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The Fundamental Theorem of Calculus in Two Dimensions Bennett Eisenberg; Rosemary Sullivan The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 806-817. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28200211%29109%3A9%3C806%3ATFTOCI%3E2.0.CO%3B2-Y The American Mathematical Monthly is currently published by Mathematical Association of America.
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The Fundamental Theorem of Calculus in Two Dimensions Bennett Eisenberg and Rosemary Sullivan 1. INTRODUCTION. Gottfried Leibniz and Isaac Newton are called the inventors of the calculus, but some of its rudimentary ideas were already known to Archimedes almost two thousand years earlier. For example, Archimedes used a technique called the method of exhaustion to find the area of a region between a parabola and a straight line. He filled the region with triangles and found the area of the region as the sum of an infinite series of areas of triangular regions. With this technique Archimedes could have told us that J; x 2 d x = 113, but not much more. Newton computed the same integral using a much more powerful technique. He generalized the problem to that of finding the integral over [0, t ] for all t . In modern terminology, he introduced the function F ( t ) = x 2 d x and showed that F ' ( t ) = t 2 and F ( 0 ) = 0. He realized that these two conditions determine F uniquely. This idea is the basis of much of differential equations and applied mathematics, where a rate of change and an initial condition are used to determine a function. Today this method for computing integrals is called the Fundamental Theorem of Calculus. It is divided into two parts:
1;
Part I. I f f is a continuous function, then lat f ( x )d x is a differentiable function and d ;i;latf 0 ) and examine dldt flDt f ( x , y) dx d y . For simplicity, we assume throughout this paper that f is a continuous function on R2 and all integrals are Lebesgue integrals. Example 1 illustrates the idea.
Example 1. Let D, be the open set bounded below by the x-axis, on the sides by the lines x = a and x = b(t), and on the top by the graph y = # ( x ) of the positive, continuous function # ( x ) . Then for f > 0, the volume V of the region that lies above D, and beneath the surface S : z = f ( x , y) is given by f ( x , y) dx dy = b(t) $ 4 ~ ) f ( x , y) dy dx. It follows that
la lo
where v ( b ( t ) ,y) = bt(t)is the velocity of the moving boundary and ds = dy is arclength measure on a* D, (see Figure 1 ) . We now extend the ideas in Example 1. Figure 2 suggests that, in a more general setting, A V = & + A t \ ~ t f ( x , y) dx dy can be approximated by a sum C f v At As,
Figure 1. The moving boundary of Dt is the line segment x = b(r).
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Figure 2. A V
-
f v At As
where v(x, y) is the normal velocity of propagation of the boundary of D, at (x, y ) with respect to t and ds is arclength measure on aD,. In other words, the distance from a point (x, y) of aD, to aD,+A, in the direction normal to D, is approximately V(X,Y) At. This suggests in turn that, under reasonable conditions,
This would be a two-dimensional analogue of (3,which itself is a direct generalization of Part I of the fundamental theorem. To give precise meaning to this, we need some conditions on our parameterized family of sets. The sets D, will be described in terms of a function g and a set A.
Definition. The function g : R2 -+ R' is called admissible if (a) g is a nonnegative, continuous function that is of class C' on ( g > 0); (b) the open set {g < t ) is bounded for each t > 0. Condition (a) is written as it is to allow for functions such as g(x, y) = -./,
Definition. The set A is compatible with g if (a) A is open and convex; (b) A n {g < t ) is simply connected for each t > 0; (c) the gradient of g is nonzero on A n { g = t ) for each t > 0. We assume throughout the remainder of this paper that the domains D, are of the form A f l {g < t ) , where g is an admissible function and A is a convex open set compatible with g. We stress that the conditions on g and A are stronger than what is strictly required to prove the main theorems. Our goal is not to prove the theorems in their greatest generality. Rather, our aim is to bring some neglected results to the attention of the MONTHLY readership and to show their relevance to the undergraduate calculus curriculum. The set a D, n { g = t ) = a* D, is called the moving boundary of D,; i.e., the moving boundary of D, is the piece of the boundary of D, that is determined by the level curve November 20021
FUNDAMENTAL THEOREM OF CALCULUS IN 2-D
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Figure 3. A is the region bounded by the x-axis, x = a, and y = @ ( x ) . The function g ( x , y ) = b-' ( x ) .
(g = t } rather than the boundary of A. Figure 3 illustrates the meaning of the moving boundary in the case of Example 1 (where the set A, although well behaved, is not convex). The rate of change of g in the direction normal to the level curve {g = t } is given by [gradgl. If one moves a distance A d in the direction normal to the level curve, the change in the value of g is approximately lgrad glA d. It follows that the perpendicular distance between the level curves {g = t A t } and {g = t } is approximately At/ lgrad g 1. Thus, in the formula (6),v should be equal to lgrad gl-' . These heuristic arguments bring us to the little coarea theorem.
+
Theorem (Little Coarea Theorem). Ifg : R2 + R is an admissiblefunction, A is a convex open set that is compatible with g, and D, = A fl (g < t }for t > 0, then for 0 < S < T and any continuousfunction f : R2 + R ds dt. I f {g = 0} is empty or consists of a single point,
dsdt. The heuristic arguments that we have given motivate the result, but we derive it here as a corollary to the coarea theorem itself. To keep the paper self-contained, we also give an elementary proof (albeit under an additional technical assumption) in Section 5. The following is a version of the coarea theorem (see Evans and Gariepy [I, p. 1171). Coarea Theorem. Let g : Rn + Rm be a Lipschitzfunction, where n >_ m. Thenfor each Lebesgue measurable subset B of Rn and each Lebesgue integrable function h:R"+R,
/, J,d x
=
lm (/,",-la)
h dHn-')
dt.
(9)
The function J, is an appropriate Jacobian for transformations between spaces of different dimensions. It equals the Euclidean norm of the gradient of g when m = 1 810
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and g is differentiable, the case of interest here. The symbols dx and d t indicate integration with respect to n- and m-dimensional Lebesgue measure, respectively. Finally, Hn-" is (n - m)-dimensional Hausdorff measure, a measure that generalizes the idea of arclength and surface area measures to irregular sets. Given all this, it is not surprising that the coarea theorem is not easily accessible even in Morgan's "Beginner's Guide" to geometric measure theory [4]. One purpose of this article is to make this intriguing result more widely appreciated. To see how the little coarea follows from this, let n = 2, m = 1, f = hlgradgl, and B = DT\ Ds. Since B has a compact closure, which does not include {g = 0}, it follows that lgrad g 1 is bounded above and away from 0 on B. Thus, g is a Lipschitz function over this set and f lgrad gl-' is integrable over the set. Now B n g-'(t) = a*D, when S < t < T and is empty otherwise. It follows that (9) can be written
Finally, the implicit function theorem and our assumptions on g imply that a*D, is a smooth curve with a finite arclength, so d H' can be replaced with ds. Returning now to the little coarea theorem, Example 1 shows that it is a generalization of Fubini's theorem for evaluating multiple integrals by foliating a region by vertical lines. The next example illustrates how the little coarea theorem can be used to evaluate a multiple integral by foliating a region by curved lines. Example 2. For t > 0 let D, be the region described in polar coordinates as ((r, 8 ) : r c t, a < 8 < B). In this case g(x, y) = and A is the set (a < Q c B). We see that, for each t, a* D, is parameterized by 8 (i.e., by x = t cos 8, y = t sine for a 5 8 5 B). Thus d s = t dB. Also lgrad g ( = 1 in {g > 0). Accordingly,
J'w,
ds
--
(gradgl
- t dB.
We then have from (8) and (10)
Reversing the order of integration, this becomes
This shows that the little coarea theorem yields as a special case the formula for computing integrals in polar coordinates. The following corollary is the two-dimensional generalization of Part I of the fundamental theorem of calculus. The integral of f over the moving boundary of D, in the right-hand side of (1 1) is the analogue of f (b(t))bl(t) in (5). Corollary 1. If g : R2 -+ R is an admissible function and A is a convex open set compatible with g, then on the interval (0, oo) d
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FUNDAMENTAL THEOREM O F CALCULUS IN 2-D
81 1
Letting H ( t ) = s,& f dx dy for some S < t in (7), we see that the right side of ( 1 1 ) is just the Radon-Nikodym derivative of H. It is thus the ordinary derivative almost everywhere in (0, m ) with respect to Lebesgue measure. To show that the right side of ( 1 1 ) equals the ordinary derivative for all t in (0, oo),it is enough to show that it is continuous. To prove this we would have to break a* D, into sections where either g, or g, is bounded away from zero and apply the implicit function theorem to the function h(x, y, t ) = g(x, y) - t . We omit the details. We can observe the continuity directly in all of our examples. If f = 1 , then &, dx dy = d ( t ) is the area of the region Dr. As a simple application of Corollary 1 we obtain: Corollary 2. Ifg : R2 + R is an admissiblefunction, A is a convex open set compatible with g, and d ( t ) is the area of D,, then on the interval (0, m)
In particular, Corollary 2 implies that d A / d t gives the length of the moving boundary as a function of t in the case where lgrad g 1 = 1 . This observation sheds light on some examples used by Strang [S, pp. 217-220 ] to illustrate certain pitfalls that one might encounter in evaluating an area by breaking a region into infinitesimal regions. The frequent mistake that students make is to give the wrong expression for an infinitesimal element of area d d by using the incorrect expression for the infinitesimal width of the region. Examples 3 and 4 indicate the subtlety of the problem.
d m ,
Example 3. Let g(x, y) = and let A = R2.For t > 0 , D, is the open disc of radius t centered at the origin, and a*D, = a D, is the bounding circle. If one were to use infinitesimal rings heuristically to compute the area of the disc, one would write d d = 2nt dt for the area of one of the rings, since the rings have inner circumference 2nt and width d t . This reasoning works because the distance between aD: and aD:+*, is At, so the velocity of the moving boundary is 1 (see Figure 4 ) . Also, since lgrad gl = 1 in R2\{(0, 0 ) ) ,Corollary 2 shows that d d l d t is the length of the boundary of D,.
Figure 4. The area of the circle is the integral of the areas of the infinitesimal rings.
Things are not so simple in the next example.
+
Example 4. Let g(x, y) = ( x l a ) ( y l b ) for a > 0 and b > 0 , and let A = { ( x ,y) : x > 0 , y > 0 ) . For t > 0 , D, is the triangular region bounded by the x- and y-axes and the line ( x l a ) + ( y l b ) = t . In this case d ( t ) = abt2/2 and d d l d t = abt. This does not equal the length of the the moving boundary (i.e., the hypotenuse of the tri812
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Figure 5. The width of the increment is ab(a2
+ b2)-'I2 At.
angle), so one cannot write the area of an infinitesimal strip along the hypotenuse as the length of the hypothenuse times dt. In this instance (see Figure 5), the distance h between a*D, and a*D,+A,is a At sin% = a b ~t (a2 b2)-'/2, making the velocity of the moving boundary ab(a2 b2)-'/'. This also follows by computing lgradgl-', which has the constant value ab(a2 b2)-'1'. Using Corollary 2, we should multiply this by the length of the moving boundary to get d d / d t . This gives dA/dt = (ab)(a2 b2)-'l2(a2 b2)ll2t = abt, which is the correct value.
