设X=e∧u·cosv,Y=e∧u·sinv,z=uv.求аz／аx,аz／аy.

Question

设X=e∧u · cos v,Y=e∧u·sin v,z = uv .求аz／аx,аz／аy.

Answer 1

x=e^u*cosv,对x求导：1=e^ucosv* u'x-e^u sinv*v'x,得：cosv*u'x-sinv*v'x=1/e^u
y=e^u*sinv,对x求导：0=e^u sinv*u'x+e^ucosv*v'x,得：sinv*u'x+cosv*v'x=0
解得：u'x=cosv/e^u,v'x=-sinv/e^u
Z'x=Z'u* u'x+Z'v*v'x=v*u'x+u*v'x=(vcosv-usinv)/e^u
x=e^u*cosv,对y求导：0=e^ucosv* u'y-e^u sinv*v'y,得：cosv*u'y-sinv*v'y=0
y=e^u*sinv,对y求导：1=e^u sinv*u'y+e^ucosv*v'y,得：sinv*u'y+cosv*v'y=1/e^u
解得：u'y=sinv/e^u,v'y=-cosv/e^u
Z'y=Z'u* u'y+Z'v*v'y=v*u'y+u*v'y=(vsinv-ucosv)/e^u