Question

求证u＝z arctan(x／y)中x,y,z的二阶偏导之和为0

Answer 1

u'x= z/(1+x^2/y^2)* 1/y=zy/(x^2+y^2)

u'y=z/(1+x^2/y^2)* (-x/y^2)=-zx/(x^2+y^2)

u'z=arctan(x/y)

u"xx=-2xyz/(x^2+y^2)^2

u"yy=2xyz/(x^2+y^2)/^2

u"zz=0

因此有u"xx+u"yy+u"zz=0