Dear Diary,
Dave, my new office mate, is busy doing something. He had an issue finding the number of particles from a density, so he asked me what could possibly be wrong.
I immediately leapt to the error that I always (and still sometimes do) make: we work in one dimension, so integrating the density requires just integrating the density. Then, all of a sudden, maybe Kieron asks a 3D question in the ABCs and you forget to put in the Jacobian (4*Pi*r^2). But once you realize that, everything is solved!*
He then promised to take me to dinner, but I'm not sure if I earned it just yet...and now he's excited dancing--sort of mosh pit style. You go, Dave.
*-Oh no...here comes a harder question about Thomas-Fermi theory and Lieb-Oxford bounds...
The Jacobian for the cartesian-spherical coordinates transformation is r^2 \sin \theta. Only if the integrand is spherically symmetric and only after integration over the complete domain of the solid angle one obtains 4\pi r^2.
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