 6

10 1 1 3 r θ O O O O O x x v v x v = 3 x v x v = 3 + x v 1 3 O ( ) v O ( v) v r v t ( x ) (t) y (t) x = r cos θ x (t) = dr dθ cos θ r sin y = r sin θ y (t) = dr dθ sin θ + r cos = dr ( cos θ sin θ ) + r dθ ( ) sin θ cos θ 7

v r = v v r = dr v T = r dθ v T = (v) v = 3v = dr/v dr = v, r dθ = 3v r dθ = 3v dr/v 3 dθ = r dr dθ = 1 3 r dr θ + C 1 = 3 log r r = C ( ) 1 θ exp 3 3 C 1 3 = C ( ) θ r = C exp 3 O r O x θ = 0 r = 1 C = 1 r = exp(θ/ 3) θ = π r = 3 ( ) 3 3 = C exp 3 ( ) π C = 3exp 3 6 r = 3exp{(θ + π)/ 3} r(t) = vt + 3 θ(t) = log (vt + 3) 3 π 8

t r(t) = vt (39) θ(t) = log(vt) 3 π (40) (40)= α t log (vt) 3 π = α 0 α < π (α = 0 π ) log (vt) = (α + π) 3 vt = exp(α + π) 3 t = 1 v exp(α + π) 3 t = 1 v exp(α + π) 3 11 x y t x(t) y(t) x(t) y(t) x(t) y(t) () x(t) y(t) t 9

x OLR y CLR x RR x(t) dx(t) y(t) dy(t) = (OLR + CLR) + RR (41) = (OLR + CLR ) + RR (4) OLR CLR RR a b c d g h (x, y) P (t) Q(t) x y x 0 y 0 x y dx(t) = ax(t) by(t) + P (t) dy(t) = cx(t) dy(t) + Q(t) A ( A ) dx(t) dy(t) = ax(t) gx(t)y(t) + P (t) = dx(t) hx(t)y(t) + Q(t) B C dx(t) dy(t) = ax(t) gx(t)y(t) + P (t) = cx(t) dy(t) + Q(t) 30

(41)(4) ax(t) dy(t) 1 dx x = a 1 dy y = d by(t) cx(t) gx(t)y(t) hx(t)y(t) A x bx(t) b y 1 dx y = b b y cx(t) b c b = r y p y c = r x p x (43) r y r x y x p y p x y x A B x(t) R x r y(t)n x gx(t)y(t) y g (43) b g r y y y A ry x A x 31

A ry g h g = r y A ry A x h = r x A rx A y (44) A (A dx(t) = by(t) dy(t) b t 0 dy(t) dx(t) = cx(t) by(t) y(t)dy(t) = c t 0 = cx(t) (45) x(t)dx(t) [ ] t [ ] t 1 1 b y (t) = c 0 x (t) 0 b[y (t) y (0)] = c[x (t) x (0)] (46) b[y (t) y 0] = c[x (t) x 0] (47) (45) by0 cx 0 K (47) by cx = K (48) K = 0 (45) (45) y x x y y x K > 0 y 3

y x y(t) = K/b y K by0 > c 0 (49) (43) (49) ( y0 x 0 ) > r x r y p x p y (50) (50) y 0 /x 0 y 0 /x 0 y 4 (48) (45) d x(t) bcx(t) = 0 (51) (51) x 1 = cos bct x = sin bct bcβ d x 1 (t) bcx 1 (t) = 0 d x (t) bcx (t) = 0 w(cos βt, sin βt) = β 0 x(t) = C 1 cos βt + C sin βt x(0) = x 0 dx(t) t=0 = by 0 dx(t) x 0 = C 1 = x 0 β sin βt + C β cos βt C = by 0 β by 0 = C β = by 0 b = bc c y 0 33

b/c = γ C = γy 0 x(t) = x 0 cos βt + γy 0 sin βt (5) y(t) = y 0 cos βt + x 0 γ sin βt (53) y y 0 x 0 γy 0 x 0 dx = gxy, dy (54) dy dx = h g = hxy (54) g[y(t) y 0 ] = h[x(t) x 0 ] (55) (54) (55) gy(t) hx(t) = gy 0 hx 0 = L gy hx = L L y L x y gy 0 hx 0 y 0 x 0 > h g y y 0 x 0 > r xa rx A x r y A ry A y (56) 34

y y 0 /x 0 A x /A y (56) A y y 0 A x x 0 > r xa rx r y A ry A y y 0 A x x 0 dx = gxy, dy = cx (57) x(t) y(t) (57) dy dx = c gy gydy = cdx 1 g[y (t) y 0 ] = c[x(t) x 0 ] gy (t) cx(t) = gy 0 cx 0 = M gy (t) cx(t) = M (58) M = gy 0 cx(t) (57) M M y 0 /x 0 1 M (y 0 /x 0 ) (c/g)x 0 1 ( y0 x 0 ) > r x r y A x p x A ry 1 x 1 m l m 35

α mv = mg(l cos θ l cos α) (59) v g s = lθ v = ds/ = l(dθ/) (59) ( ) l dθ = g(cos θ cos α) (60) ( ) dθ = g (cos θ cos α) l dθ g = ± (cos θ cos α) l θ t dθ/ < 0 (60) dθ g = (cos θ cos α) l l dθ = g cos θ cos α T θ(t) = θ(t + T ) T 4 = l g l T = 4 g α 0 0 α dθ cos θ cos α dθ (61) cos θ cos α α α (61) cos θ = 1 sin θ cos α = 1 sin α l T = g α 0 dθ l = sin (α/) sin (θ/) g α 0 dθ k sin (θ/) (6) 36

k = sin(α/) θ ϕ sin(θ/) = k sin ϕ θ 0 α ϕ 0 π/ 1 cos θ dθ = k cos ϕdϕ... dθ = k cos ϕdϕ cos(θ/) (6) l π/ k cos ϕdϕ/ cos(θ/) T = g 0 k sin (θ/) = l g l = g l = 4 g π/ 0 π/ 0 π/ 0 k cos ϕdϕ/ cos(θ/) k k sin ϕ k cos ϕdϕ/ cos(θ/) k cos ϕ dϕ cos(θ/) cos(θ/) = 1 k sin ϕ l T = 4 g π/ 0 dϕ 1 k sin ϕ l T = 4 F (k, π/) g F (k, ψ) = ψ 0 dϕ 1 k sin ϕ E(k, ψ) = ψ 0 1 k sin ϕdϕ 37

(60) t d θ g + k sin θ = 0 k = l 13 13.1 17 Circular x r θ M x = OS = OP OS, y = MS = CP CN OP = MP = rθ, SP = MN = r sin θ, CP = r, CN = r cos θ y = r r cos θ x = r(θ sin θ), y = r(1 cos θ) (63) θ = arc cos r y r Oxy x = rarc cos r y r = rarc cos r y r r sin ry y ( arc cos r y ) r 38

cusp vertex πr base 13. lim θ π ry y sin θ = r dy dy dx = dθ r sin θ = dx r(1 cos θ) sin θ 1 cos θ dθ cos θ = lim θ π 0 sin θ = = lim θ π+0 cos θ sin θ = + 13..1 S S = π 0 S = πr 0 ydx dx = r(1 cos θ)dθ x θ 0 πr 0 π r(1 cos θ)r(1 cos θ)dθ π = r (1 cos θ + cos θ)dθ 0 = r π 0 ( 1 cos θ + ) 1 + cos θ dθ = r [ θ sin θ + 1 θ + 1 4 sin θ ] π = 3πr πr 0 39

13.. l l = π (dx ) ( ) dy + dθ 0 dθ dθ π = r (1 cos θ) + r sin θdθ = r = r 0 π 0 π 0 cos θdθ sin θ dθ 0θ π 0 θ 90 sin θ > = 0 l = r = 8r π 0 sin θ dθ [ = r cos θ π ] π 0 13.3 Christian Huygens,169-1695 1673 t = t 0 40

isochronous eqaul time tautochronous identical tautos 13.3.1 M K M (x 0, y 0 ) θ 0 N(θ) h (63) x = r(θ + sin θ), x 0 = r(θ 0 sin θ 0 ) y = r(1 cos θ), y 0 = r(1 cos θ 0 ) h = y y 0 = r(1 cos θ) r(1 cos θ 0 ) = r(cos θ 0 cos θ) v = gh = gr(cos θ 0 cos θ) s T s = θ θ 0 v = dt ds = gr(cos θ 0 cos θ) (dx ) + dθ dt =...ds = r sin ( ) dy θ dθ = r dθ ( ) θ dθ r sin (θ/)dθ gr(cos θ0 cos θ) θ 0 sin θ dθ T = r g r = g π 0 [ sin 1 r = g (sin 1 1) r = π g sin (θ/)dθ cos θ0 cos θ ] cos (θ0/) t cos (θ 0 /) 41

M K T r T = π g T θ 0 M M N K K M M 1 T 0 = 4π r/g 14 brachistochrone A B A B A B v 1 P B v A B T a + x T = b + (c x) + v 1 A B T dt/ T x v dt dx... = ( a + x b + (c x) + v 1 v ) x = v 1 a + x + x c v b + (c x) = 0 x v 1 a + x = c x v b + (c x) 4

sin α 1 v 1 = sin α v Snell sin α 1 / sin α = a (a ) A B A B sin α v a = a (64) v = gy (65) sin α = cos β = 1 sec β = 1 1 + tan β = 1 1 + (y ) (64)(65) y[1 + (y ) ] = C (66) (66) y = dy/dx (66) y[1 + (y ) ] = C ( ) dy y + y = C dx ( ) dy = C y dx y 43