+
+
+
+
+
Examples 5 and 6 illustrate how to apply Corollary 2 in some cases where the velocity of the moving boundary is not constant. Example 5. Let g(x, y) = y - h(x), where h is a positive C1-function on R, and let A = {(x, y) : a < x < b, y > O}. Then D, is the region bounded below by the x-axis, t. If the on the sides by the lines x = a and x = b, and on the top - by- -y = h(x) . . moving boundary is parameterized by 9 = x for a 5 x 5 b, we have lgradgl = and d s = , / w d x , whence dsllgrad gl = dx. It follows that d d l d t = jabdx = (b - a), i.e., d d = (b - a ) d t . Figure 6 shows why this result holds.
+
Figure 6. The areas of the two shaded regions are equal.
+
+
Example 6. Let g(x, y) = J(x/a)2 (y/b)2 = r ,/(cos %/a)2 (sin %/b)2 and A = R2. For t > 0, D, = {g < t ) is the region enclosed by the ellipse (x2/a2) (y2/b2) = t2. A simple calculation using polar coordinates then shows that
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FUNDAMENTALTHEOREM OF CALCULUS IN 2-D
+
In this example we know that implies the identity
A = n a b t 2 ,giving d A / d t
= 2nabt. Corollary 2 then
In the case where f is constant on a* D, for each t > 0, we define f * : [0,oo) + R by setting f * ( t )= f ( x , y ) for any ( x ,y ) in a*Dr. Corollary 3 then gives a generalization of the shell method for computing volumes of revolution. Corollary 3 (Generalized Shell Method). I f g : R2+ R is an admissible function and A is a convex open set compatible with g, then
This follows by combining Corollaries 1 and 2. Example 7 shows why it is called the generalized shell method. = t centered at Example 7. In Example 3, if f is constant on each circle f ( x . y) d x dp = 2 n &?t f * ( t )d t , the same formula obtained by the origin, then using the shell method in calculus courses.
hT
The generalization of a volume of revolution is the volume of a region over DT \ DS and under the graph of a function that is constant on a* D, for S < t < T . Example 8. In Example 4, if f is constant on the lines ( x l a ) f ( x . y ) d x d y = ab J: t f * ( t )d l .
JLT,6s
+ ( y l b ) = t , then
Example 9. In Example 5 , let f be constant on the curves y = h ( x ) b h(x)+T Ja S,,),, f ( x . y) d y d x = ( b - a ) J: f * ( t )d t .
+ t . Then
+
Example 10. In Example 6, if f is constant on the ellipses ( x 2 / a 2 ) ( y 2 / b 2 )= t 2 , then f ( x . y ) d x d y = ?nab J: tf * ( t )d t .
llo,,,,
As was stated earlier, the little coarea theorem and its corollaries have analogues in higher dimensions. With the intuition that we have developed, we can perhaps see how to extend the shell method to higher dimensions. For example, let D, be the ball of radius t centered at the origin in R? Let f ( x , y, z ) = f * ( t )be constant on the sphere aDr of radius t . We can think of revolving the graph of the function w = f * ( x )around just the w-axis in four dimensions to get a function f ( x , y, z ) = f as we revolve the graph of z = f * ( x )around the z-axis in three dimensions in the shell Replacing d A / d t by d V / d t , the method to get a function f ( x , y ) = f rate of change of the volume of D,, we find that the four-dimensional volume of revolution JJJDblba f ( x . y. z ) d x d y dz = 4 n Jab t 2f * ( t )d t , a result that can be verified by using sphencal coordinates.
*(dm)
*(dm').
