Differentiation theorems for multiple integralsWe begin with a result which is closely related to the Fundamental Theorem of Cal-culus.THEOREM 1. Suppose that f(x) is a continuous function defined on an open intervalJ, and let x0∈ J be arbitrary. Then there is some r1> 0 such that (x0− r, x0+ r) iscontained in J for 0 < r ≤ r1, andlimr→012r·Zx0+rx0−rf(t) dt = f(x0) .Derivation of Theorem 1. Define the antiderivative function to f byg(x) =Zxx0f(t) dtif x ≥ x0and byg(x) = −Zx0xf(t) dtif x ≤ x0; both definitions yield g(x0) = 0, so the functions fit together to form a differen-tiable function on the interval (x0−r1, x0+r1). By the Fundamental Theorem of Calculuswe also know that g0(t) = f(t) for all t in the given interval.Now let 0 < r < r1. Then by the Mean Value Theorem there is some Yrin (x0−r, x0+ r) such thatg(x0+ r) − g(x0− r)2r= f(Yr)and the identities g(x0+ r) =Rx0+rx0f(t) dt, g(x0− r) = −Rx0x0−rf(t) dt allow us torewrite this as follows:12r·Zx0+rx0−rf(t) dt =12r·Zx0+rx0f(t) dt +Zx0x0−rf(t) dt=g(x0+ r) − g(x0− r)2r= f(Yr)Since f is continuous and |Yr− x0| < r, it follows that the limit of the right hand side asr → 0 is equal to f(x0), and therefore the limit of the left hand side as r → 0 is also equalto f(x0).It turns out that Theorem 1 extends to functions of two, three, and even more realvariables; we shall only consider the cases with two and three variables since these are thecentral objects in the first seven chapters of the course text. In order to state these results,we need one piece of notation.Definition. Let x be a point in coordinate n-space (where we may restrict to n = 2, 3 ifwe wish), and let r > 0. The closed disk of radius r with center x, written Dr(x), is theset of all y in coordinate n-space such that |y − x| ≤ r.We shall state the 2-dimensional and 3-dimensional versions of the Multivariable Dif-ferentiation Theorem separately. It is possible to state all the higher dimensional versionsof this theorem in a unified fashion, but we shall pass on doing so here (details can befound in graduate level courses on integration theory).THEOREM 2. Suppose that f(x) is a continuous function defined on an open region Uin the coordinate plane, let x0∈ U be arbitrary, and let A(r) = πr2. Then there is somer1> 0 such that Dr(x0) is contained in U for 0 < r ≤ r1, andlimr→01A(r)·Z ZDr(x0)f(u) dA = f (x0) .THEOREM 3. Suppose that f(x) is a continuous function defined on an open regionU in coordinate 3-space, let x0∈ U be arbitrary, and let V (r) =43πr3. Then there is somer1> 0 such that Dr(x0) is contained in U for 0 < r ≤ r1, andlimr→01V (r)·Z Z ZDr(x0)f(u) dV = f(x0) .Both of these results can be established using mean value theorems for multiple inte-grals like those formulated for double and triple integrals on pages 473 and 445 (respec-tively) of the course text. However, we shall use a different and more direct (but also moreabstract) approach, which uses the classical definition of continuity:WEIERSTRASS DEFINITION OF CONTINUITY. A real valued function f is continuousat a point x if and only if for each ε > 0 there is some δ > 0 such that |f(y) − f(x)| < εprovided |y − x| < δ.We shall also need the following standard upper estimates for integrals:Let f be a continuous function defined on the closed region Dr(x0) in the coordinate planeor coordinate 3-space. Then there is a positive constant M > 0 such that |f (y)| < M forall y in the given region, and the associated double or triple integral satisfies the respectiveinequalityZ ZDr(x0)f(x) dA< M · A(r) ,Z Z ZDr(x0)f(x) dV< M · V (r)where A(r) and V (r) are defined as before.The 2-dimensional version of this inequality is mentioned in one of the commentarieson Chapter 5 in this directory.Proof of Theorem 2. This argument is at the level of an introductory real variablescourse such as Mathematics 151A.Let ε > 0. We need to find some δ > 0 such that1A(r)·Z ZDr(x0)f(u) dA − f(x0)< εprovided r < δ. We can rewrite the expression inside the absolute value sign as1A(r)·Z ZDr(x0)f(u) dA −1A(r)·Z ZDr(x0)f(x0) dA =1A(r)·Z ZDr(x0)f(u) − f (x0)dA .Since f is continuous and ε > 0, we know that there is some δ > 0 such that |f (y)−f (x0)| <ε provided |y − x| < δ. Therefore if r < δ then we have1A(r)·Z ZDr(x0)f(u) − f (x0)dA<1A(r)· ε · A(r) = εand therefore we have the desired limit formula.Proof of Theorem 3. This argument is a fairly straightforward modification of theprevious one.Let ε > 0. We need to find some δ > 0 such that1V (r)·Z Z ZDr(x0)f(u) dA − f(x0)εprovided r < δ. We can rewrite the expression inside the absolute value sign as1V (r)·Z Z ZDr(x0)f(u) dV −1V (r)·Z Z ZDr(x0)f(x0) dV =1V (r)·Z Z ZDr(x0)f(u) − f (x0)dV .Since f is continuous and ε > 0, we know that there is some δ > 0 such that |f (y)−f (x0)| <ε provided |y − x| < δ. Therefore if r < δ then we have1V (r)·Z Z ZDr(x0)f(u) − f (x0)dV<1V (r)· ε · V (r) = εand therefore we have the desired limit formula.A related result for vector fieldsIn certain contexts (like the derivations of Faraday’s Law and Amp`ere’s Law) oneneeds a version of the Multidifferentiation Theorem for 3-dimensional vector fields insteadof scalar valued functions. We shall formulate the statement of such a result and make afew comments on how it can be recovered from Theorem 2.Let F be a vector field defined in a region of 3-dimensional coordinate space (as usual,the coordinate functions are assumed to have continuous partial derivatives). Given apoint p = (c1, c2, c3) in the region, a typical plane through p has a parametrization of theformX(u, v) = (a1u + b1v + c1, a2u + b2v + c2, a3u + b3v + c3)where the vectors X1= (a1, a2, a3) and X2= (b1, b2, b3) are linearly independent; thelatter is equivalent to assuming that their cross product X1× X2, which is just the normalvector N to the plane, is nonzero.We shall be interested in parametrized pieces of the plane which are obtained byrestricting X to disks of the form u2+ v2≤ r2for r sufficiently close to zero (specifically,we want r so small that the image of this piece of the plane will lie inside the region onwhich the vector field F is defined). The parametrized surface obtained in this