z=z(x,y)定义在全平面上,(1)若f'x(x,y)=0,试证z=f(y); (2)若f'xy(x,y)=0,试证z=g(x)+f(y)
问题描述：
z=z(x,y)定义在全平面上,(1)若f'x(x,y)=0,试证z=f(y); (2)若f'xy(x,y)=0,试证z=g(x)+f(y)
是偏导数和全微分这一章的题
答
1、由单变元的微分中值定理,有f(x,y)－f(x0,y)＝f'x(c,y)*(x－x0)＝0,于是f(x,y)的值只与y有关,故z＝f(y).2、由1知道,当f'xy(x,y)＝0时,f'y(x,y)是y的函数,即f'y(x,y)＝h(y),于是f(x,y)＝h(y)的不定积分＝s(y)+C(x),...