If `z=x+y^x`, Prove that `(del^2z)/(delxdely)=(del^2z)/(delydelx)`

If H=f(y-z,z-x,x-y) Prove that (del H)/(del x)+(del H)/(del y)+(del H)/(del z)=0

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 u=log(x^(2)+y^(2)+z^(2)) prove that (del^(2)u)/(del x^(2))+(del^(2)u)/(del y^(2))+(del^(2)u)/(del z^(2) = 2/(x^(2)+y^(2)+z^(2)

If u = x ^ (2) tan ^ (- 1) ((y) / (x)) - y ^ (2) tan ^ (- 1) ((x) / (y)), prove that (del ^ (2) u) / (del x del y) = (x ^ (2) -y ^ (2)) / (x ^ (2) + y ^ (2))

If z (x + y) = x ^ (2) + y ^ (2) show that [(del z) / (del x) - (del z) / (del y)] ^ (2) = 4 [1 - (del z) / (del x) - (del z) / (del y)]

If y=f(x+ct)+g(x-ct), then show that (del^(2)y)/(del t^(2))=c^(2)(del^(2)y)/(del x^(2))

If u=x^y then find (del u)/(del x) and (del u)/(del y) .

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

If x = r cos theta and y = r sin theta, prove that (del ^ (2) r) / (del x ^ (2)) + (del ^ (2) r) / (del y ^ (2)) = (1) / (r) [((del r) / (del x)) ^ (2) + ((del r) / (del y)) ^ (2)]

If z=ax^(3)+by^(3), where a and b are arbitrary constants, then X((del z)/(del x))+y((del z)/(del y)) is equal to: