問一題vector field

Determine whether or not Ｆ is a conservative vector.



【Ｆ(x,y)=[ y(e^x) (siny)]i ＋ [(e^x) x(cosy)]j 】



If it is, let ▽f(x,y)=Ｆ. Find f(x,y)=?





Solve:



f(x,y)對x偏微分 = [ y(e^x) (siny)]



∴f(x,y) = [ y(e^x) x(siny) ] g(y)



f(x,y)對y偏微分 = e^x x(cosy) g'(y) = (e^x) x(cosy)



g'(y) = 0 g(y) = K





f(x,y) = y(e^x) x(siny)



and F is conservative #