dy/dx = tan β 0 < β < π tan β > 0... dy dx =... dx = dy dx > 0 ( C y ) 1/ y ( ) 1/ y dy C y ϕ ( ) 1/ y = tan ϕ C y y C y = tan ϕ... y = (C y) tan ϕ y = C tan ϕ 1 + tan ϕ = C sin ϕ cos ϕ 1 cos ϕ = C sin ϕ dy = C sin ϕ cos ϕdϕ dx = tan ϕdy = tan ϕc sin ϕ cos ϕdϕ = C sin ϕdϕ = C(1 cos ϕ)dϕ x = C (1 cos ϕ)dϕ = 1 C(ϕ sin ϕ) + C 1 ϕ = 0 x = y = 0 C 1 = 0 x = C (ϕ sin ϕ) 44

y = C sin ϕ = C (1 cos ϕ) C/ = r ϕ = θ 15 15.1 Carl F.Gauss, 1777-1855 m 0 n 0 m 0 > n 0 m 0 n 0 m 1 n 1 m 0 n 0 m 1 = m 0 + n 0, n 1 = m 0 n 0 m 1 n 1 m 0 n 0 m = m 1 + n 1, n = m 1 n 1 m k n k k = 0, 1,, 15. Carl W.Borchar,1817-1880 a m 0 n 0 a = f(m 0, n 0 ) f a a = f(m 1, n 1 ) m 0 n 0 k m 1, n 1, m, n, a k a m 0 n 0 a = m 0 f(1, n 0 /m 0 ) = m 1 f(1, n 1 /m 1 ) x = n 0 /m 0 x 1 = n 1 /m 1 y = 1/f(1, n 0 /m 0 ) y 1 = 1/f(1, n 1 /m 1 ) y = 1 f(1, n 0 /m 0 ) = m 0 a = m 0 m 1 /y 1 = y 1 m 0 m 1 = m 0 m 0 + n 0 y 1 = y 1 1 + x (67) 45

x 1 x x 1 = n 1 m 1 = m 0 + n 0 m 0 n 0 = m 0 + n 0 m 0 m 0 m 0 n 0 = m0 n 0 = 1 + n 0 /m 0 m 0 1 + x x = x 1 + x dx 1 dx (1 + x)/ x x = (1 + x) = (1 + x x)/ x (1 + x) = 1 x (1 + x) x = (x 1 x 3 1 )(1 + x) (x x 3 ) (67) dy dx = (1 + x) y 1 + 1 + x = (1 + x) y 1 + 1 + x dy 1 dx 1 dx 1 dx (x 1 x 1 3 )(1 + x ) (x x 3 ) = (1 + x) y 1 + (x 1 x 3 1 )((1 + x) dy 1 (x x 3 ) dx 1 (x x 3 ) dy 1 dx 1 (x x 3 ) dy dx = y 1 (1 + x) (x x3 ) + (x 1 x 3 1 ) + (x 1 x 3 1 )(1 + x) dy 1 dx 1 x(x 1) = 1 + x y 1 + (1 + x)(x 1 x 3 1 ) dy 1 dx 1 x [ d (x x 3 ) dy ] d[x(x 1)/(1 + x)] = y 1 dx dx dx [ d (x x 3 ) dy dx dx ] xy = a (y) = + x(x + 1) 1 + x dy 1 dx 1 dx 1 dx 1 x ( d (1 + x) [x 1 x 3 1 ] dy ) 1 x 1 y 1 x dx 1 dx 1 [ d (x x 3 ) dy ] xy = a (y) dx dx 1 x (1 + x) x 1 x 1 (1 + x 1 ) x 1 1 x (1 + x ) x 1 x n (1 + x n ) x n a (y n ) n 1 x n y a (y) = 0 (x x 3 ) d y dx + (1 3x ) dy dx xy = 0 46

a = f(m 0, n 0 ) = y f (m 0, n 0 ) m 0 (68) y (68) y 0 m k n k 16 α v 0 M[x(t), y(t)] m P = mg x y m m d x = 0 y md = mg d x = 0 d y = g (69) t = 0 x = 0 y = 0 dx = v 0 cos α dy = v 0 sin α (70) (70) (69) x = (v 0 cos α)t y = (v 0 sin α)t gt (71) (71) y = 0 t (71) [ t v 0 sin α gt ] = 0 47

t = 0 t = (v 0 sin α)/g t x (71) x = (v 0 cos α)(v 0 sin α) g = v 0 sin α g α = 90 α = 45 v 0/g y dy/ y (71) t dy = gt + v 0 sin α gt + v 0 sin α = 0 t = v 0 sin α g t (71) h h = (v 0 sin α)v 0 sin α g = v 0 sin α v 0 sin α g = v 0 sin α g g v0 sin α g (71) y = x tan α gx v0 sec α 48

17 17.1 g 17. 17..1 P ω = αg 0 < α < 1 g P Q F = ma m d x = P Q (7) d x/ = ω = αg m = P/g (7) Q = P m d x = P mαg = P (1 α) (73) 0 < α < 1 Q < P Q = P (1 α) ω = αg 0 < α < 1 Q = P (1 + α) 17.. (73) Q 0 α = 1 Q g 49

