高维微分学——曲线向量值映照 谢锡麟 将式|(t)ls=((t),(t)gs的两端求对t的导数得 2()a()3=2((o,r() 所以有 (r(t), r(t))23 此时 r"'(s(t)= r(t) (r(t),r(t))R (t)12|(t) F()-(e(),r() r(t) r(t)R3 Ir(t)lRs/R3 i(t)l23 根据内蕴正交分解,对于向量T,有 T=(T,e)ke-e×e×T) 令T=计(,e=m1可得 Ir(tlr3 r"(s(t)= r(t)×((t)×f(t) Fr(t)13i(t)lR3(/r (t) x r(t) r(tI 因此主法向量为 m(t r"(s()) r(t)×((t)×(t) r(t)×(r(t) r"(s(t)l3|i(t)×(T(t)×(t)8|()Rs(t) rola 此时曲率可以表示为 k(t)=|r"(s(1)=1(t)×(r()×(t)k3|r()×(t)k3 r(t)193 r(t)1 副法向量即为 b()=T(t)×n(t) (t) (t)×(t) p(t)Ie3 (t)123 F(t)-(F(t), r(t)l/gsP()」-1r() 挠率为 db dt b(s(t)),n(t) a()as(),n() 1d(t)×(t)(t)×(r r( t)lR3 dt i(1)xr(t)l3r(t)2sr(1)×r(t)3丿 r(t)×((t)×(t) r(t)lgs|(t)×(t)k3'|(t)l(t)×(t)/3 T(t)×(t) ) r(t)×(t) r(t) r(t) P(tIR3(i(t)lr3 (t) 、x:(0010+(m0)0 (t)R3/R3|r(t)IR3/r3 r(t),(t),(t) r(t)×(t) (f(t)×T(t),f(t)g3 (t)×r(t)l微积分讲稿 谢锡麟 高维微分学—— 曲线向量值映照 谢锡麟 将式 |r˙(t)| 2 R3 = (r˙(t), r˙(t))R3 的两端求对 t 的导数得 2|r˙(t)|R3 d dt |r˙(t)|R3 = 2 (r¨(t), r˙(t))R3 , 所以有 d dt |r˙(t)|R3 = 1 |r˙(t)|R3 (r¨(t), r˙(t))R3 , 此时 r ′′(s(t)) = r¨(t) |r˙(t)| 2 R3 − (r¨(t), r˙(t))R3 |r˙(t)| 4 R3 r˙(t) = 1 |r˙(t)| 2 R3 [ r¨(t) − ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ] . 根据内蕴正交分解, 对于向量 T , 有 T = (T , e)R3 e − e × (e × T ). 令 T = r¨(t), e = r˙(t) |r˙(t)|R3 可得 r ′′(s(t)) = − 1 |r˙(t)| 2 R3 r˙(t) |r˙(t)|R3 × ( r˙(t) |r˙(t)|R3 × r¨(t) ) = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)| 4 R3 . 因此主法向量为 n(t) = r ′′(s(t)) |r ′′(s(t))|R3 = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t) × (r˙(t) × r¨(t))|R3 = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 . 此时曲率可以表示为 κ(t) = |r ′′(s(t))|R3 = |r˙(t) × (r˙(t) × r¨(t))|R3 |r˙(t)| 4 R3 = |r˙(t) × r¨(t)|R3 |r˙(t)| 3 R3 . 副法向量即为 b(t) = τ (t) × n(t) = r˙(t) |r˙(t)|R3 × 1 |r˙(t)| 2 R3 [ r¨(t) − ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ] = r˙(t) × r¨(t) |r˙(t)| 3 R3 . 挠率为 σ(t) = − ( d ds b(s(t)), n(t) ) R3 = − ( db dt (t) dt ds (s), n(t) ) R3 = − ( 1 |r˙(t)|R3 d dt r˙(t) × r¨(t) |r˙(t) × r¨(t)|R3 , − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 ) R3 = ( r˙(t) × ... r (t) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 , r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 ) R3 = 1 |r˙(t) × r¨(t)| 2 R3 ( r˙(t) × ... r (t), r˙(t) |r˙(t)|R3 × ( r˙(t) |r˙(t)|R3 × r¨(t) )) R3 = 1 |r˙(t) × r¨(t)| 2 R3 ( r˙(t) × ... r (t), −r¨(t) + ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ) R3 = − 1 |r˙(t) × r¨(t)| 2 R3 (r˙(t) × ... r (t), r¨(t))R3 = [r˙(t), r¨(t), ... r (t)] |r˙(t) × r¨(t)| 2 R3 , 5