二阶微分,若x是自变量,那么dx^2是否等于2xdx,或者dx^2=(dx)^2?
问题描述:
二阶微分,若x是自变量,那么dx^2是否等于2xdx,或者dx^2=(dx)^2?
还有d^2x如何定义?对于y=uv,u=u(x),v=v(x),d^2y如何表示,请写的详细一些.
答
dx^2=(dx)^2,而不是d(x^2).两者是不同的.dy=udv+vdu=(uv'+vu')dxd^2y=d((uv'+vu')dx)=d(uv'+vu')*dx=(v'du+udv'+u'dv+vdu')*dx=(v'u'+uv''+u'v'+vu'')*dx*dx=(uv''+2u'v'+vu'')*dx^2.