17.3 r h r + h Q Q = 0 x n mv = ρ ρ = r + h n k=1 F kn = F F n i=1 F kn mv r + h = F = x md d x = v r + h d x/ (7) mv r + h = P Q (74) P F F r + h F = km (r + h) m k h = 0 F mg k = gr P = F = mgr (r + h) 50

g P (74) mv r + h = mgr (r + h) g v = r r + h 18 r m M G F = GmM r (75) m M t (x, y) F x F cos ϕ y F sin ϕ (75) mẍ = F cos ϕ = GmM r cos ϕ (76) mÿ = F sin ϕ = GmM r sin ϕ (77) sin ϕ = y/r cos ϕ = x/r (76) (77) k GM ẍ = kx r 3 ÿ = ky r 3 r = x + y kx ky ẍ = ÿ = (78) (x + y ) 3/ (x + y ) 3/ t = 0 x = a y = 0 ẋ = 0 ẏ = v 0 (79) 51

(78) (79) (78) x = r cos ϕ y = r sin ϕ ẋ = ṙ cos ϕ (r sin ϕ) ϕ ẏ = ṙ sin ϕ + (r cos ϕ) ϕ ẍ = r cos ϕ (ṙ sin ϕ) ϕ (r sin ϕ) ϕ (r cos ϕ) ϕ ÿ = r sin ϕ + (ṙ cos ϕ) ϕ + (r cos ϕ) ϕ (r sin ϕ) ϕ (80) ẍ = ( r r ϕ ) cos ϕ (ṙ ϕ + r ϕ) sin ϕ ÿ = ( r r ϕ ) sin ϕ + (ṙ ϕ + r ϕ) cos ϕ (78) (80) ( r r ϕ ) cos ϕ (ṙ ϕ + r ϕ) sin ϕ = k cos ϕ r (81) ( r r ϕ ) sin ϕ + (ṙ ϕ + r ϕ) cos ϕ = k sin ϕ r (8) (81) cos ϕ (8) sin ϕ r r ϕ = k r (83) (81) sin ϕ (8) cos ϕ ṙ ϕ + r ϕ = 0 (84) (79) t = 0 r = a + 0 = a (a > 0) ϕ = tan 1 y x = tan 1 0 a = 0 (85) ṙ = 0 1 ϕ = 1+(x/y) ẏx yẋ x = av0 a = v0 a (78) (79) (83) (84) (85) (84) (86) d (r ϕ) = 0 (86) r ϕ = C 1 (87) 5

C 1 P Q P Q OP OQ P Q S S = 1 ϕ 0 r dϕ ds = 1 r dϕ ds = 1 dϕ r = 1 r ϕ (88) ds/ (87) r ϕ (83) (84) (85) t = 0 r = a dotϕ = v 0 /a (87) C 1 = av 0 r ϕ = av 0... ϕ = av 0 r (89) (83) r = a v 0 r 3 k r ṙ = p dp = dp dr dr = pdp dr = a v0 r 3 k r p dp dr = a v0 r 3 k r ( a v0 p dp = r 3 k ) r dr p = k r a v 0 r + C 53

r = a p = ṙ = 0 C = v 0 k a ṙ = k r a v 0 r + v 0 k a ( dr = v0 k ) + k a r a v0 r (90) (90) (89) dr dϕ = r αr + βr 1 α = 1 a k a 3 v0 r = 1/u β = k a v0 dr dϕ = d(1/u) = du/dϕ ϕ u dr dϕ = 1 α u u + β u 1 du α dϕ = u u + β u 1... = α + βu u du dϕ = α + βu u 1 du = dϕ α + βu u α + βu u = (u β) + β + α [ = (β + α) 1 ] β + α (u β) + 1] ( = (β u β + α) + 1 β + α) 54

1 β + α du 1 [(u β)/( β + α)] = dϕ (u β)/ (β + α) = v du = β + αdv dv = 1 v dϕ arccos v = ϕ + C 3 v = cos(ϕ + C 3 ) (1/r) β β + α = cos(ϕ + C 3) r = 1 r = β + β + α cos(ϕ + C 3 ) = = 1 β + β + α cos(ϕ + C 3 ) 1/β 1 + [ β + α cos(ϕ + C 3 )]/β a v 0/k 1 + e cos(ϕ + C 3 ) e = β + α/β = av 0 /k 1 ϕ = 0 r = a a v a = 0/k 1 + e cos +C 3 av 0 cos C 3 k cos C 3 = 1 = av 0 k... C 3 = 0 r = a v 0/k 1 + e cos ϕ (91) e (1) e < 1 v0 < k/a () e > 1 v0 > k/a 55

(3) e = 1 v 0 = k/a (4) e = 0 v 0 > k/a v0 k/a e e V x y ẋ ẏ V v v = ẋ + ẏ (80) v = r ϕ + ṙ m 1 mv = 1 m(r ϕ + ṙ ) (9) km r r dr = [ km r ] r = km r (93) E E (9) (93) ϕ = 0 (91) (94) 1 m(r ϕ + ṙ ) km r = E (94) r = a v 0/k 1 + e mr a v0 r 4 km r = E r e = 1 + E a v 0 mk 56