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4. T H E R E L A T I O N B E T W E E N G R E E N ' S T H E O R E M A N D T H E C O A R E A T H E O R E M . There is still one thing that is missing in the generalization of the fundamental theorem of calculus to two dimensions. In the one-dimensional theorem there is an intimate connection between Parts I and 11. Part I essentially says d l d t J: f ( x ) d x = f ( t ) , while Part I1 says Jd f f ( x ) d x = f ( t ) - f ( a ) . This close relationship does not hold between ( 1 1) and Green's theorem. Indeed, the divergence form of Green's theorem involves JS, div F d x d y , the double integral of the divergence of a vector field, whereas ( 1 1) involves J&, f ( x , y ) d x d y , the double integral of a real-valued function. However, with the proper substitutions a connection can be established. In what follows D, is a bounded, open, simply connected set, and the set A in the little coarea theorem is R2, making a*Dl = a D,. In the version of Green's theorem given in formula ( 2 ) , assume that D = D l , where Dl = { g < t ] for t > 0 and g is an admissible function with nonvanishing gradient. Then N can be written lgrad g I - ' grad g and Green's theorem becomes
Next, in formula (1 1) set f = F . grad g . For this choice of f , ( 1 1 ) asserts that
Thus, when f and F are related in this way, the right-hand sides of ( 2 ) and (1 1 ) match, and the left-hand sides both involve the same vector field F . In doing this, we have derived the following interesting corollary to ( 2 ) and ( 1 1): for any smooth vector field F on R2,
This result has an interesting one-dimensional analogue. For example, if g ( x ) = Ix 1 , we have Dl = { g < t ] = ( - t , t ) , g l ( x ) = 1 for x > 0 , and g ' ( x ) = - 1 for x < 0 . The one-dimensional analogue of ( 1 3 ) in this case says that
Finally, let us note that there are ways to construct increasing families of domains other than as the interiors of level sets of a function. We could start with a smoothly bounded set D and locally form the family D, = D t V for t 3 0, where V is a smooth, never vanishing vector field that we assume always points out from D. We f ( x , y ) d x d y lIzo.Heuristic arguments similar to those could then consider d l d t given in Section 3 indicate that, under appropriate regularity conditions, this derivative is equal to J,, f ( x , y ) V . N d s , where N = N ( x , y ) signifies the unit normal vector to a D at ( x , y ) that points outward from D . Combining this fact with Green's theorem and taking f = 1 , we conclude that
+
[Ll
yet another interesting connection between Green's theorem and the derivatives of integrals over an expanding family of domains. Moreover, formula ( 1 4 ) has a physical November 20021
FUNDAMENTAL THEOREM OF CALCULUS IN 2-D
815
interpretation when V is the velocity field of a fluid of density one. In that case it shows that the rate of flow of fluid out of a region is equal to the rate of change of the area of the region covered by the fluid; i.e., the loss of fluid from a region causes an equivalent change in the area covered by the fluid. If F = V , D, = D t V , and t = 0 in the left side of (13), then (14) appears to contradict (13).Of course, this is not the case. Let V ( x ,y ) = V,( x , y)i V 2 ( x ,y)j. If we define a function g so that g ( x , y ) = t on aD,, then
+
+
for t 2 0. Assume that g is a C1-function. Differentiating both sides of (15) with respect to t yields
for t > 0 and ( x , y ) in aD. In other words, V . grad g = 1 on a D (when t = 0 ) indidx dy = d l d t &I V . grad g d x d y thus reconciling the cating that d l d t differences between the right-hand sides of formulas (13)and (14).
hr
It=,
It=,
5. AN ELEMENTARY PROOF OF THE LITTLE COAREA THEOREM. An elementary proof of the little coarea theorem of Section 3 can be based on the familiar change of variable theorem for multiple integrals if we are allowed to add an additional technical assumption that is not stated in the little coarea theorem itself. The extra assumption is that an appropriate parametric description of DT\ Ds exists. It is satisfied in all of our examples. Definition. Let g : R 2 + IR be an admissible function, and let A be a convex open subset of R 2 that is compatible with g. We say that A admits smoothparameterizations with respect to g if, whenever 0 < S < T , the open set G = DT \ Ds has the following properties: (i) there exist continuous functions u and v on ( S , T ) and C1-functions x = x ( t , 8 ) a n d y = y ( t , 8 ) on E = { ( x ,y ) : u ( t ) 5 8 5 v ( t ) , S < t < T ) suchthat G = { ( x ,y ) : ( t , 0 ) E E ) ; (ii) the transformation ( t ,0 ) + ( x , y ) is one-to-one with nonvanishing Jacobian determinant J = x,ye - xey, in the interior of E; (iii) for fixed t in ( S , T ) , a*D, = { ( x ( t ,8 ) , y ( t , 8 ) ) : u ( t ) 5 8 5 v ( t ) } . In other words, each moving boundary is parameterized by 8 , the family of boundaries is parameterized by t , and the mapping from ( t , 8 ) to ( x , y ) is one-to-one and smooth. In Example 2, ( t , 8 ) are just the polar coordinates of ( x , y). ProoJ Assume that A admits smooth parameterizations with respect to g. By the change of variable theorem for multiple integrals
Since g ( x ( t , Q ) ,y ( t , 8 ) ) = t for ( t ,8 ) in E , it follows that
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Solving for g, and gi,we find that lgrad g 1 = Thus,
,/= = J=xth
-
xsR 1 - I .
Conclusion (7) follows by integrating (16) with respect to t . We obtain (8) by taking limits as S approaches 0. ยค ACKNOWLEDGMENT. We would like to thank Frank Morgan for an important idea used in Section 4.
REFERENCES 1. L. C. Evans and R. F. Gariepy, Measure Theory and the Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. 2. H. Federer, Curvature measures, Trans. Amer Math. Soc. 93 (1959) 418491. 3. J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, A Unijied Approach, Prentice Hall, Upper Saddle River, NJ, 1999. 4. F. Morgan, Geometric Measure Theory, A Beginner's Guide, 2nd ed., Academic Press, Boston, 1995. 5. G. Strang, Calculus, Wellesley Cambridge Press, Wellesley, MA, 1991. 6. S. Weintraub, Differential Forms: A Complement to Vector Calculus, Academic Press, San Diego, 1996.
