高维微分学—向量值映照的可微性 谢锡麟 等价于 dA (t+△t)=A(t)+(t)△t+o(△t) 故有 (aA+BB)(t+△t)≡aA(t+△t)+BB(t+△t) dA dB =aA()+a(t)△t+o△t)+B|B()+可(t)△t+o(△) aA+BB)(t) da dB (t)+B-1(t) o(△t) 亦即3a(a4+BB)()=aa(t)+Bat() 类似地,有 (A,B)3(t+△t)≡(A(t+△t),B(t+△t)g3 dA A(t)+x(t)△t+o(△),B(t)+a(t)△t+o(△t dA =(A,B)3(t)+ dB (t),B(t)+(A(),=(t) (A×B)(t+△)≡A(t+△t)×B(t+△t) LA dB 4(t)+x()△t+o(△t) ()+x(t)△t+o(△t) dA dB (A×B)(1)+x()×B(t)+A(1)×x(t)△t+o(△) 由此即有彐(A,B)R(t) dB (t), B(t)+(A(t).at 另外,可有彐(A×B)(t)=-(t)×B(t)+A(t)×=(t) 性质3.2.1.设A∈Rm×n为常数矩阵,b(t)∈Rn,有 R3t++Ab(t)∈Rn 则有 (Ab)(t)=Ax(t)∈R 设A(t)∈Rmxn,b(t)∈Rn,有 IRt+(Ab)(t)= A(tb(t)ER 则有 dt (ab)(t)=dt(t)b(t)+A()d(ER 此处 dAl (t) da dA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 等价于 A(t + ∆t) = A(t) + dA dt (t)∆t + o(∆t), 故有 (αA + βB)(t + ∆t) ≡ αA(t + ∆t) + βB(t + ∆t) = α [ A(t) + dA dt (t)∆t + o(∆t) ] + β [ B(t) + dB dt (t)∆t + o(∆t) ] = (αA + βB)(t) + [ α dA dt (t) + β dB dt (t) ] ∆t + o(∆t), 亦即 ∃ d dt (αA + βB)(t) = α dA dt (t) + β dB dt (t). 类似地, 有 (A, B)R3 (t + ∆t) ≡ (A(t + ∆t), B(t + ∆t))R3 = ( A(t) + dA dt (t)∆t + o(∆t), B(t) + dB dt (t)∆t + o(∆t) ) R3 = (A, B)R3 (t) + [(dA dt (t), B(t) ) R3 + ( A(t), dB dt (t) ) R3 ] ∆ + o(∆t) (A × B)(t + ∆t) ≡ A(t + ∆t) × B(t + ∆t) = [ A(t) + dA dt (t)∆t + o(∆t) ] × [ B(t) + dB dt (t)∆t + o(∆t) ] = (A × B)(t) + [ dA dt (t) × B(t) + A(t) × dB dt (t) ] ∆t + o(∆t), 由此即有 ∃ d dt (A, B)R3 (t) = ( dA dt (t), B(t) ) R3 + ( A(t), dB dt (t) ) R3 . 另外, 可有 ∃ d dt (A × B)(t) = dA dt (t) × B(t) + A(t) × dB dt (t). 性质 3.2. 1. 设 A ∈ R m×n 为常数矩阵, b(t) ∈ R n , 有 R ∋ t 7→ Ab(t) ∈ R m 则有 d dt (Ab)(t) = A db dt (t) ∈ R m. 2. 设 A(t) ∈ R m×n，b(t) ∈ R n , 有 R ∋ t 7→ (Ab)(t) = A(t)b(t) ∈ R m 则有 d dt (Ab)(t) = dA dt (t)b(t) + A(t) db dt (t) ∈ R m, 此处 dA dt (t) :=       dA11 dt · · · dA1n dt . . . . . . dAm1 dt · · · dAmn dt       (t). 9