(91) a v r = 0/k 1 + 1 + E(a v0 /mk ) cos ϕ E < 0 E > 0 E = 0 E = mk /(a v 0) E E x ξ + y η = 1 e = C/ξ C = ξ η e = ξ η ξ η = ξ (1 e ) (95) (91) ξ = 1 ( a v0/k 1 + e + a v0/k ) = a v0 1 e k(1 e ) = a v0ξ kη η = a v 0ξ k (96) T πξη (88) (89) πξη = av 0 T/ (96) π ξ η = 1 4 a v 0T a v 0 k π ξ 3 = 1 4 a v 0T... T = 4π ξ 3 k 57

19 19.1 19.1.1 AB 19.1. OA O x y O F 1 F x y x sag 19. 19..1 x M(x) x 19.. x y EJ [1 + (y ) ] 3 = M(x) (97) / 58

E Young J x EJ flexural rigidity y (97) EJy = M(x) (98) (98) l pkgf (pl/) M(x) Q Q OA O x Q Q M(x) = pl ( x ) x + px = px plx (99) M(x) = p(l x) l x pl px (l x) = plx (100) (99) (100) (98) EJy = px plx (101) O A (101) y x lx = T x = 0 y = 0 x = l y = 0 T = x4 1 l 6 x3 + C 1 x + C y = p ( x 4 EJ 1 l ) 6 x3 + C 1 x + C 59

y = p ( x 4 lx 3 + l 3 x ) 4EJ x = l/ 5pl 4 /(384EJ) E = 1 10 5 kgf/cm J = 3 10 4 cm 4 0 A B h XY AX = λh C AC = ah a a = 0.3 a F F P P QQ RR SS L L W X βh O h/α 60

A A AC χ Oxy θ BAC u u = χ θ u h OX OA + W O AB OP OA + L OP AC A OA AC χ A 30m N N = 180 π 30α h (10) h h = 9m α = 0.1 N =..19 h = 1m α = 1.0 N =..14 α 0 1 α λh h 3h λ 1 3 a 0 a < 0.5 h 9m 1m A x = h α sin χ y = h α cos χ B X = h α sin χ h cos θ Y = h cos χ + h sin θ (103) α B BC dy dx = tan ψ (104) ψ BC χ = u + θ ABC 61

sin(χ ψ) h = sin(θ ψ) ah = sin u bh (105) 0 < b < 1 ψ θ u χ (103) (104) ( h ) dθ sin χ + h α dχ cos θ cos ψ + h α ( h dθ cos χ + h α a χ = u + θ χ 1 = du dχ + dθ dχ ) dχ sin θ sin ψ = 0 dθ (sin χ cos ψ sin ψ cos χ) = h (cos θ cos ψ + sin θ sin ψ) dχ sin(χ ψ) = α dθ cos(θ ψ) (106) dχ (106) (105) du dχ du dχ = 1 dθ dχ sin(χ ψ) = 1 α cos(θ ψ) (sin u)/b = 1 α cos(θ ψ) (107) (105) sin(θ ψ) = a b sin u 1 cos (θ ψ) = a b (1 cos u) cos (θ ψ) = 1 a b (1 cos u) cos(θ ψ) > 0 cos(θ ψ) = 1 a b + a b cos u (107) du dχ = 1 sin u α b a + a cos u χ = 0 θ = 0 u(0) = 0 BASIC 6

II 1 v x θ Galileo θ v x y v x v y v v x v = cos θ v y v = sin θ x x y Galileo x x v x = v cos θx v x t x x = v x t y t v y t Galileo t g t g t y y = g t + v y t x = v x t y = g t + v y t 1.1 v y g 63

y = g t + v y t y = 0 0 = t (v y g ) t t = 0, v y g t = v y g R x x v x v y = gr R = v x t = v x v y g = v xv y g v x v y = gr R 0 x y v x v y v y = gr v x g = 3, /sec R 5000 y v y = 400 /sec R = 5000 v x v y = 80500 v x 400 = 80500 v x v x = 01, 5 v x 01, 5 /sec 64

R R = v g sin θ R = v xv y g = v cos θ sin θ g = v g sin θ sin θ = 1 θ = π θ = π 4 = 45 ( π ) R = v 4 g v dy = gt + v sin θ = 0 t = v sin θ g ( ) ( ) v sin θ v sin θ + v sin θ y = g g = v sin θ g = v sin θ g + v sin θ g g 65

R > 0 v R = v sin θ g gr v = sin θ sin θ = 1 θ = π 4 v = gr v R = v g sin θ sin θ = gr v 0 < R < v g 0 < gr v < 1 θ 1 + θ = π θ 1 + θ = π v θ α < θ v θ 66

y = g t + v y t R = v x t R tan α = g t + v y t v x tan α = g g t + v y g = v y v x tan α t = g (v y v x tan α) = v (sin θ cos θ tan α) g v α R R = v x t = v x g (v y v x tan α) = g (v xv y v x tan α) R v x v y R = g v xv y g v x tan α g v xv y = R g v x tan α v y = gr v x + v x tan α v x + v y = 1 = 1 { ( ) } gr v x + + v x tan α v x ( v x cos α + g R ) + gr tan α 4v x 67

