[zeshkani]

Find the result of operating ∇2 on ( x2 + y2 + z2 ). Is ( x2 + y2 + z2 ) an eigenfunction of ∇2 ? Try
the same problem in polar coordinates; the answer should be the same.

iam totally lost on this question, dont even know were to start

[Yggdrasil]

Do you know what the ∇² operator is?  If not, a good starting point will be the following wikipedia article, especially the section about the Laplacian in three-dimensions:

http://en.wikipedia.org/wiki/Laplacian
http://en.wikipedia.org/wiki/Laplacian#Three_dimensions

[FeLiXe]

just do the derivations and summations in cartesian coordinates
then change x^2+y^2+z^2 to polar coordinates and apply the Laplace from wikipedia

[Martingale]

Find the result of operating ∇2 on ( x2 + y2 + z2 ). Is ( x2 + y2 + z2 ) an eigenfunction of ∇2 ? Try
the same problem in polar coordinates; the answer should be the same.

...

Isn't polar coordinates a two-dimensional coordinate system? 

[FeLiXe]

in this case they are certainly talking about 3-dimensional spherical polar coordinates
the definition is for example here: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates