首先注明式中d表示微分.A,B皆为向量.*为叉乘.需要证明一下d(A*B)/dt=(dA/dt)*dB+A*(dB/dt).

Question

首先注明式中d表示微分.A,B皆为向量.*为叉乘.
需要证明一下d(A*B)/dt=(dA/dt)*dB+A*(dB/dt).

Answer 1

用坐标法证明之.
设A(t)=(x(t),y(t),z(t)),
B(t)=(a(t),b(t),c(t)).
则A*B=(yc-zb, za-xc, xb-ya)
d(A*B)/dt=(y'c+yc'-z'b-zb', z'a+za'-x'c-xc',
x'b+xb'-y'a-ya')
=(y'c-z'b, z'a-x'c, x'b-y'a)
+(yc'-zb', za'-xc', xb'-ya')
=A'*B+A*B'