BENNETT EISENBERG attended Dartmouth College and MIT and taught at Cornell University and the University of New Mexico before coming to Lehigh. He twice gave a series of lectures on his work in probability and statistics at the University of Osnabmck in Germany. His work with Rosemary Sullivan led to papers on random triangles, Crofton's differential equation, and the fundamental theorem of calculus, all of which appeared in this MONTHLY. Lehigh University Bethlehem, PA 18015 be01 @Lehigh.edu ROSEMARY SULLIVAN received her B.S. from Penn State and her M.S. and Ph.D. from Lehigh University. Before coming to West Chester University she taught at Widener University and Muhlenberg College. She is an MAA Project NexT Fellow. Her main area of interest is geometric probability. She enjoys gardening, travelling, skiing, and hiking. West Chester UniversiQ, West Cheste~PA 19383 [email protected]
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FUNDAMENTAL THEOREM OF CALCULUS IN 2-D
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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http://www.jstor.org Mon Mar 10 19:34:44 2008
The Fundamental Theorem of Calculus in Two Dimensions Bennett Eisenberg and Rosemary Sullivan 1. INTRODUCTION. Gottfried Leibniz and Isaac Newton are called the inventors of the calculus, but some of its rudimentary ideas were already known to Archimedes almost two thousand years earlier. For example, Archimedes used a technique called the method of exhaustion to find the area of a region between a parabola and a straight line. He filled the region with triangles and found the area of the region as the sum of an infinite series of areas of triangular regions. With this technique Archimedes could have told us that J; x 2 d x = 113, but not much more. Newton computed the same integral using a much more powerful technique. He generalized the problem to that of finding the integral over [0, t ] for all t . In modern terminology, he introduced the function F ( t ) = x 2 d x and showed that F ' ( t ) = t 2 and F ( 0 ) = 0. He realized that these two conditions determine F uniquely. This idea is the basis of much of differential equations and applied mathematics, where a rate of change and an initial condition are used to determine a function. Today this method for computing integrals is called the Fundamental Theorem of Calculus. It is divided into two parts:
1;
Part I. I f f is a continuous function, then lat f ( x )d x is a differentiable function and d ;i;latf 0 ) and examine dldt flDt f ( x , y) dx d y . For simplicity, we assume throughout this paper that f is a continuous function on R2 and all integrals are Lebesgue integrals. Example 1 illustrates the idea.
Example 1. Let D, be the open set bounded below by the x-axis, on the sides by the lines x = a and x = b(t), and on the top by the graph y = # ( x ) of the positive, continuous function # ( x ) . Then for f > 0, the volume V of the region that lies above D, and beneath the surface S : z = f ( x , y) is given by f ( x , y) dx dy = b(t) $ 4 ~ ) f ( x , y) dy dx. It follows that
la lo
where v ( b ( t ) ,y) = bt(t)is the velocity of the moving boundary and ds = dy is arclength measure on a* D, (see Figure 1 ) . We now extend the ideas in Example 1. Figure 2 suggests that, in a more general setting, A V = & + A t \ ~ t f ( x , y) dx dy can be approximated by a sum C f v At As,
Figure 1. The moving boundary of Dt is the line segment x = b(r).
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Figure 2. A V
-
f v At As
where v(x, y) is the normal velocity of propagation of the boundary of D, at (x, y ) with respect to t and ds is arclength measure on aD,. In other words, the distance from a point (x, y) of aD, to aD,+A, in the direction normal to D, is approximately V(X,Y) At. This suggests in turn that, under reasonable conditions,
This would be a two-dimensional analogue of (3,which itself is a direct generalization of Part I of the fundamental theorem. To give precise meaning to this, we need some conditions on our parameterized family of sets. The sets D, will be described in terms of a function g and a set A.
Definition. The function g : R2 -+ R' is called admissible if (a) g is a nonnegative, continuous function that is of class C' on ( g > 0); (b) the open set {g < t ) is bounded for each t > 0. Condition (a) is written as it is to allow for functions such as g(x, y) = -./,
Definition. The set A is compatible with g if (a) A is open and convex; (b) A n {g < t ) is simply connected for each t > 0; (c) the gradient of g is nonzero on A n { g = t ) for each t > 0. We assume throughout the remainder of this paper that the domains D, are of the form A f l {g < t ) , where g is an admissible function and A is a convex open set compatible with g. We stress that the conditions on g and A are stronger than what is strictly required to prove the main theorems. Our goal is not to prove the theorems in their greatest generality. Rather, our aim is to bring some neglected results to the attention of the MONTHLY readership and to show their relevance to the undergraduate calculus curriculum. The set a D, n { g = t ) = a* D, is called the moving boundary of D,; i.e., the moving boundary of D, is the piece of the boundary of D, that is determined by the level curve November 20021
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Figure 3. A is the region bounded by the x-axis, x = a, and y = @ ( x ) . The function g ( x , y ) = b-' ( x ) .
(g = t } rather than the boundary of A. Figure 3 illustrates the meaning of the moving boundary in the case of Example 1 (where the set A, although well behaved, is not convex). The rate of change of g in the direction normal to the level curve {g = t } is given by [gradgl. If one moves a distance A d in the direction normal to the level curve, the change in the value of g is approximately lgrad glA d. It follows that the perpendicular distance between the level curves {g = t A t } and {g = t } is approximately At/ lgrad g 1. Thus, in the formula (6),v should be equal to lgrad gl-' . These heuristic arguments bring us to the little coarea theorem.
+
Theorem (Little Coarea Theorem). Ifg : R2 + R is an admissiblefunction, A is a convex open set that is compatible with g, and D, = A fl (g < t }for t > 0, then for 0 < S < T and any continuousfunction f : R2 + R ds dt. I f {g = 0} is empty or consists of a single point,
dsdt. The heuristic arguments that we have given motivate the result, but we derive it here as a corollary to the coarea theorem itself. To keep the paper self-contained, we also give an elementary proof (albeit under an additional technical assumption) in Section 5. The following is a version of the coarea theorem (see Evans and Gariepy [I, p. 1171). Coarea Theorem. Let g : Rn + Rm be a Lipschitzfunction, where n >_ m. Thenfor each Lebesgue measurable subset B of Rn and each Lebesgue integrable function h:R"+R,
/, J,d x
=
lm (/,",-la)
h dHn-')
dt.
