Solid State Question: A Confusing Integral in Ashcroft

[kq6up]

I am trying to undersand Ashcroft's derivation of energy density on page 38. I am following most of the arguments save EQ 2.30. Where did the vector go? I did follow my professors argument for converting to polar/spherical when the argument of a function is a modulus of a vector — exploiting symmetry to make a nice easy integral. However, it is a whole other matter two have an actual vector in the integrand. I would expect it to be dotted into something in order to have it equal a scalar (other than zero). I am scratching my head here. The steps leading up to this are a little long, so I am just leaving it as a reference and hoping you guys have access to this textbook.

Thanks,
Chris

 

[MisterX]

I think it's not a vector in the integrand.
$$d\mathbf{k} = dk_x \, dk_y\,dk_z $$
This may be confusing since the total derivative of $\mathbf{k}$ would also be denoted $d\mathbf{k}$, but I think represents something different. However I believe in this case it just means an integral over k space. So just a somewhat confusing notation.

 

[kq6up]

Perfect, thank you.

Chris