高维微分学—向量值映照的可微性 谢锡麟 证明方法一按映照观点(aA+BB)(t)∈R3可理解为 A1+BB1)(t) R3t→(aA+B()=a42()+B|B2()=(aA2+BB2)t)∈R3 A3+BB3)(t) 由此,可有 dal dB dA dB dt (aA+BB(t) d A2 dB da (t)+B dB2 d a3 dB d A3 d B3 (t) dB dt (t)+B(t) ∈ dt 类似地,有 R彐t B)g(t)=(41B1)(t)+(A2B2)(t)+(A3B3)(t)∈R 由此,可有 (A,B)g3(t) (aB2)(0+(8B2)()+ dal das BI B3)()+ dt )(t) 3dB d (t) dA (t,B(t)+(A(t),(t)∈R 另有 A2B3-A3B R3t→(A×B)(t) A2A()=A3B1-4B3(t)∈R BI B2B B2-A2Bl 可有 da3 2dB dal d Bl d B3 (A×B)(t) B3()+|A dA1n,d∠ dB2 dBl BI dt dB (t)×B(t)+A(t)×=(t)∈R 方法二按极限分析.考虑到 dA 全im (t+△t)-A(t)微积分讲稿 谢锡麟 高维微分学—— 向量值映照的可微性 谢锡麟 证明 方法一 按映照观点. (αA + βB)(t) ∈ R 3 可理解为 R ∋ t 7→ (αA + βB)(t) = α     A1 A2 A3     (t) + β     B1 B2 B3     (t) =     (αA1 + βB1 )(t) (αA2 + βB2 )(t) (αA3 + βB3 )(t)     ∈ R 3 . 由此, 可有 d dt (αA + βB)(t) =          ( α dA1 dt + β dB1 dt ) (t) ( α dA2 dt + β dB2 dt ) (t) ( α dA3 dt + β dB3 dt ) (t)          = α         dA1 dt dA2 dt dA3 dt         (t) + β         dB1 dt dB2 dt dB3 dt         (t) = α dA dt (t) + β dB dt (t) ∈ R 3 . 类似地, 有 R ∋ t 7→ (A, B)R3 (t) = (A 1B 1 )(t) + (A 2B 2 )(t) + (A 3B 3 )(t) ∈ R. 由此, 可有 d dt (A, B)R3 (t) = ( dA1 dt B 1 ) (t) + ( dA2 dt B 2 ) (t) + ( dA3 dt B 3 ) (t) + ( A 1 dB1 dt ) (t) + ( A 2 dB2 dt ) (t) + ( A 3 dB3 dt ) (t) = ( dA dt (t), B(t) ) R3 + ( A(t), dB dt (t) ) R3 ∈ R. 另有 R ∋ t 7→ (A × B) (t) ,         i j k A1 A2 A3 B1 B2 B3         (t) =     A2B3 − A3B2 A3B1 − A1B3 A1B2 − A2B1     (t) ∈ R 3 , 可有 d dt (A × B)(t) =         dA2 dt B3 − dA3 dt B2 dA3 dt B1 − dA1 dt B3 dA1 dt B2 − dA2 dt B1         (t) +         A2 dB3 dt − A3 dB2 dt A3 dB1 dt − A1 dB3 dt A1 dB2 dt − A2 dB1 dt         (t) = dA dt (t) × B(t) + A(t) × dB dt (t) ∈ R 3 . 方法二 按极限分析. 考虑到 ∃ dA dt (t) , lim ∆t→0 A(t + ∆t) − A(t) ∆t ∈ R 3 8