Laplacian of a function under different coordinates

Please assist with the attached problem.
(a) Calculate the Laplacian of function u(x,y,z) = x^3 - 3xy^2 + z^2 in 3D Cartesian coordinates.
(b) Convert the formula for u into formula for u involving cylindrical polar coordinates. Then compute the Laplacian using the cylindrical polar form. Show that your answer here is the same as your answer from (a).
(c) For this section take u(x,y,z) =x ^2, Convert the formula for u into a formula for u involving spherical polar coordinates. Then compute the Laplacian using the spherical polar form. Check that your answer is 2.

The solution shows detailed calculation of Laplacian in three different 3D coordinate systems: Cartesian (rectangular), cylindrical polar, and spherical.

... Since the Bessel functions are a function of r only, they ... polar integral, and vice versa (the trigonometric functions will not ... The Laplace equation: ∇2Φ = 0. ...

...Laplace equation in cylindrical coordinates is: (1.1) The ... is called the mth modified Bessel function of the ... relations of the trigonometric functions: (1.34) And ...

... 2ψ (r ,θ ) = 0 Is the generalized Laplace equation for polar coordinates. ... to 1 although r=1 is not allowed in the domain of these functions as the ...

... The functions are angular-independent ... We know that the delta function is zero everywhere except at the ... Appendix: The Laplacian of 1/r We would like to calculate ...

... If we multiply the function by sin m ... B1 C 1a k Laplace equation in polar ... of two single-variable completely independent functions: ...

... assumption, W 0 and the vector Laplacian is defined ... all just alpha-numeric representations of functions we need ... y (r) is a scalar function such that ...

Required to solve the attached Laplace Equation problems using ... 1.3) We start by writing the function as a product of two independent functions: (1.4) Then ...

... Writing out the Laplacian in spherical coordinates, this equation ... Since the term to the left is a function of r only and the term to the right is a function...

... In cylindrical coordinates the Laplacian is given by: ... Since the system is in cylindrical coordinates, we require ... for example, if the wave function must vanish ...

... 3. Consider Laplace's equation [see the attachment for the full equation] a ... is the nth root of the Bessel function of order ... The first three functions look like: ...