y = x ; x + y b = ; 3 t = cosh t, b sinh t; 4 t =t, cosh t >. t =xt, yt st = s = s = t / s t = θ θ, t = s =st = s = t / s t = t t = s =st = x t +y t, y t, x t x t +y t, st = κtst x t, y t x t +y t y tx ty t x ty t, x tx ty t x ty t {x t +y t }, κt = x ty t x ty t {x t +y t } 3/, t =t, t, t = + 4 t st = + 4 θ θ = θ + 4 θ + 4 lnθ + + 4 θ = t + 4 t + 4 lnt + + 4 t, θ=t θ=
κt = + 4 t 3/. t = cos t.b sin t, t [, π, t = sin t + b cos t st = θ θ = sin θ + b cos θθ, κt = b sin t + b cos t 3/. 3 t = sinh t + b cosh t st = θ θ = sinh θ + b cosh θθ, b κt = sinh t + b cosh t. 3/ 4 t = + sinh t t st = θ θ = + sinh θ θ, κt = cosh t + sinh = t 3/ cosh t. C r, θ r = f θ C κθ = f θ+ f θ f θ f θ { } 3. f θ+f θ C θ =xθ, yθ = r cos θ, r sin θ =f θ cos θ, f θ sin θ x θ = f θ cos θ f θ sin θ, x θ = f θ cos θ f θ sin θ f θ cos θ, y θ = f θ sin θ + f θ cos θ, y θ = f θ sin θ + f θ cos θ f θ sin θ, x θ +y θ = f θ+f θ, x θy θ x θy θ = f θ+ f θ f θ f θ, t = cosh t, sinh t, bt > ; t =3t t,3t,3t + t ; 3 t = sin t, cos t, bt > ; 4 t =t, ln t, t >.
E 3 t t t t, t, t κt = t 3, τt = t t. st = θ θ, = s = / s t t = t t, = t t = t 4 t, t t t, t t t = t t t 4, =,,. t t t t t κt = = t 3, = t t t κ = t t t, t t t t = = t t t t, t t t t, t t t = t t t = t t t t, = t t t t t t t t, t t t t = t t t 3, = t, t t t, t t t t t,
τt = t, t t t, t = t t t, t, t = t t. E 3 t t t t, t, t κt = t 3, τt = t t. κt = t = sinh t, cosh t, b, t = cosh t, sinh t,, t = sinh t, cosh t,, + b cosh t cosh t + b 3/, τt = b + b cosh t. t =3 t,6t,3+ t, t =, 6,, t =,,, κt = 3 t, τt. + 9 3/ 3 t = cos t, sin t, b, t = sin t, cos t,, t = cos t, sin t,, κt = b, τt = + b + b.
4 κt = 3 + s s = 3 t =,, t t, t =, t, t 3, t =, t 3, 6 t 4, t + t, τt =, s 3 3 s Frenet s + s s =, s s,, t + t. < s < =, s = + s s s =,,, s =,,, 4 + s s κs = s = 4 s, s = s s = s, + s,. s =s s = + s s,, Frenet {s; s, s, s}, τs = s, s = 4 s. T X =X + P E 3 et = t E 3 = t t s s st = θ θ = θ et θ = θ θ = st,,,, κ, τ,,,, κ, τ = + P = s s = =, s
t κ = s = s = = κ, s κ = κ, =. = = =et = =, T T =et T T, T E 3 t τ = = s s, = s t = s t t = s, t t st =st, κt =κt, τt = τt. = τ. C : s κ > τ >, s C C s = s s C κ = τ, τ = κ C Frenet uu, s = s = s C κs =τs, s = s = s, s = s κs = s, s = s = s = τss, s = s s = s s =s, s = s = κss, τs = s, s = κs. C Frenet s C Frenet { s; s, s, s} = { s; s, s, s}, s κ, τs s S s s =s τ, κ = + τ = s τ κ κ κ[ + τ κ ].
