Answer

问题及解答

r=x2+y2+z2, 证明: 2(lnr)x2+2(lnr)y2+2(lnr)z2=1r2.

Posted by haifeng on 2023-03-25 08:03:12 last update 2023-03-25 08:03:12 | Edit | Answers (1)

r=x2+y2+z2, 证明: 2(lnr)x2+2(lnr)y2+2(lnr)z2=1r2. 即 Δ(lnr)=1r2.

 

1

Posted by haifeng on 2023-03-25 08:11:39

x(lnr)=1rrx=1rxx2+y2+z2=1rxr=xr2

类似的,

y(lnr)=yr2,z(lnr)=zr2

 

2x2(lnr)=x(xr2)=1r2x2rrxr4=r22rxxrr4=r22x2r4.

类似的, 有

2y2(lnr)=r22y2r4,2z2(lnr)=r22z2r4.

于是

Δ(lnr)=(2x2+2y2+2z2)(lnr)=r22x2r4+r22y2r4+r22z2r4=3r22r2r4=1r2.