Math 510 – SchultzFall 1995Applications to the Heat EquationThe purpose of this section is to explain how the Divergence Theorem can be used to derivethe partial differential equation for heat conduction. A substantial portion of this material isadapted from the book, Introduction to Partial Differential Equations with Applications, by E. C.Zachmanoglou and D. W. Thoe, especially page 156.In dealing with applications of mathematics it is often necessary to develop some theoreticalmaterial that may be of independent interest. We shall begin with some items that will be neededin the derivation of the Heat Equation.PreliminariesThe first thing we need is a mean value theorem for multiple integrals.Mean Value Theorem. Suppose that the function f is continuous on a Type I region D definedby inequalities of the form a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x), h1(x, y) ≤ z ≤ h2(x, y), where thepreceding functions are continuous at the appropriate points. Then there is a point (X, Y, Z) in Dfor whichZ Z ZDf(x, y, z) dx dy dz = f(X, Y, Z) · Volume(D).The description of Type I regions in the theorem corresponds to the usage in the book, VectorCalculus, by J. Marsden and A. Tromba; actually the definition in the latter also allows similarsets of inequalities where the roles of x and y are reversed.Sketch of derivation. Basic theoretical results on continuous functions imply that f takes aminimum value m and a maximum value M on the set D. By the properties of integrals on pages319 of Marsden and Tromba we havem · Volume(D) ≤Z Z ZDf(x, y, z) dx dy dz ≤ M · Volume(D)and if we divide these inequalities by the volume of D we obtain the inequalitiesm ≤R R RDf(x, y, z) dx dy dzVolume(D)≤ M.Another basic fact about continuous functions on connected sets like D states that f takes everyvalue between its minimum and maximum. Choose (X, Y, Z) so that f(X, Y, Z) takes the value ofthe middle term in the second string of inequalities; if both sides of the associated equation aremultiplied by the volume of D, the equation in the theorem will be obtained.This leads to another useful result that is sometimes called the Differentiation Theorem formultiple integrals.Theorem. Let f (x, y, z) be continuous on an open set U containing v0= (x0, y0, z0) and for allr > 0 let D(r) be the closed disk defined by |v − v0| < r. Then we havef(v0) = limr→01Volume(D(r))Z Z ZD(r)f(x, y, z) dx dy dz.Sketch of derivation. According to the preceding Mean Value Theorem the expression on theright hand side is equal to f (Xr, Yr, Zr) for some point (Xr, Yr, Zr) in D(r). Since the points inquestion satisfylimr→0(Xr, Yr, Zr) = (x0, y0, z0)by continuity we also havelimr→0f(Xr, Yr, Zr) = f(x0, y0, z0)and if we make the substitutionf(Xr, Yr, Zr) =1Volume(D(r))·Z Z ZD(r)f(x, y, z) dx dy dzwe obtain the formula in the theorem.We need one more piece of mathematical input.Differentiation under the integral sign. Suppose that F (s, t) is a function of several variableswith continuous partial derivatives for appropriate values of s and t, and letf(t) =ZDF (s, t) ds(the integral can be an ordinary or multiple integral depending on the number of variables repre-sented by s). Then f has a continuous derivative that is given byf0(t) =ZD∂F (s, t)∂tds.The derivation of this formula is not difficult and follows from the principles discussed in Section5.5 of Marsden and Tromba; details can be found in many undergraduate texts on functions of realvariables.Derivation of the Heat EquationLet U be a region in R3, and let u(x, y, z, t) be a function corresponding to the temperaturedistribution as a function of the space coordinates (x, y, z) ∈ U and a real time coordinate t.Given a point (x, y, z) ∈ U, let D be a small closed disk that lies entirely in U, and let S beits boundary sphere. The outward normal at a point of S will be denoted by n. If q(t) denotesthe thermal energy in D at a given time t, the following physical law relates the thermal energyfunction q to the temperature distribution function u:−dqdt= −Z ZSk(x, y, z)∂u∂ndS = −Z ZSk(x, y, z)(∇u) · dSwhere k is the thermal conductivity of the material filling the region U and ∇ is taken with respectto x, y, z. we are assuming that the material is ISOTROPIC; in other words, k does not depend onthe normal direction to S but only on the x, y, z coordinates.Consider the change in q from time t to time t + ∆t:∆q =Z Z ZDc(x, y, z)ρ(x, y, z)[u(x, y, z, t + ∆t) − u(x, y, z, t)] dx dy dzwhere c represents the specific heat of the material and ρ represents the density.If we form the difference quotient∆q∆t, take limits as ∆t → 0, and apply differentiation underthe integral sign we obtain the equationdqdt=Z Z ZDc(x, y, z)ρ(x, y, z)∂∂tu(x, y, z, t) dx dy dz.On the other hand, the previously mentioned physical law and the Divergence Theorem com-bine to imply thatdqdt=Z Z ZD∇ · (k∇u) dx dy dzand standard identities involving ∇ imply that the integrand in the latter expression is equal to∇k · ∇u + k∇2u. Therefore we have shown that the triple integrals of the functionscρ∂u∂tand ∇k · ∇u + k∇2uare equal over the disk D, and therefore the quotients of these integrals by the volume of D arealso equal. Since the radius r of D is arbitrary, the limits of these expressions as r → 0 are equalto the integrands. Thus the equality of the triple integrals implies equality of the integrands; moreprecisely, we havecρ∂u∂t= ∇k · ∇u + k∇2u.If we assume that c, ρ and k are constant this reduces to the standard Heat Equationcρk∂u∂t= ∇2u.Steady state temperature distributions. Physical considerations suggest that a temperaturedistribution will stabilize to some equilibrium distribution after a while. Mathematically, thismeans that we expect the function u to satisfy a relationship of the form limt→∞u(x, y, z, t) =w(x, y, z) where w(x, y, z) represents the equilibrium temperature distribution. This well knownphysical phenomenon is also predicted theoretically by the theory of partial differential equationsand boundary problems, but such topics are beyond the scope of this course. A little formalmanipulation suggests that the equilibrium temperature satisfies the Laplace equation ∇2w = 0.Here is a STANDARD EXERCISE motivated by the preceding discussion: Use Green’s FirstIdentity to show that if ∇2u = 0 on a reasonable closed region R and u = 0 on the boundary Σ ofR, then u = 0 on all of R.This has the