s rs r s = s = κ, r s = κ, r s = s κ =κ + κ s = κ + κ κ + τ = κ + κ + κτ, r s =κ + κ s κκ κ s + κ τ + κτ + κτ s = κ + κ κ + τ κκ κ κ+κ τ + κτ + κτ τ = 3κκ +κ κ 3 κτ +κ τ + κτ, s s =κ τ + κ 3, s s = κ κ + τ, s, s, s = s s, s = κ τ 3κκ +κ 3 κ τ + κτ = κ 4 τ κ 3 κ τ, s = s s τ, s 3 = + κ s, s, s τ = s s = κ4 τ κ 3 κ τ κ 4 κ + τ = s τ κ κ[ + τ κ ]. κs = +s s κs = s s s =xs, ys κs s x s +y s = s = s =x s, y s, s = y s, x s, s = s =x s, y s = κss, { x s = κsy s, y s = κsx s, { x s = cosθs, y s = sinθs,
θ = θs s θs ODE θ s =κs, s s θs =θ+ κtt = κtt, κs = θs = s s = rctn s +s +s xs = x+ s x tt = x+ s cosrctn t t = x+ s t = x+lns + + s, +t ys = y+ s y tt = y+ s sinrctn t t = y+ s s t = y+ + s, +t s = lns + + s, + s κs = θs = s s s = rcsin s s xs ys s = = x+ s x tt = x+ s cos rcsin t t = x+ s t t = x+ s t + rcsin s = y+ s y ss = y+ s sin rcsin t = y+ s t s t = y+, s t + rcsin s E 3 s s, s, 3 s 3 = κ τ s, s, 3 s 3 = κ 3 κ τ κτ =κ 5 s τ κ,, s s =, s = = κ, 3 s 3 = κ + κ = κ + κ κ + τ = κ + κ + κτ, s, s, 3 s 3 = s s, 3 s 3 = κ, κ + κ + κτ = κ, κ + κ + κτ = κ τ. s = s = κ, s = 3 s 3 = κ + κ + κτ,
3 = κ κ + κ + κ + κτ + κ τ + κτ s3 = κ κ + κ + κ κ + τ+ κτ + κ τ + κτ τ = 3κ κ + κ κτ + κτ + κ τ, s, s, 3 s 3 = s s, 3 s 3 = κ κ + κ + κτ, 3κ κ + κ κτ + κτ + κ τ = κ τ + κ 3, 3κ κ + κ κτ + κτ + κ τ = κ 4 τ κ 3 κτ = κ 5 τ. s κ [ ] + = κ τ s κ κ κ κ κ s s s C s s s = C, s s, s =, + s s, κss =, s, κss + s s, κ ss+κs κss+τss =, s s Sn{s, s} s s, s = κs, s s, s = κ s τs κ s =, τs s κs s s = s s, s s+ s s, s s s =s+ κs s+ τs s, s κs
rs s =s+ s κs + s τs s κs { =τs κs + τs s κs + s κs τs κs + τs s+ κs s+ [ τs s κss+τss τss τs s κs ]} s, κs s κs s τs s τs =, s κs s, s = s = s =s+ κs s+ s, τs s κs s s =, s =, s κs t = t + 3 sin t, cos t, 3t sin t rt = cos t, sin t, t t t s s E 3, b κ s =κ s >, τ s =τ s, s, b E 3 T = T t = + 3 cos t, sin t, 3 cos t, t = sin t, cos t, ; t = 3 sin t, cos t, sin t, t = cos t, sin t,; t = 3 cos t, sin t, cos t, t = 4 sin t, 4 cos t,; κt = t t t 3 = 4, κt = t t t 3 = 4 ; τt = t, t, t t t = 4, τt = t, t, t t t = 4 ; t t
