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函数u=ln1/r,r=(x^2+y^2+z^2)^1/2,求U对x,y,z的二次偏导数之和,过程.谢谢!
题目详情
函数u=ln1/r,r=(x^2+y^2+z^2)^1/2,求U对x,y,z的二次偏导数之和,过程.谢谢!
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答案和解析
u=-lnr=-1/2*ln(x^2+y^2+z^2)
du/dx=-1/2*1/(x^2+y^2+z^2)*2x=-x/(x^2+y^2+z^2),
du/dy=-1/2*1/(x^2+y^2+z^2)*2y=-y/(x^2+y^2+z^2),
du/dz=-1/2*1/(x^2+y^2+z^2)*2z=-z/(x^2+y^2+z^2),
d^2u/dx^2=-1/(x^2+y^2+z^2)+2*x^2/(x^2+y^2+z^2)^2
d^2u/dy^2=-1/(x^2+y^2+z^2)+2*y^2/(x^2+y^2+z^2)^2
d^2u/dz^2=-1/(x^2+y^2+z^2)+2*z^2/(x^2+y^2+z^2)^2
则U对x,y,z的二次偏导数之和=-3/(x^2+y^2+z^2)+2/(x^2+y^2+z^2)=-1/(x^2+y^2+z^2)
du/dx=-1/2*1/(x^2+y^2+z^2)*2x=-x/(x^2+y^2+z^2),
du/dy=-1/2*1/(x^2+y^2+z^2)*2y=-y/(x^2+y^2+z^2),
du/dz=-1/2*1/(x^2+y^2+z^2)*2z=-z/(x^2+y^2+z^2),
d^2u/dx^2=-1/(x^2+y^2+z^2)+2*x^2/(x^2+y^2+z^2)^2
d^2u/dy^2=-1/(x^2+y^2+z^2)+2*y^2/(x^2+y^2+z^2)^2
d^2u/dz^2=-1/(x^2+y^2+z^2)+2*z^2/(x^2+y^2+z^2)^2
则U对x,y,z的二次偏导数之和=-3/(x^2+y^2+z^2)+2/(x^2+y^2+z^2)=-1/(x^2+y^2+z^2)
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