( 1 v x cos α + g R ) + gr tan α 4v x g R 4cos α +1 gr tan α = gr ( 1 ) cos α + tan α v x v x cos α = g R 4v x v x = tan θ = vy v x gr cos α tan θ = 1 cos α + tan α R v g θ α R = g (v xv y v x tan α) = v g (cos θ sin θ cos θ tan α) v α R cos θ sin θ cos θ tan α = cos α cos θ sin θ cos θ sin α cos α sin θ cos α sin α cos θ sin α = cos α sin(θ α) sin α = cos α R θ α = π θ = π 4 + α 68

R < v g ( sin(θ α) sin α cos α R = v g ) ( ) 1 sin α cos α R = ( ) v sin(θ α) sin α g cos α gr v = sin(θ α) sin α sin(θ α) = gr cos α + sin α < 1 v θ 1 α + θ α = π θ 1 + θ = π + α θ 1 α = π θ.1 Newton Principia Newton 69

v x θ θ R(θ) = v g sin θ g R 0 R < v /g R Tartaglia 1537 θ = π/4rad k (x, y) k [ ẋ ẏ ] [ 0 ] g ẍ = kẋ ÿ = g kẏ ẋ(0) = v cos θ x(0) = 0 ẏ(0) = v sin θ y(0) = 0 u = ẋ u = ku u + ku = 0 0 = e kt + ke kt u 0 = d (ekt u) 70

ẋ(0) = C u = ẋ = v cos θe kt x(0) = v cos θ( 1 k ) + D = 0 D = 1 k v cos θ x = 1 k v cos θ( e kt + 1) s = ẏ ṡ = g ks ṡ + ks = g e kt ṡ + ke kt s = ge kt d (ekt s) = ge kt e kt s = g e kt e kt s = g k ekt + C s(0) = ẏ(0) = v sin θ = g k + C C = v sin θ + g k s = g k + v sin θe kt + g k e kt ( = e kt v sin θ + g ) g k k y = 1 ( k e kt v sin θ + g ) g k k t + D y(0) = 1 ( v sin θ + g ) + D = 0 k k D = 1 ( v sin θ + g ) k k y = 1 ( k e kt v sin θ + g ) g k k t + 1 ( v sin θ + g ) ( k k v sin θ = + g ) k k (1 e kt ) g k t 71

x 1 e kt = kx v cos θ t e kt = 1 kx v cos θ kt = log(1 kx v cos θ ) t = 1 ( k log 1 kx ) v cos θ y ( v sin θ y = + g ) kx k k v cos θ + g ( k log 1 kx ) v cos θ ( k x g tan θ + k ) v sec θ ( + log 1 kx ) = 0 v cos θ x 1 kx v cos θ = e A(θ)x A(θ) = k g tan θ + k v sec θ R(θ) R(θ) = cos θ a (1 e A(θ)R(θ) ) A(θ) = a sec θ + b tan θ,a = k/v,b = k /g R(θ) θ R 0 R n+1 = cos θ a (1 e A(θ)Rn ) R(θ) R(θ) 7

R(θ) R(θ) Tartaglia Tartaglia. c,d(c > 0, d > 1/c) x 0 f(x) = c(1 e dx ) g(x) = x f(x) = c(1 e dx ) f (x) = cde dx f (0) = cd > 1 lim f(x) = c x 7 f f(x) = c(1 e dx ) c, d c > 0, cd > 1 f p f(p) = p p f 0 f 73

f(0) = 0 f (0) > 1 s f(s) > s f (0) = cd > 1 F (s) = f(s) s F (s) > 0 F (s) = f (s) 1 F (0) = f (0) 1 > 0 F (0) = 0 F (s) F (0) = F (c)(s 0) (0 < c < s) c F (s) = f (c)s c F F (s) = F (c)s > 0 F (s) > 0 f(s) > s f(c) < c f (0, c) f(c) < c F (c) = f(c) c = ce cd < 0 f(s) > s 74

F (X) = f(x) x F (c) < 0 F (s) > 0 F (x) = 0 (s, c) (0, c) x f (x) f p f (x) = cd e dx < 0 (α, β, α β) f(x) = x F (x) = f(x) x F (α) = 0 F (β) = 0 f(0) = 0 0 x c 0 < ξ < α, F (ξ) = 0 ξ α x β α < η < β, F (η) = 0 η ξ x η ξ < γ < η, F (γ) = 0 γ F (γ) = f (γ) = 0 f (x) < 0 75

x(x > p) f(x) > x f q > p q x f(x) > x x < p x f(x) < x x > p 1. f(x) > x x < p. f(x) < x x > p 1. f(r) < r r < p F (x) = f(x) x F (r) > 0 F (c) < 0 (r, c) F (q) = 0 q f(q) = q q p, q f(x) > x x < p. f(r) < r r < p F (r) < 0 F (s) > 0 (s, r) F (q) = 0 q f(q) = q p, q f(x) < x x < p e x > 1 + x(x > 0) x > 0 (x + 1)(1 e x ) > x f((cd 1)/d) > (cd 1)/d d(c p) < 1 76