. ` ò ê.. Ë ' Úø ÂÚø P B µñ : x + c Ñü : x x + c + c logx + x + c ee`ò. le h = = 4 x + 3 = 3 x + y 4 b b 4 e 3 e4 4 3 b 3 = sh t + b ch t 3 3 5 e6 4 6 5, 4 = ch t... ÇÊ @: ÿ 9 8Ñ ø øô / le øô ' v le ; ÑÚø/ Ñ ø B øñ hæ øñúá, 3 AP µñ' Ú OA OB Ñ9 :.. ~t = x,y x + y, s t = ~r t = x + y  :Úø xtñ9 Çú π x 6= Ÿ πñªflö x 6=, = rctn y x = s = t t s = y x x + + y x y x x + y 4
 πp x = y 6= æÿπ:üπ Ô Úø øùøl lÿãl = {x = y} lóñúø : e, e Ñ h æ :ert =yt,xt e ÑÚá e t = t,çmb ú bx, y ó e t = t = y x + y x x + y 3..3 e..4 = + b cht b 3, = cht + b 3 b cht + =, = t + 9 3 3 = + b, = b + b 4 = t t + t, = + t..5 ~r t = ~t s ~r t = ~r s s t = ~t ~r t s + ~t t t ~r t = ~ b ~r t ~r t = ~n ~r t + ~n + ~t + ~t t ~r t  hæ e v~t, ~n, ~ b=sô ee`ò. $ Ç $I /&cn s t = ~r t, s t = ~r t...6 +s ~t =, s ~n =, s, +s, ~ +s b =, ee`ò. ÂÚø= * %b b = = s s, 5
..7 ée t Ñ 6 =. t!, {~t, ~n, ~ b}! {,,,,,,,, }, t! +, {~t, ~n, ~ b}! {,,,,,,,, }, ee`ò. /&X  ÅBÑöI, ] ÑI zùúø~rt, ÂX wsñ  X Ù..8. St! ˆ,!, ~rt! üπ, leúø P s Z ~rt t le, @ >. St! ˆ!,!leÅBX8 >, s.t > t, ~rt! üπ. Õ6ÅBÚø P 3. St! ˆ!,! leåb >, ~rt! üπ. Õ6ÅBÚø P 4. St! ˆ!,!, ~rt! üπ, FÅBÚø P s Z ÂX >, ó~rt{t,t + } öi ~rt t!, s! = ~r P t = t t < ~r P, ~r P > t =< t ~rt, ~rt P >=< ~tt s t, ~rt P >..9 æx,ó~r + ~t =, B ó ~t + ~n + ~t = =, =. E 6= =, é/fiì le Í Àπ ˆ @Â. æx,ó~r + ~n =, B ó ~t + ~t + ~ b+ ~n = =, =, = E Constnt, le 6= : =., = Constnt ee`ò. ÏlezÙÚø π Ñ; ~n åo ~ b Ñsb:ÚøÂπ Ñ sb  Úø 6= π Ñ sb«öπ ÂÚb/ búø ee`ò. /&X I ÑzÙÚø, o ø«öπ? ee`ò. ~rt:r 3 -c Úø X8,, 3, óùøœ{~rt + ~tt + ~nt + 3 ~ bt R}«öπ ÓŸö,, 3,ÇúÚø Ï/&ÔÂnöŸÚøÑ Ub ee`ò. $ Ç }ò/&cn Çcn cn >ãù ÂR 3 - Úø; ø«ÿöùø ÂÚø: Ò ø.. ' ÿ ent =nt, e t = t 6
.. HB ersñfrenet é ~ b =Es _/ er Ñ' ets = ers s = ~ bs ens = ets s / ets s = ~ bs s / ~ bs s = ~n = ~n e bs = et ^ en = e b ^ ~n =~t e = ets s = ~ bs s =, e =< en s,e bs >=< ~t ~ b,~t >=.. ñh Ï óers =~tsñfrenet erñ' : ets = er = ers s s = ~ts s s s = ~ns éet : E s =, le et = ~n ens = et / et é en =, E et = ~ns s s / et = ~ts+ ~ bs et = e = r + e = +, en = ~t + ~ b + ~ b = ~t ^ ~n, ó ~ b ^ ~t = ~n, ~n ^~b = ~t e b = et ^ en = ~ b + ~t + e =< en, e b > =< s s le = s + = s + ~t + ~! b, + ~ b + ~t + > le + + s + + ee`ò. æ 6=, 6= ˆr = ~ b óúá á 7
