Generalizations of Pappus' Centroid Theorem Via Stokes' Theorem

inv lve a journal of mathematics

Generalizations of Pappus’ centroid theorem via Stokes’ theorem Cole Adams, Stephen Lovett and Matthew McMillan

msp

2015 vol. 8, no. 5 INVOLVE 8:5 (2015) msp dx.doi.org/10.2140/involve.2015.8.771

Generalizations of Pappus’ centroid theorem via Stokes’ theorem Cole Adams, Stephen Lovett and Matthew McMillan (Communicated by Kenneth S. Berenhaut)

This paper provides a novel proof of a generalization of Pappus’ centroid theorem on n-dimensional tubes using Stokes’ theorem on manifolds.

1. Introduction

The (second) Pappus centroid theorem or the Pappus–Guldin theorem states that the volume of a solid of revolution generated by rotating a plane region R with piecewise-smooth boundary about an axis L is 2r Area.R/, where r is the distance from the centroid of R to L. This result generalizes considerably to the following main theorem. Theorem 1.1 (main theorem). Let C be a simple, regular, smooth curve in ޒn. Let n 1 R be a region in ޒ whose boundary is an embedding of the .n 2/-dimensional n 2 n sphere ޓ . Let W be a region in ޒ whose boundary is a generalized tube around C such that the cross-section normal to C of W at each point P of C is the region R with centroid at P. Assuming the cross-section R rotates smoothly as it “travels” along C , then

Voln.W/ length.C / Voln 1.R/: D The Pappus centroid theorem follows from this main theorem by taking n 3, C D to be a circle in ޒ3, and R to remain ﬁxed with respect to the principal normal to C in the normal plane. This theorem recently was proved by Gray, Miquel, and Domingo- Juan in[Domingo-Juan and Miquel 2004] and[Gray and Miquel 2000] using parallel transport. However, Goodman and Goodman[1969] proved this theorem in a special case for ޒ3 using elementary methods related to Stokes’ theorem. This article proves the main theorem in full generality using Stokes’ theorem on manifolds. In this

MSC2010: 53A07, 58C35. Keywords: Stokes’ theorem on manifolds, volume, manifolds, tubes. The authors thank Wheaton College’s Faculty Student Mentoring Initiative, which made possible the collaboration that led to this article.

771 772 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN regard, we can consider the proof elementary compared to those in[Domingo-Juan and Miquel 2004] and[Gray and Miquel 2000]. Before proving the main theorem in full generality, we sketch the proof of it in ޒ3 found in[Goodman and Goodman 1969], leaving the reader to consult that work for details. The description of the generalized tube and the method involving the divergence theorem motivate the situation for arbitrary n.

2. Generalized tubes in dimension 3

Definition 2.1. Let C be a simple, regular, smooth space curve and let R be a compact planar region with one boundary component @R, a piecewise smooth simple closed curve. Select a marked point P in R. C has a normal plane at each point. Let W be a region in ޒ3 such that the intersection of W with the normal plane to C at any point is isometric to the region R, with the corresponding marked point P lying on the curve C . We assume R rotates smoothly in the normal plane to C as it travels along C . Such a region W is called a generalized tube along C with cross-section R and center P. This definition allows for rotational freedom of R around the marked point P in the normal planes to C . However, this rotational varies smoothly. We may also describe the generalized tube as a fiber-bundle over C with fiber R, that is a subbundle of the normal bundle over C . Figure 1 depicts two generalized tubes around a portion of a helix. More precisely, the figure depicts the tube boundary excluding the “caps”, or cross-sections at the end points of C . The planar curve shows its generating region R where the marked point of R is the origin. Let S be the boundary @W of a generalized tube excluding the caps. (If C is a closed curve, then @W has no caps.) Suppose that ˛ Œ0; ` ޒ3 gives W ! a parametrization by arclength of C . Also suppose that ˇ Œ0; c ޒ2 is a E W ! parametrization of @R placing the marked point P at the origin. We write ˇ.u/ E D .x.u/; y.u// for the coordinate functions. A parametrization for S is X .s; u/ ˛.s/ cos..s//x.u/ sin..s//y.u/P.s/ E DE C E sin..s//x.u/ cos..s//y.u/ B.s/ (1) C C E for some function .s/, where P.s/ and B.s/ are respectively the principal normal E E and binormal vector functions to ˛.s/. E Recall that .T .s/; P.s/; B.s//, where T , P, and B are the usual tangent, princi- E E E E E E pal normal, and binormal vectors to ˛.s/, is called the Frenet frame to ˛.s/. The E E function .s/ determines the rotation of the region R around the origin with respect to the Frenet frame. The stipulation that R rotates smoothly as it moves along C implies that .s/ is a smooth function. GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 773

Figure 1. A generalized tube with its generating region.