(x + 1)(1 e x ) x = (x + 1)e x + 1 = ex (x + 1) e x > 0... (x + 1)(1 e x ) > x f((cd 1)/d) > (cd 1)/d ( ) cd 1 f cd 1 = c ce (cd 1) cd 1 d d d { = c 1 e (cd 1) cd 1 } cd (x + 1)(1 e x ) > x 1 e x > x x + 1 { c 1 e (cd 1) cd 1 } cd.. ( ) cd 1. f d ( cd 1 > c cd > cd 1 d cd 1 ) = 0 cd d(c p) < 1 f(x) > x x < p 0 < f (p) < 1 cd 1 d < p cd 1 < dp d(c p) < 1 f(p) = p p = c ce dp e dp = c p c f (p) = cde dp f (p) = cd c p c = d(c p) < 1... 0 < f (p) < 1 77

0 < θ < π/,r > 0 F (θ, r) = cos θ a (1 e A(θ)r ) A(θ) = a sec θ + b tan θ F F θ F r R(θ) = F (θ, R(θ)) F θ (θ, r) = sin θ a (1 + A (θ)re A(θ)r ) A (θ) = a sin θ + b cos θ F r (θ, r) = cos θ a A(θ)e A(θ)r R(θ) = cos θ a (1 e A(θ)R(θ) ) = F (θ, R(θ)) F r (θ, R(θ)) < 1 F r (θ, r) = cos θ a A(θ)e A(θ)r c = cos θ a, d = A(θ), f(x) = c(1 e dx ) p = R(θ) cd = cos θ a A(θ) = cos θ (a sec θ + b tan θ) a = 1 (a + b sin θ) a = 1 + b a sin θ > 1 f (p) = cde dp F r (θ, R(θ)) < 1 78

R (0, π/) [0, π/] R(θ) g(θ, r) = F (θ, r) r r R(θ) g(θ, r) = 0 g r (θ, R(θ)) = F r (θ, r) 1 < 0... g r (θ, R(θ)) R(θ) (0, π/) [0, π/] R(θ) = cos θ a (1 e A(θ)R(θ) ) t = R(θ) sec θ ( g(t) = at 1 + e at+b t R(θ) ) = 0 ( g (R(θ) sec θ) = ar(θ) sec θ 1 + e ar(θ) sec θ+b R(θ) sec θ R(θ) ) (ar(θ) sec θ+br(θ) tan θ) = ar(θ) sec θ 1 + e = ar(θ) sec θ 1 + e A(θ)R(θ) = 0 R(θ) = 1 e A(θ)R(θ) a sec θ = cos θ a (1 e A(θ)R(θ) ) R = max θ [0, π ] R(θ) 0 R < R R(θ) = R θ [0, π ] 79

R(0) = r R(θ) = cos θ a (1 e A(θ)R(θ) ) A(0) = a r = 1 r (1 ear ) ar = 1 e ar e ar = 1 ar ar = 0 r = 0 R(0) = 0 ( π ) R = 0 [0.π/] R(θ) R [0, c], R(c) = R > R, R(0) = 0 < R 0 < ξ < c, R(ξ) = R ξ c < η < π, R(η) = R η R(θ) = R θ g(t) = 0 t g (α) = 0 α g(t) = at 1 + e (at+b t R(θ) ) B = at + b t R(θ) g (t) = a B e B g (t) = B e B + (B ) e B = e B {(B ) B } B t = a + b t R(θ) 80

B br(θ) = (t R(θ) ) t R(θ) < 0 g (t) > 0 R(θ) = R θ f x n+1 = f(x n ) p f 0 < x 0 < p x n < x n+1 < p p < x 0 p < x n+1 < x n x 0 > 0 p x n+1 = f(x n ) p = f(p) 1. 0 < x 0 < p x n < x n+1 < p. p < x 0 p < x n+1 < x n 1. x 0 < x 1 < < x n < p x n < x n+1 < p x 0 < x 1 < p x n < x n+1 < px 0 < x 1 < p x 0 = x 1 x 0 = p 0 < x 0 < p x 0 > x 1 x 1 = f(x 0 ) x 0 > f(x 0 ) x f(x) < x x > p x 0 > p 0 < x 0 < p x 0 < x 1 x 1 = f(x 0 ),p = f(p) x 1 p = f(x 0 ) f(p) 81

f(x 0 ) f(p) = f (ξ)(x 0 p) x 0 < ξ < p ξ f (ξ) > 0,x 0 p < 0 f(x 0 ) f(p) < 0 x 1 p < 0 x 1 < p x 0 < x 1 < p x n < x n+1 < p. 1 θ R (θ) = 0 θ sin θ = (sin θ + A (θ)r(θ) cos θ)e A(θ)R(θ) R (θ) = sin θ a = sin θ a R (θ) = 0 (1 ea(θ)r(θ) ) + cos θ a [A(θ)R(θ)] e A(θ)R(θ) (1 e A(θ)R(θ) ) + cos θ a [A (θ)r(θ) + A(θ)R (θ)]e A(θ)R(θ) 0 = sin θ a + sin θ a e A(θ)R(θ) + cos θ a A (θ)r(θ)e A(θ)R(θ) 0 = sin θ + e A(θ)R(θ) sin θ + A (θ)r(θ)e A(θ)R(θ) cos θ sin θ = (sin θ + A (θ)r(θ) cos θ)e A(θ)R(θ) e A(θ)R(θ) = 1 a sec θr(θ) θ R(θ) = (c cos θ)/a sin θ + c c = vk/g 8

sin θ = (sin θ + A (θ)r(θ) cos θ)(1 a sec θr(θ)) R(θ) 0 = sin θ ar(θ) tan θ + A (θ)r(θ) cos θ aa (θ)r(θ) R(θ) = tan θ A (θ) + cos θ a A (θ) = (a sin θ + b)/ cos θ R(θ) = sin θ cos θ cos θ a sin θ + b + cos θ a a sin θ cos θ + a sin θ cos θ + b cos θ = a sin θ + ab b cos θ = a sin θ + ab (b cos θ)/a = sin θ + b/a a = k/v,b = k /g,c = vk/g = b/a θ R(θ) = (c cos θ)/a sin θ + c A(θ)R(θ) = c + c sin θ sin θ + c s = sin θ sin s = (s + c)e (c+c s)/(s+c) (c cos θ)/a A(θ)R(θ) = (b tan θ + a sec θ) sin θ + c (bc sin θ)/a + c = sin θ + c = c sin θ + c sin θ + c s = sin θ A(θ)R(θ) = c s + c s + c 83

s = = ( s + a sin θ + b cos θ (s + c sin θ + c sin θ + c = (s + c)e (c+c s)/(s+c) s (c cos θ)/a sin θ + c ) e (c+c s)/(s+c) x = e hx ) cos θ e (c+c s)/(s+c) h = (1 c )/e,x = es/(s + c) es s + c = e 1 c e es s+c = e s(1 c )/(s+c) s = (s + c)e (s sc s c)/(s+c) = (s + c)e (c+c s)/(s+c) x = e hx h < 1/e h x = e hx x(h) (0, e) a (0, e) e ha < a x(h) < a f(x) = x e hx (h < 1/e, 0 < x < e) f(0) = 1 < 0 f(e) = e e he > 0 (eh < 1) 84

0 < x < e (a, b) a = e ha b = e hb 0 < a < e 0 < b < e F (x) = x e hx F (a) = F (b) = 0 a < b F (ξ) = 0, a < ξ < b ξ F (x) = 1 he hx F (ξ) = 1 he hξ = 0 e hξ = 1 h > e 0 < ξ < e 0 < hξ < he < 1 e < e hξ < e 1 0 < a < e e ha < a x(h) < a x(h) < a p < a f(x) = x e hx e ha a < 0 f(a) > 0 p < a p > a f(0) = 1 < 0 f (ξ) = 0 0 < ξ < a ξ 0 < a < e e ha < a x(h) < a 85

j [1, ) j(1) = j (1) = 1 w > 1 j (w) > 0 w > 1 j(w) > w Taylor j(1) = j (1) = 1 j(w) = j(1) + j(1)(w 1) 1! + j (c)(w 1)! > w (1 < c < w)...j(w) > w s s < 1/ x(h) x(h) < e/(1 + (1 eh)) w = 1 + (1 eh) e h(e/w) < e w s < 1/ { s = (s + c)e c+c s s+c s < 1 x = e hx x < e 1+ (1 eh) x < e 1 + (1 eh) x(1 + (1 eh)) < e x (1 eh) + x < e x 1 eh < e x x 1 eh e x < 1 86

h = 1 c e c = 1 eh x = s = es s + c s = x 1 eh e x w = 1 + (1 eh) x < e w x ha < a x < a e ha < e w x < a w x < e w s < 1 e ha < e/w s < 1/ w = 1 + (1 eh) eh = 1/ + w + w / w = 1 + (1 eh) w w + 1 = (1 eh) eh = 1 + w w eh = 1 + w + w w > 1 w < e (w w 1 )/ e h(e/w) w < e (w w 1 )/ < e w w > 1 w < e w w 1 e h e e w < w eh = 1 + w w w w 1 = 1 eh w 87

w < e 1 eh w e h( e w ) < e w w < e (w w 1 )/ j(w) = e w/ w 1 / w < e (w w 1 )/ j(1) = j (1) = 1 j (w) > 0 j(w) > w j (w) > 0 j(1) = j (1) = 1 j(w) = e w/ w 1 / j (w) = ( w w 1 ) e w/ w 1 / = ( 1 + w 1 )ew/ w / j (w) = ( w 3 e w w 1 ) + ( 1 + w ) e w w 1 = e w w 1 {( 1 + w ) w 3 } ( 1 + w ) w 3 > 0 ( 1 + w ) w 3 = w4 + w + 1 4w 4w 4 f(w) = w 4 + w + 1 4w f (w) = 4(w 3 + w 1) > 0 f(1) = 0 f f(w) > f(1)...f(w) > 0 j (w) > 0 j(w) > 0 88

[1] W [] V V 89

TEX 90