If `x=rcosthetaandy=rsintheta` , prove that `(del^2r)/(delx^2)+(del^2r)/(dely^2)=1/r[((delr)/(delx))^2+((delr)/(dely))^2]`

If u= x^3-3xy^2 , show that (del^2u)/(delx^2)+(del^2u)/(dely^2) =0

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 x= r cos theta, y = r sin theta , then (del r)/(del x) =_________

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 = x/y^2- y/x^2 , show that (del^2u)/(delxdely)=(del^2u)/(delydelx) .

If x = r cos theta, y = r sin theta then prove that (del (x, y)) / (del (r, theta)) xx (del (r, theta)) / (del (x, y)) = 1

If w(x,y) = xy + sin(xy) , then prove that (del^(2) w)/(del y del x) = (del^(2) w)/(del x del y)

If V(x,y) = e^(x) (x cos y - y siny) , then prove that (del^(2)V)/(del x^(2)) + (del^(2) V)/(del y^(2)) =0

If u= tan^-1((y^5)/(x^2)) , then find i. x((delu)/(delx)) + y((delu)/(dely)) ii. x^2((del^2u)/(delx^2))+2xy((del^2u)/(delxdely))+y^2((del^2u)/(dely^2))