Laplacian in spherical, glomar coordinates

[alkaline]

I have calculated the laplacian in glomar coordinates. That was quite the task.

The laplacian in spherical coordinates:
(L)[sup]2[/sup] = (d/dr)[sup]2[/sup] + 1/r[sup]2[/sup]*(d/dθ)[sup]2[/sup] + 1/r[sup]2[/sup]sin[sup]2[/sup]θ*(d/dφ)[sup]2[/sup]

The laplacian in glomar coordinates:
(L)[sup]2[/sup] = (d/dr)[sup]2[/sup] + 1/r[sup]2[/sup]*(d/dθ)[sup]2[/sup] + 1/r[sup]2[/sup]sin[sup]2[/sup]θ*(d/dφ)[sup]2[/sup] + 1/r[sup]2[/sup]sin[sup]2[/sup]θsin[sup]2[/sup]φ*(d/dω)[sup]2[/sup]

[alkaline]

actually, it looks like these might be wrong. The spherical coordinate version is supposed to be:

(L)[sup]2[/sup] = 2/r*d/dr + (d/dr)[sup]2[/sup] + 1/r[sup]2[/sup]*cosθ/sinθ*d/dθ + 1/r[sup]2[/sup]*(d/dθ)[sup]2[/sup] + 1/r[sup]2[/sup]sin[sup]2[/sup]θ*(d/dφ)[sup]2[/sup]

[Aale de Winkel]

(L)[sup]2[/sup] = 2/r*d/dr + (d/dr)[sup]2[/sup] + cosθ/sinθ*d/dθ + 1/r[sup]2[/sup]*(d/dθ)[sup]2[/sup] + 1/r[sup]2[/sup]sin[sup]2[/sup]θ*(d/dφ)[sup]2[/sup]

I don't know why Eric Weisstein put it into the form shown in:
http://mathworld.wolfram.com/Laplacian.html
a quick further derivation of that version shows you are probably correct with the above "quoted version" A further look though, derivation of wolframs third term your third term looks more like: [1/(r[sup]2[/sup]tan(θ))] δ/δθ (so you missed deviding by r[sup]2[/sup])
the rest of your terms I could reobtain from Eric's more compact version!

I don't know though whether it is correct to write δ[sup]2[/sup]/δθ[sup]2[/sup] as (δ/δθ)[sup]2[/sup]

Boy you have me reaquainting with stuff I have forgotten while back , It'l take me a while to understand these things enough to figure the glomar laplacian, or perhaps the more general n-spherical.

I thought the upsidedown delta was called nabla but that doesn't seem to be recognized

[alkaline]

yes you're right, i forgot the 1/r[sup]2[/sup] before the third term. I have fixed it above. The form that Eric Weisstein has the Laplacian is the same form that appears in my nonclassical physics book. I would assume that it is prefered because it is more compact.

As for (δ/δθ)[sup]2[/sup], I think you can write as δ[sup]2[/sup]/δθ[sup]2[/sup] by convention (i'm not really sure).