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A college student wakes up from a night of partying and regales his friends with tales of the girl he met last night. "She was amazing. She was continuous everywhere... and her volume evaluated over any arbitrary region vanishes." His friends have themselves a good laugh and reply, "forget it. She's a zero!"

My PDEs prof called this the basic lemma. Though it is very simple and very easy to prove it is very important in deriving the partial differential equations that govern things like vibration, electromagnetism, fluid flow, and diffusion of heat.

The lemma states that a function that is continuous everywhere and whose volume integral over any arbitrary region vanishes is by identity zero.

Proof: Suppose f(x) is continuous everywhere and V is an arbitrary volume in space, then consider ∫∫∫Vf(x)dτ. Suppose the function of f(x) is not zero at some point, then to satisfy continuity, f(x) must be non-zero around that point (i.e. the function will have some sort of hump) and the volume over this region will not be zero. Hence the only function that satisfies these conditions is the zero function so f(x) ≡ 0.

Application: Diffusion of heat:

Given a body of arbitrary shape, bounded by a surface S, we can define the rate of heat transfer from the body to the environment as the sum of the net heat flux through S and the net heat generated or absorbed by the body.

Let us generalize this as the relationship dQv/dt = dQflux/dt + dQgenerated/absorbed/dt

Now we can define a function ψ(x,y,z,t) to be the temperature at a point (x,y,z) on the body and time t, and given the specific heat capacity (c), density (ρ), and thermal conductivity constant (κ) we can "simplify" the above equation using the following relationships:

dQv/dt = (d/dt)∫∫∫Vcρψdτ which by Leibniz's rule is, ∫∫∫Vcρ(∂/∂t)ψdτ

dQflux/dt = ∯Sκ∇ψndS which by the divergence theorem, becomes dQflux/dt = ∫∫∫Vκ2ψdτ

dQgenerated/absorbed/dt = ∫∫∫VQ(x,y,z,t)dτ

So by substituting we obtain the equation:

∫∫∫Vcρ(∂/∂t)ψdτ = ∫∫∫Vκ∇2ψdτ + ∫∫∫VQ(x,y,z,t)dτ

Since all integrals are over V which is the volume of the body in question, remember that V is totally arbitrary, we can combine all of these terms into one volume integral:

∫∫∫V{cρ(∂/∂t)ψ - κ∇2ψ - Q(x,y,z,t)} dτ = 0

Now since all these functions are continuous everywhere in V and V is arbitrary, by the basic lemma:

cρ(∂/∂t)ψ - κ∇2ψ - Q(x,y,z,t) = 0

Supposing that no heat is generated or absorbed by the body then this becomes the more familiar heat equation which is also Poisson's equation:

(1/α2)(∂ψ/∂t) = ∇2ψ where α2 = κ/cρ is the diffusivity constant.

In one dimension this simplifies to ψt = α2ψxx

Here is an application of the basic lemma to reduce an equation involving integration to one much simpler. A very similar approach is seen in deriving equations for vibrating strings, electromagnetism, and incompressible fluid flow among others.

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