..3 æúøx, ysb Úø xtñ9 : ÚáöI,òî > = s = + s Ô ó = rctn s + C, tn C =s ÑC s@ ÚøÀl E,' ÏÔ C. r, s s Ÿˆ ~t = + s, Ô ó ~r = log ++ s,! + s = s = Ô ó = rcsin s s + C, sin C = s! s Ô C, Ÿˆ ~t =, s, ŸÃGæ > Ô ó ~r = s r! s + rcsin s, s ee`ò. B  $ÚøÙÄ ee`ò. sbúø ~rt, Ñ^' hæ ~r t = +t, = t + t + 3 B ~rt hæ ee`ò. sb' Úø ~rs, = s, B ~rs hæ...4 π : ~t, ~n, ~ b=< ~t ^ ~n, ~ b>= Frenet Ñ π 4 ~r s 4 = 3 ~t + 3 ~n + + ~ b π å: é,åó,çòñ e, e, e,ò, Ó vë Ñ ee`ò. ó ~! b s, ~ b s, 3~ b s 3 8
..5 Ç~r: búø ~rñ sb«öπ,ç,9òee`ò ÔæX,µ, ó Ï ˆ ~r + ~n + µ ~ b B ó ~t + ~n + ~t + ~ b+µ ~ b + µ ~n = + s =, ~r + ~n + = s = = le ~r + s ~n + = µ, + µ = ~ b : s ÔÂ ~r + ~n + " + + " + s s " ~ b = s #!! # ~ b:8<! # + ~ b @ÂÇú /8< ~r + ~n +..6 s ~ b:8< æp :üπ l:zt P πd ÔæÚø: ~rt =xt,yt,t, x = y = x = y = xt =x t + Ot3, yt =y t + Ot3 óó = ~r t = x,y, x + y + = x + y æp = ~rt, P, l lim P!P P,P =lim x t+y t x t! x t+y t+t =lim t 4 + y t 4 + Ot 5 t! t + Ot 3 = x + y = ee`ò. Õ,òÑ æl/p π ÚøCÑ ; Ñsb : 3P, L lim P!P P, lp, P = P 9
ee`ò. : «s Àl, c ÚøÔ@Ëh::..7 ~rt =t, t + Ot 3, 6 t3 + Ot 4 π = clele 6= e,çò ó e = +c, e =, E~ts/ * ~ bs/8 œfë ˆs /ers =~tsñ' Ÿ Ñ * ~t, : ˆÑÚø π, ~t @ ~n s ~ b A = @ @ c c - O = @ E s O @ ~t ~n ~ b Z s ~tt, ~rs A @ A = O T @ c c + c + A = @ ~t ~n ~ b A = @ c + c + c c + c + c A @ c c + A O A, O T O = OO T =I c + A O @ ~t ~n ~ b A ~t ~n ~ b A ó s s c~t + ~! b =, c + s ~n = c + c~t + ~! b = c +~n c + c~t + ~! b, c + c~t + ~ b c~t + ~ b c~t + ~ Z b c + = sin c + ~n + cos Z c + c~t + ~! b c + ~t = c c~t + ~ b + c +sin c + Z s ~n cos c + Z s c~t + ~ b
~tô Ôó~r ee`ò. Ë,Ç5ò-  + le s =, = s q @ = const :8 Çú = s =, ó =, = =, = s s wsô ÂÚø ee`ò. zùúøâ öπ ö ÂÚø = c ÕK 6 sâ Úø = c ÂÚøÑ öπ ö : bñπ,. Ë,Ç,mò..8,föIÚáˆÅB ö~nó{~t, ~n} {@ x, @ y }ö sô «Àls {~t, ~n} {@ x, @ y }Õ e t = t e t = t, e t = t..9.. ~v = ~ b + ~t ~r t øπ,, 3 w ertøπ,, w :~$ÚøÕ Ñ S ÏH ~r t y tàl ~r t, ert  w ~r t B @ 3 3 C A = sin t, cos t, t 6 ç sin t, cos t, t ztàl Ôv ertõ.. ee`ò. Ÿö, :x, y, z, s ÑI ó = ~rs,s, = ~rs,s XzÙÚø~rs :R! R 3 = {x, y, z},