(9)
The function J, is an appropriate Jacobian for transformations between spaces of different dimensions. It equals the Euclidean norm of the gradient of g when m = 1 810
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and g is differentiable, the case of interest here. The symbols dx and d t indicate integration with respect to n- and m-dimensional Lebesgue measure, respectively. Finally, Hn-" is (n - m)-dimensional Hausdorff measure, a measure that generalizes the idea of arclength and surface area measures to irregular sets. Given all this, it is not surprising that the coarea theorem is not easily accessible even in Morgan's "Beginner's Guide" to geometric measure theory [4]. One purpose of this article is to make this intriguing result more widely appreciated. To see how the little coarea follows from this, let n = 2, m = 1, f = hlgradgl, and B = DT\ Ds. Since B has a compact closure, which does not include {g = 0}, it follows that lgrad g 1 is bounded above and away from 0 on B. Thus, g is a Lipschitz function over this set and f lgrad gl-' is integrable over the set. Now B n g-'(t) = a*D, when S < t < T and is empty otherwise. It follows that (9) can be written
Finally, the implicit function theorem and our assumptions on g imply that a*D, is a smooth curve with a finite arclength, so d H' can be replaced with ds. Returning now to the little coarea theorem, Example 1 shows that it is a generalization of Fubini's theorem for evaluating multiple integrals by foliating a region by vertical lines. The next example illustrates how the little coarea theorem can be used to evaluate a multiple integral by foliating a region by curved lines. Example 2. For t > 0 let D, be the region described in polar coordinates as ((r, 8 ) : r c t, a < 8 < B). In this case g(x, y) = and A is the set (a < Q c B). We see that, for each t, a* D, is parameterized by 8 (i.e., by x = t cos 8, y = t sine for a 5 8 5 B). Thus d s = t dB. Also lgrad g ( = 1 in {g > 0). Accordingly,
J'w,
ds
--
(gradgl
- t dB.
We then have from (8) and (10)
Reversing the order of integration, this becomes
This shows that the little coarea theorem yields as a special case the formula for computing integrals in polar coordinates. The following corollary is the two-dimensional generalization of Part I of the fundamental theorem of calculus. The integral of f over the moving boundary of D, in the right-hand side of (1 1) is the analogue of f (b(t))bl(t) in (5). Corollary 1. If g : R2 -+ R is an admissible function and A is a convex open set compatible with g, then on the interval (0, oo) d
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FUNDAMENTAL THEOREM O F CALCULUS IN 2-D
81 1
Letting H ( t ) = s,& f dx dy for some S < t in (7), we see that the right side of ( 1 1 ) is just the Radon-Nikodym derivative of H. It is thus the ordinary derivative almost everywhere in (0, m ) with respect to Lebesgue measure. To show that the right side of ( 1 1 ) equals the ordinary derivative for all t in (0, oo),it is enough to show that it is continuous. To prove this we would have to break a* D, into sections where either g, or g, is bounded away from zero and apply the implicit function theorem to the function h(x, y, t ) = g(x, y) - t . We omit the details. We can observe the continuity directly in all of our examples. If f = 1 , then &, dx dy = d ( t ) is the area of the region Dr. As a simple application of Corollary 1 we obtain: Corollary 2. Ifg : R2 + R is an admissiblefunction, A is a convex open set compatible with g, and d ( t ) is the area of D,, then on the interval (0, m)
In particular, Corollary 2 implies that d A / d t gives the length of the moving boundary as a function of t in the case where lgrad g 1 = 1 . This observation sheds light on some examples used by Strang [S, pp. 217-220 ] to illustrate certain pitfalls that one might encounter in evaluating an area by breaking a region into infinitesimal regions. The frequent mistake that students make is to give the wrong expression for an infinitesimal element of area d d by using the incorrect expression for the infinitesimal width of the region. Examples 3 and 4 indicate the subtlety of the problem.
d m ,
Example 3. Let g(x, y) = and let A = R2.For t > 0 , D, is the open disc of radius t centered at the origin, and a*D, = a D, is the bounding circle. If one were to use infinitesimal rings heuristically to compute the area of the disc, one would write d d = 2nt dt for the area of one of the rings, since the rings have inner circumference 2nt and width d t . This reasoning works because the distance between aD: and aD:+*, is At, so the velocity of the moving boundary is 1 (see Figure 4 ) . Also, since lgrad gl = 1 in R2\{(0, 0 ) ) ,Corollary 2 shows that d d l d t is the length of the boundary of D,.
Figure 4. The area of the circle is the integral of the areas of the infinitesimal rings.
Things are not so simple in the next example.
+
Example 4. Let g(x, y) = ( x l a ) ( y l b ) for a > 0 and b > 0 , and let A = { ( x ,y) : x > 0 , y > 0 ) . For t > 0 , D, is the triangular region bounded by the x- and y-axes and the line ( x l a ) + ( y l b ) = t . In this case d ( t ) = abt2/2 and d d l d t = abt. This does not equal the length of the the moving boundary (i.e., the hypotenuse of the tri812
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Figure 5. The width of the increment is ab(a2
+ b2)-'I2 At.
angle), so one cannot write the area of an infinitesimal strip along the hypotenuse as the length of the hypothenuse times dt. In this instance (see Figure 5), the distance h between a*D, and a*D,+A,is a At sin% = a b ~t (a2 b2)-'/2, making the velocity of the moving boundary ab(a2 b2)-'/'. This also follows by computing lgradgl-', which has the constant value ab(a2 b2)-'1'. Using Corollary 2, we should multiply this by the length of the moving boundary to get d d / d t . This gives dA/dt = (ab)(a2 b2)-'l2(a2 b2)ll2t = abt, which is the correct value.
