We now derive the heat equation in three space dimensions. Suppose we have a solid three-dimensional piece of material, with constant mass density ρ, specific heat σ and thermal conductivity k (all as defined in Section 2.2). Suppose also that there is a heat source/sink throughout the material, given by

f(x, y, z, t) = rate at which heat is added/removed, per unit volume,

at point (x, y, z) at time t.