设函数u=f(r),r=√(x^2+y^2+z^2),则э^2u/эx^2+э^2u/эy^2+э^2u/эz^2=

1个回答

  • эu/эx=f'(r)*эr/эx

    =f'(r)*x/r

    э^2u/эx^2=f''(r)*(x/r)^2+f'(r)*(r-x*x/r)/r^2

    =f''(r)*(x/r)^2+f'(r)*(r^2-x^2)/r^3

    同理

    э^2u/эy^2=f''(r)*(y/r)^2+f'(r)*(r^2-y^2)/r^3

    э^2u/эz^2=f''(r)*(z/r)^2+f'(r)*(r^2-z^2)/r^3

    所以

    э^2u/эx^2+э^2u/эy^2+э^2u/эz^2

    =f''(r)*(x/r)^2+f'(r)*(r^2-x^2)/r^3+f''(r)*(y/r)^2+f'(r)*(r^2-y^2)/r^3+f''(r)*(z/r)^2+f'(r)*(r^2-z^2)/r^3

    =f'(r)*(x^2+y^2+z^2)/r^2+f''(r)*[3r^2-(x^2+y^2+z^2)]/r^3

    =f'(r)+2f'(r)/