设有数量场u=ln(x^2+y^2+z^2),则div(gradu)=

题目

设有数量场u=ln(x^2+y^2+z^2),则div(gradu)=

答案

əu/əx=2x/(x^2+y^2+z^2),
əu/əy=2y/(x^2+y^2+z^2),
əu/əz=2z/(x^2+y^2+z^2),
gradu=(əu/əx, əu/əy, əu/əz)=(2x/(x^2+y^2+z^2), 2y/(x^2+y^2+z^2), 2z/(x^2+y^2+z^2))
div(gradu)=ə²u/əx²+ ə²u/əy²+ ə²u/əz²
=2[3(x²+y²+z²)-2(x²﹢y²﹢z²)] /(x²+y²+z²)²,
=2/(x²+y²+z²)