þ l lúª m¡ Û¼ê¡ Û¼ê §9.7 m¡ 9.7.1 ëê§L«m m¥^ L dëê§ ~r = ~r(t) = x(t)~i + y(t)~j + z(t)~k : [α, β] ⊂ R −→ R 3 L«. 1 ◦ þ þü: M0 Ú M, §»©O´ ~r(t0) Ú ~r(t). u´ −−−* M0M = ~r(t) − ~r(t0), ~r(t)−~r(t0) t−t0 Ò −−−* M0M , ¿ëêO\. t → t0 , XJuþ ~r(t)−~r(t0) t−t0 43 ~r0 (t0) = lim t→t0 ~r(t) − ~r(t0) t − t0 Kd4Ò¡3 M0 þ. M0 M 1/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê ØJwÑ, þëêO\. 3 t0 þ3, du ~r(t) ©þ x(t), y(t), z(t) 3 t0 , ~r0 (t0) = x 0 (t0)~i + y 0 (t0)~j + z 0 (t0)~k XJ x 0 (t), y0 (t), z0 (t) Ñ3 [α, β] þëY, ØÓ", K¡ L ´ 1w . XJ L ±©¤kã, zãÑ´1w, K L ¡ Åã1w. XJ L g Ø, =é?¿ α 6 t1 < t2 < β, Ñk ~r(t1) 6= ~r(t2), K¡ L ´{ü½ Jordan . e ~r(α) = ~r(β), K¡ L ´4. k þ, Ò3 M0 § x − x(t0) x0(t0) = y − y(t0) y0(t0) = z − z(t0) z 0(t0) . 3 M0 {²¡§ x 0 (t0)(x − x(t0)) + y 0 (t0)(y − y(t0)) + z 0 (t0)(z − z(t0)) = 0. 2/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê 2 ◦ l O²¡l{ , (½ml¤¦^{´: ±SòlCq, -Sò>êÃO\, Ù4ҽ l. äkëê§L«, ù 4±|^½È©?1O. A M M M M M M 1 2 3 n-1 k k-1 B äNL§Xe. 3 L þSò AM1M2 · · · Mn−1B, ©: T : α = t0 < t1 < t2 < · · · < ti−1 < ti < · · · < tn−1 < tn = β ©:´º:¤éAëê, Kù^òoÝ l(T ) = X n i=1 |~r(ti) − ~r(ti−1)| = X n i=1 p [x(ti) − x(ti−1)]2 + [y(ti) − y(ti−1)]2 + [z(ti) − z(ti−1)]2 . 3/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê éz©þ©O¦^©¥úª l(T ) = X n i=1 q x02 (ξi) + y02 (ηi) + z 02 (ζi) ∆ti, Ù¥ ti−1 0, 3 δ1 > 0. kT k := max 16i6n ∆ti < δ1 , é?Û (ξi, ηi, ζi) ∈ [ti−1, ti ] 3 (i = 1, 2, · · · , n) Ñk p x02(ξi) + y02(ηi) + z 02(ξi) − p x02(ti) + y02(ti) + z 02(ti) < ε. 4/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê Ï |~r0 (t)| = p x02(t) + y02(t) + z 02(t), kT k 0, kT k < δ2 X n i=1 |~r0 (ti)|∆ti − Z β α |~r0 (t)|dt < ε. Ïd, δ = min(δ1, δ2). u´ kT k < δ Òk l(T ) − Z β α |~r0 (t)|dt < ε(β − α + 1). 5/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê = L l½Â s = Z β α |~r0 (t)|dt = Z β α p x02(t) + y02(t) + z 02(t) dt. (9.1) L ´Åã1w, ±ÅãÄl, éAþªm> È©, Ò´3 [α, β] þ©ãÈ©. XJ²¡´dwL« y = f(x), x ∈ [a, b] Ñ, Äk±r§w¤ ´ëêL«AϹ ~r(x) = (x, f(x), 0), x ∈ [a, b] Ù¥, x ´ëê, Klúª s = Z b a |~r0 (x)|dx = Z b a q 1 + f 02 (x)dx. ·3c¡Ù!¥Q²íÑù(J. 6/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê 3 ◦ lëê 3 L þÄ: M(t), Klå: A Ä: M(t) l AM s(t) = Z t α q x02 (τ ) + y02 (τ ) + z 02 (τ ) dτ = Z t α |~r0 (τ )| dτ. M 3 L þCz, l s = s(t) ´CþÈ©, Ïd(½ t ¼ê. ù¼êXe5µ Ù´ s(α) = 0, s(t) > 0£Ýo´¤. Ùé t ¦ ds dt = q x02 (t) + y02 (t) + z 02 (t) = |~r0 (t)|. Ïd, |~r0 (t)| 6= 0, ¼ê´îüNO£lAÛþw, XÄ:? ò, Ýo´îO\¤. 7/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê ¤±¼ê s = s(t) k¼ê t = t(s), Ò´ëê t ±L«¤ s ¼ ê. \ëê§L«, ~r = ~r(s) = ~r(t(s)) ½ö x = x(s), y = y(s), z = z(s), Ò´`, ±^±l s ëêëê§5L«. Ïlëê s ´AÛþ, §´ ¤k, ØëêL« ØÓ UC. Ïd·rdkAÛþlëêëê §L«¡mg,§. 8/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê ,¡, ÏL©úª ds = q x02 (t) + y02 (t) + z 02 (t) dt = |~r0 (t)| dt, k ds2 = dx2 + dy2 + dz2 . ùÒ´ml©úª. XJr ds w¤´lÝ, K dx, dy, dz ©O´ ds 3nI¶þÝK, Ïdþª±w¤´/ ª½n. Ï d~r = (dx, dy, dz) ¤± |d~r| = |~r0 (t)| · |dt|, XJ ~r = ~r(s) é s ¦, K d~r ds = d~r dt dt ds = 1. 9/22 kJ Ik J I £ ¶ '4 òÑ
þ l lúª m¡ Û¼ê¡ Û¼ê ½¤ dx ds2 + dy ds2 + dz ds2 = 1. Ò´`, 3g,ëê§e, ~r(s) él s û´ L 3: M(s) ?ü þ, ¿l s O\. ePùþn{ u (cos α, cos β, cos γ), Kk dx ds = cos α, dy ds = cos β, dz ds = cos γ. ²¡ ~r = (x(t), y(t)) ±w¤AÏm ~r = ~r(x(t), y(t), 0), dd2gã L : ~r = (x(t), y(t)), t ∈ [α, β] lúª s = Z β α q x02 (t) + y02 (t) dt. éuÅã1w, Ùl½Âã1wlÚ. 10/22 kJ Ik J I £ ¶ '4 òÑ