+
+
+
+
+
Examples 5 and 6 illustrate how to apply Corollary 2 in some cases where the velocity of the moving boundary is not constant. Example 5. Let g(x, y) = y - h(x), where h is a positive C1-function on R, and let A = {(x, y) : a < x < b, y > O}. Then D, is the region bounded below by the x-axis, t. If the on the sides by the lines x = a and x = b, and on the top - by- -y = h(x) . . moving boundary is parameterized by 9 = x for a 5 x 5 b, we have lgradgl = and d s = , / w d x , whence dsllgrad gl = dx. It follows that d d l d t = jabdx = (b - a), i.e., d d = (b - a ) d t . Figure 6 shows why this result holds.
+
Figure 6. The areas of the two shaded regions are equal.
+
+
Example 6. Let g(x, y) = J(x/a)2 (y/b)2 = r ,/(cos %/a)2 (sin %/b)2 and A = R2. For t > 0, D, = {g < t ) is the region enclosed by the ellipse (x2/a2) (y2/b2) = t2. A simple calculation using polar coordinates then shows that
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FUNDAMENTALTHEOREM OF CALCULUS IN 2-D
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In this example we know that implies the identity
A = n a b t 2 ,giving d A / d t
= 2nabt. Corollary 2 then
In the case where f is constant on a* D, for each t > 0, we define f * : [0,oo) + R by setting f * ( t )= f ( x , y ) for any ( x ,y ) in a*Dr. Corollary 3 then gives a generalization of the shell method for computing volumes of revolution. Corollary 3 (Generalized Shell Method). I f g : R2+ R is an admissible function and A is a convex open set compatible with g, then
This follows by combining Corollaries 1 and 2. Example 7 shows why it is called the generalized shell method. = t centered at Example 7. In Example 3, if f is constant on each circle f ( x . y) d x dp = 2 n &?t f * ( t )d t , the same formula obtained by the origin, then using the shell method in calculus courses.
hT
The generalization of a volume of revolution is the volume of a region over DT \ DS and under the graph of a function that is constant on a* D, for S < t < T . Example 8. In Example 4, if f is constant on the lines ( x l a ) f ( x . y ) d x d y = ab J: t f * ( t )d l .
JLT,6s
+ ( y l b ) = t , then
Example 9. In Example 5 , let f be constant on the curves y = h ( x ) b h(x)+T Ja S,,),, f ( x . y) d y d x = ( b - a ) J: f * ( t )d t .
+ t . Then
+
Example 10. In Example 6, if f is constant on the ellipses ( x 2 / a 2 ) ( y 2 / b 2 )= t 2 , then f ( x . y ) d x d y = ?nab J: tf * ( t )d t .
llo,,,,
As was stated earlier, the little coarea theorem and its corollaries have analogues in higher dimensions. With the intuition that we have developed, we can perhaps see how to extend the shell method to higher dimensions. For example, let D, be the ball of radius t centered at the origin in R? Let f ( x , y, z ) = f * ( t )be constant on the sphere aDr of radius t . We can think of revolving the graph of the function w = f * ( x )around just the w-axis in four dimensions to get a function f ( x , y, z ) = f as we revolve the graph of z = f * ( x )around the z-axis in three dimensions in the shell Replacing d A / d t by d V / d t , the method to get a function f ( x , y ) = f rate of change of the volume of D,, we find that the four-dimensional volume of revolution JJJDblba f ( x . y. z ) d x d y dz = 4 n Jab t 2f * ( t )d t , a result that can be verified by using sphencal coordinates.
*(dm)
*(dm').
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4. T H E R E L A T I O N B E T W E E N G R E E N ' S T H E O R E M A N D T H E C O A R E A T H E O R E M . There is still one thing that is missing in the generalization of the fundamental theorem of calculus to two dimensions. In the one-dimensional theorem there is an intimate connection between Parts I and 11. Part I essentially says d l d t J: f ( x ) d x = f ( t ) , while Part I1 says Jd f f ( x ) d x = f ( t ) - f ( a ) . This close relationship does not hold between ( 1 1) and Green's theorem. Indeed, the divergence form of Green's theorem involves JS, div F d x d y , the double integral of the divergence of a vector field, whereas ( 1 1) involves J&, f ( x , y ) d x d y , the double integral of a real-valued function. However, with the proper substitutions a connection can be established. In what follows D, is a bounded, open, simply connected set, and the set A in the little coarea theorem is R2, making a*Dl = a D,. In the version of Green's theorem given in formula ( 2 ) , assume that D = D l , where Dl = { g < t ] for t > 0 and g is an admissible function with nonvanishing gradient. Then N can be written lgrad g I - ' grad g and Green's theorem becomes
Next, in formula (1 1) set f = F . grad g . For this choice of f , ( 1 1 ) asserts that
Thus, when f and F are related in this way, the right-hand sides of ( 2 ) and (1 1 ) match, and the left-hand sides both involve the same vector field F . In doing this, we have derived the following interesting corollary to ( 2 ) and ( 1 1): for any smooth vector field F on R2,
This result has an interesting one-dimensional analogue. For example, if g ( x ) = Ix 1 , we have Dl = { g < t ] = ( - t , t ) , g l ( x ) = 1 for x > 0 , and g ' ( x ) = - 1 for x < 0 . The one-dimensional analogue of ( 1 3 ) in this case says that
