If `u=f(x/y,y/z,z/x)`, prove that `x(delu)/(delx)+y(delu)/(dely)+z(delu)/(delz)=0`

If U=(y)/(z)+(z)/(x), then find x(del u)/(del x)+y(del u)/(del y)+z(del u)/(del z) .

If u = (x^2+y^2+z^2)^(-1/2) then prove that (del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2) =0

If |(a,y,z),(x,b,z),(x,y,c)|=0 , then prove that (a)/(a-x)+(b)/(b-y)+(c)/(c-z)=2

For conservative force field, vec(F)=-(delU)/(delx)hat(i)-(delU)/(dy)hat(j)-(delU)/(delz)hat(k) where Frarr Maginitude of Force. Urarr Potential energy and (delU)/(delx)= Differentiation of U w.r.t. x keeping y and z constant and so on. Choose incorrect matching :-

If (log x)/(y-z)=(log y)/(z-x)=(log z)/(x-y), then prove that: x^(x)y^(y)z^(z)=1

If x,y,z gt 0 then prove that (x+y+z)(1/x+1/y+1/z) geq 9

If x,y,z gt 0 and x + y + z = 1, the prove that (2x)/(1 - x) + (2y)/(1 - y) + (2z)/(1 - z) ge 3 .

If x=y^z,y=z^x,z=x^y then

(If(y+z-x))/((x(y+z-x))/(log y))=(y(z+x-y))/(log y)(z(x+y-z))/(log z), prove that x^(y)y^(x)=z^(x)y^(z)=x^(z)z^(x)

If (log x)/(y-z)=(log y)/(z-x)=(log z)/(x-y) then prove that x^(y)+z^(z)+xx^(y+z)+y^(x+x)+z^(x+y)>=3