Using the general definition of the density of states ( ν(ϵ) = 2Vk δ(ϵ - ϵₖ) ), calculate this quantity for the three-dimensional electron gas.

a) ( ν(ϵ) = 2Vk )
b) ( ν(ϵ) = 2Vk³ )
c) ( ν(ϵ) = 2V )
d) ( ν(ϵ) = 2k³ δ(ϵ - ϵₖ) )