Finally, let us note that there are ways to construct increasing families of domains other than as the interiors of level sets of a function. We could start with a smoothly bounded set D and locally form the family D, = D t V for t 3 0, where V is a smooth, never vanishing vector field that we assume always points out from D. We f ( x , y ) d x d y lIzo.Heuristic arguments similar to those could then consider d l d t given in Section 3 indicate that, under appropriate regularity conditions, this derivative is equal to J,, f ( x , y ) V . N d s , where N = N ( x , y ) signifies the unit normal vector to a D at ( x , y ) that points outward from D . Combining this fact with Green's theorem and taking f = 1 , we conclude that
+
[Ll
yet another interesting connection between Green's theorem and the derivatives of integrals over an expanding family of domains. Moreover, formula ( 1 4 ) has a physical November 20021
FUNDAMENTAL THEOREM OF CALCULUS IN 2-D
815
interpretation when V is the velocity field of a fluid of density one. In that case it shows that the rate of flow of fluid out of a region is equal to the rate of change of the area of the region covered by the fluid; i.e., the loss of fluid from a region causes an equivalent change in the area covered by the fluid. If F = V , D, = D t V , and t = 0 in the left side of (13), then (14) appears to contradict (13).Of course, this is not the case. Let V ( x ,y ) = V,( x , y)i V 2 ( x ,y)j. If we define a function g so that g ( x , y ) = t on aD,, then
+
+
for t 2 0. Assume that g is a C1-function. Differentiating both sides of (15) with respect to t yields
for t > 0 and ( x , y ) in aD. In other words, V . grad g = 1 on a D (when t = 0 ) indidx dy = d l d t &I V . grad g d x d y thus reconciling the cating that d l d t differences between the right-hand sides of formulas (13)and (14).
hr
It=,
It=,
5. AN ELEMENTARY PROOF OF THE LITTLE COAREA THEOREM. An elementary proof of the little coarea theorem of Section 3 can be based on the familiar change of variable theorem for multiple integrals if we are allowed to add an additional technical assumption that is not stated in the little coarea theorem itself. The extra assumption is that an appropriate parametric description of DT\ Ds exists. It is satisfied in all of our examples. Definition. Let g : R 2 + IR be an admissible function, and let A be a convex open subset of R 2 that is compatible with g. We say that A admits smoothparameterizations with respect to g if, whenever 0 < S < T , the open set G = DT \ Ds has the following properties: (i) there exist continuous functions u and v on ( S , T ) and C1-functions x = x ( t , 8 ) a n d y = y ( t , 8 ) on E = { ( x ,y ) : u ( t ) 5 8 5 v ( t ) , S < t < T ) suchthat G = { ( x ,y ) : ( t , 0 ) E E ) ; (ii) the transformation ( t ,0 ) + ( x , y ) is one-to-one with nonvanishing Jacobian determinant J = x,ye - xey, in the interior of E; (iii) for fixed t in ( S , T ) , a*D, = { ( x ( t ,8 ) , y ( t , 8 ) ) : u ( t ) 5 8 5 v ( t ) } . In other words, each moving boundary is parameterized by 8 , the family of boundaries is parameterized by t , and the mapping from ( t , 8 ) to ( x , y ) is one-to-one and smooth. In Example 2, ( t , 8 ) are just the polar coordinates of ( x , y). ProoJ Assume that A admits smooth parameterizations with respect to g. By the change of variable theorem for multiple integrals
Since g ( x ( t , Q ) ,y ( t , 8 ) ) = t for ( t ,8 ) in E , it follows that
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@ THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 109
Solving for g, and gi,we find that lgrad g 1 = Thus,
,/= = J=xth
-
xsR 1 - I .
Conclusion (7) follows by integrating (16) with respect to t . We obtain (8) by taking limits as S approaches 0. ยค ACKNOWLEDGMENT. We would like to thank Frank Morgan for an important idea used in Section 4.
REFERENCES 1. L. C. Evans and R. F. Gariepy, Measure Theory and the Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. 2. H. Federer, Curvature measures, Trans. Amer Math. Soc. 93 (1959) 418491. 3. J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, A Unijied Approach, Prentice Hall, Upper Saddle River, NJ, 1999. 4. F. Morgan, Geometric Measure Theory, A Beginner's Guide, 2nd ed., Academic Press, Boston, 1995. 5. G. Strang, Calculus, Wellesley Cambridge Press, Wellesley, MA, 1991. 6. S. Weintraub, Differential Forms: A Complement to Vector Calculus, Academic Press, San Diego, 1996.
BENNETT EISENBERG attended Dartmouth College and MIT and taught at Cornell University and the University of New Mexico before coming to Lehigh. He twice gave a series of lectures on his work in probability and statistics at the University of Osnabmck in Germany. His work with Rosemary Sullivan led to papers on random triangles, Crofton's differential equation, and the fundamental theorem of calculus, all of which appeared in this MONTHLY. Lehigh University Bethlehem, PA 18015 be01 @Lehigh.edu ROSEMARY SULLIVAN received her B.S. from Penn State and her M.S. and Ph.D. from Lehigh University. Before coming to West Chester University she taught at Widener University and Muhlenberg College. She is an MAA Project NexT Fellow. Her main area of interest is geometric probability. She enjoys gardening, travelling, skiing, and hiking. West Chester UniversiQ, West Cheste~PA 19383 [email protected]
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FUNDAMENTAL THEOREM OF CALCULUS IN 2-D
