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. ...

... 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 ...

...Laplace's equation in polar coordinates is: (1.1) We ... the equation: (1.3) Since the functions are completely ... This function is single values, which means that. ...

... We are supposed to convert this function to Cartesian coordinates. ... means that we are to consider the function g ( x ... e) The Laplacian can be determined from its ...

... 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 ...

... the Laplace's equation solution in cylindrical coordinates, find the ... This in itself does satisfy the Laplace equation ... The Z_0 is constant as a function of theta ...