Figure 1, middle, depicts a generalized tube where the x-axis in the depiction of R always lies along the principal normal vector of ˛.s/, and Figure 1, right, depicts E a generalized tube with the same cross-section region but having some rotation with respect to the basis .P; B/ in the normal plane. For brevity, we write E E X ˛ .x cos y sin /P .x sin y cos /B; E DE C E C C E where functional dependence is understood from (1).

Theorem 2.2 [Goodman and Goodman 1969, Corollary 2]. The volume of a generalized tube as described in Definition 2.1 is V length.C / Area.R/. D The Goodmans’ method to calculate the volume uses the fact that the position vector field r.x; y; z/ .x; y; z/ has divergence equal to 3 everywhere. So, using E D the notation defined above, the volume of the generalized tube is

1 • 1 • 1 “ Vol.W/ 3 dV r dV r dA; D 3 W D 3 W r E D 3 @W E E where dA is the outward pointing surface element. Note that @W consists of the E tube’s outward surface S, parametrized by X , and the end caps (if C is not a closed E curve). Over S, dA is given by dA .Xu Xs/ du ds with .u; v/ Œ0; c Œ0; `, E E D E E 2 while on the end caps, dA T .0/ dA when s 0 and dA T .`/ dA when E D E D E D E s `. The caps, like any cross-section at s, are parametrized by D

Ys.p; q/ ˛.s/ pP.s/ qB.s/ for .p; q/ Rs; E DE C E C E 2 where Rs is the region R rotated about the origin (the marked point P) by the angle .s/. Thus, since T .s/ is perpendicular to both P.s/ and B.s/, we have E E E 774 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN

3 Vol.W/ ` c Z Z “ “ X .Xu Xs/ du ds ˛.`/ T .`/ dp dq ˛.0/ T .0/ dp dq D E E E C E E C E E s 0 u 0 R` R0 D D ` c Z Z X .Xu Xs/ du ds Area.R/ ˛.`/ T .`/ ˛.0/ T .0/ : (2) D E E E C E E E E s 0 u 0 D D The problem of calculating the volume of W reduces to calculating the double integral in (2). Recall that vectors of the Frenet frame (parametrized by arclength) differentiate according to T P; E 0 D E P T B; (3) E0 D E C E B P; E0 D E where .s/ and .s/ are the curvature and torsion functions of the space curve ˛.s/. E Then the tangent vectors to X are given (after simpliﬁcation) by E

Xu .x cos y sin /P .x sin y cos /B; E D 0 0 E C 0 C 0 E Xs .1 x cos y sin /T . /.x sin y cos /P E D C E 0 C C E . /.x cos y sin /B: C 0 C E So

Xu Xs . /.xx yy /T .1 x cos y sin /.x sin y cos /P E E D 0C 0C 0 EC C 0 C 0 E .1 x cos y sin /.x cos y sin /B: C 0 0 E

The dot product X .Xu Xs/ involves many terms. However, all of the additive E E E terms involved in the integrals are multiplicatively separable, which, by the usual corollary to Fubini’s theorem, allows us to separate the double integral. Many of the integrals involving u vanish or evaluate to a simple constant, namely the area of the cross-section. Consider the following integrals. By substitution, Z c 2ˇc xx0 du x ˇ0 0 u 0 D D D because .x.u/; y.u// with u Œ0; c parametrizes a closed curve @R. By similar 2 reasoning, the following integrals are all 0: Z c Z c Z c Z c x0 du 0; y0 du 0; xx0 du 0; yy0 du 0: (4) u 0 D u 0 D u 0 D u 0 D D D D D GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 775

By Green’s theorem for the area of the interior of a simple closed piecewise smooth curve, Z c Z c “ xy0 du yx0 du 1 dA Area.R/: (5) u 0 D u 0 D R D D D Also by Green’s theorem, Z c Z c “ 1 2 2 x y0 du xyx0 du x dA 0 (6) u 0 D u 0 D R D D D because this integral is the y-moment of Rs and by hypothesis, the centroid of Rs is .0; 0/ for all s. By the same reasoning but for the x-moment, we also have Z c Z c “ 1 2 2 y x0 du xyy0 du y dA 0: (7) u 0 D u 0 D R D D D Upon applying these integrals, only a few terms remain in (2). Setting A D Area.R/, we get Z ` 3 Vol.W/ . ˛ T 0A 2A/ ds A ˛.`/ T .`/ ˛.0/ T .0/ : D s 0 E E C C E E E E D Using integration by parts on the dot product, we obtain 3 Vol.W/ Z ` ˇ` A.˛ T /ˇ0 A ˛0 T ds 2A` A ˛.`/ T .`/ ˛.0/ T .0/ D E E C s 0 E E C C E E E E D A˛.`/ T .`/ ˛.0/ T .0/ D E E E E Z ` A T T ds 2A` A˛.`/ T .`/ ˛.0/ T .0/ C s 0 E E C C E E E E D A` 2A` 3A`: D C D We conclude that Vol.W/ Area.R/ length.C /. D Theorem 2.2 establishes the main theorem of the paper for generalized tubes in ޒ3. In order to prove the main theorem in full generality, we will need to use differential forms along with Stokes’ theorem on manifolds. However, a key component to the main theorem is a set of integral formulas for the general case similar to (4), (5), (6), and (7).

3. Volumes, moments, and zero integrals for solids in ޒm

Recall that Stokes’ theorem on manifolds states that if M is an m-dimensional, ori- ented manifold with boundary @M , and ! is a differential .m 1/-form on M , then Z Z ! d!; (8) @M D M 776 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN where @M has the boundary orientation inherited from the orientation on M . Definition 3.1. We define a solid in ޒm as a compact embedded m-dimensional submanifold of ޒm with boundary @M . We assume the pull-back orientation on M . We define the .m 1/-form i in ޒm by i . 1/i 1dy1 dy2 dyi dym; D C ^ ^ ^ ^ ^ 1 2 m m where .y ; y ;:::; y / is a coordinate system on ޒ and b denotes removal of that term. Lemma 3.2. The m-dimensional volume of a solid M is Z b i i Volm.M / y D @M for any i 1; 2;:::; m. D Proof. The differential of yii is d.yii/ . 1/i 1dyi dy1 dy2 dyi dym dy1 dy2 dym: D C ^ ^ ^ ^ ^ ^ D ^ ^ ^ This form is precisely the volume form on ޒm, and thus on the solid M as well. Hence, by Stokes’ theorem, Z Z i i 1 2 m y dy dy dy Volm.M /: @M D M ^ b^ ^ D This lemma immediately implies the following corollary: 1 m Corollary 3.3. Let P yii. The m-dimensional volume of M is D m i 1 D Z Volm.M / : D @M In this article, if F M N is a differentiable map between differentiable W ! manifolds, we will denote by ŒdF the matrix of functions of the differential dF in reference to given coordinate systems on M and on N . Furthermore, when the dimension of M is one less than the dimension of N and when coordinate systems on neighborhoods of M and N are implied, we denote by dj F the determinant j j of ŒdF in which the j -th row is removed. Proposition 3.4. Let M be an m-dimensional solid such that the boundary @M m 1 m is the embedding of a continuous map H ޓ ޒ that is smooth except m 1 W ! on a subset of measure 0 in ޓ . Suppose also that H induces an orientation on @M that is compatible with the boundary orientation induced from M . Let be the .m 1/-form as in Corollary 3.3 and let ! be the .m 1/-form on ޓm 1 GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 777 given by ! dx1 dx2 dxm 1 for coordinates .x1; x2;:::; xm 1/. The D ^ ^ ^ m-dimensional volume of M is Z Z 1 Z Volm.M / H det.H;ŒdH/!; (9) D m 1 D m 1 D m m 1 H .ޓ / ޓ ޓ where in det.H;ŒdH/ we write the components of H as a column vector. If H induces the opposite orientation, the second two integrals change sign.

Proof. The equality Z Volm.M / D m 1 H .ޓ /

1 2 m 1 follows immediately from Corollary 3.3. Let .x ; x ;:::; x / be coordinates m 1 1 2 m m on ޓ and .y ; y ;:::; y / on ޒ . Notice that the pullback of by H is m 1 X H H i. 1/i 1dH 1 dH 2 dH i dH m: D m C ^ ^ ^ ^ ^ i 1 D Or, writing in xi coordinates, and using the fact that

@H i dH i dxj D @xj b (assuming the Einstein summation convention), we ﬁnd

m 1 Â @H 1 Ã Â @H 2 Ã X i i 1 j1 j2 H H . 1/ C dx dx D m @xj1 ^ @xj2 ^ i 1 D Â @H i Ã Â @H m Ã dxji dxjm : ^ @xji ^ ^ @xjm By Theorem C.5.22 in[Lovett 2010], this is equivalent to

ˇ 1 1 1 ˇ ˇ @H @H ::: @H ˇ ˇ @x1 @x2 @xm 1 ˇ ˇ 3 ˇ ˇ @H 2 @H 2 @H 2 ˇ ˇ 1 2 ::: m 1 ˇ ˇ @x @x @x ˇ ˇ : : : : ˇ m ˇ : : :: : ˇ 1 X i i 1 ˇ : : : ˇ 1 2 m 1 H H . 1/ C ˇ ˇ dx dx dx : D m ˇ i i i ˇ ^ ^ ^ i 1 ˇ @dH @dH @dH ˇ D ˇ 1 2 ::: m 1 ˇ ˇ @x @x @x ˇ ˇ ˇ ˇ : : :: : ˇ ˇ : : : : ˇ ˇ ˇ ˇ@H m @H m @H m ˇ ˇ 1 2 ::: m 1 ˇ @x @x @x 778 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN

Taking the summation and recognizing the Laplace expansion of a determinant down the ﬁrst column, we see that 1 H det.H;ŒdH/ dx1 dx2 dxm 1: D m ^ ^ ^ Then (9) follows. Note that the second integral changes sign if H induces the opposite orientation on @M , so the third integral changes sign as well. Lemma 3.2, Corollary 3.3, and Proposition 3.4 are generalizations to higher dimensions of Green’s theorem for area. For example, suppose that S is a solid in ޒ3 such that the boundary @S is parametrized by X .u; v/ .x.u; v/; y.u; v/; z.u; v// E D with .u; v/ D such that Xu Xv is outward-pointing. Then by Proposition 3.4, 2 E E the volume of S is ˇx x x ˇ “ ˇ u vˇ 1 ˇ ˇ Vol.S/ ˇy yu yvˇ du dv: D 3 D ˇz zu zv ˇ Because of the ﬂexibility in Stokes’ theorem, as in Green’s area theorem, this formula still applies when @S is piecewise smooth. In that case, we interpret the above integral as a sum of integrals taken over domains D1; D2;:::; Dr such that the parametrizations for the smooth pieces of @S have domains Di. The same principle applies in (9). We will encounter other integrals that cancel. We list them here. Proposition 3.5. Let M be a solid and let .y1; y2;:::; ym/ be a coordinate system on M . Then for i and q in 1; 2;:::; m , f g Z q i q y ıi Volm.M /; @M D where ıq is the Dirac delta in which ıq 1 if i q and ıq 0 if i q. i i D D i D ¤ Proof. The case with i q is Lemma 3.2. If i q, then D ¤ d.yqi/ dyq dy1 dy2 dyi dym 0 D ^ ^ ^ ^ ^ ^ D because one differential is repeated. Then by Stokes’ theorem, we have Z Z Z yqi d.yqi/ 0 0: @M D M D M D Corollary 3.6. Let M , H , and ! be as in Propositionb 3.4. Then Z i 1 q q . 1/ C H diH ! ıi Volm.M /: m 1 j j D ޓ Proof. This follows immediately from the fact that i 1 q q i . 1/ H diH ! H .y /: C j j D GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 779

Proposition 3.7. Let M , H , and ! be as in Proposition 3.4. Let a .a1;a2;:::;am/ ED be a constant vector, listed as a column vector. Then Z det.a;ŒdH/! 0: m 1 E D ޓ Proof. By the reasoning in the proof of (9), we see that

m X det.a;ŒdH/! . 1/i 1ai dH 1 dH 2 dH i dH m E D C ^ ^ ^ ^ ^ i 1 D m ÂX Ã H . 1/i 1ai dy1 dy2 dyi dym : D C ^ ^ ^ ^ ^ i 1 D Hence, by a pull-back and then Stokes’ theorem, b Z Z m X i 1 i 1 2 i m det.a;ŒdH/ . 1/ C a dy dy dy dy ޓm 1 E D H .ޓm 1/ ^ ^b ^ ^ ^ i 1 D Z Â m Ã X i 1 i 1 2 i m d . 1/ C a dy dy dy dy D M ^ ^ ^ ^ ^ i 1 Z D 0 0: b D M D In the proof of Theorem 2.2, certain integrals vanished by virtue of the cross- section always having its centroid on the curve C , and the same thingb occurs in higher dimensions. The following proposition establishes the centroid generalizations needed later:

Proposition 3.8. Let M be an m-dimensional solid as given in Deﬁnition 3.1. Let .y1; y2;:::; ym/ be a coordinate system covering M . Let .y1; y2;:::; ym/ be the N N N center of mass of M . Then 8 Z 0 if p i and q i; p q i < p ¤ ¤ y y y Volm.M / if p i and q i; @M D : N i ¤ D 2y Volm.M / if p q i. N D D Proof. By Stokes’ theorem, Z Z ypyqi d.ypyqi/: @M D M However,

d.ypyqi/ .yqdyp ypdyq/ i yqdyp i ypdyq i: D C ^ D ^ C ^ 780 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN

If neither p i nor q i, then dyp i 0 and dyq i 0. If q i and p i, D D ^ D ^ D D ¤ then d.ypyqi/ yp dy1 dy2 dym and D ^ ^ ^ Z p 1 2 m p y dy dy dy y Volm.M / M ^ ^ ^ DN by deﬁnition of the center of mass. Finally, if p q i, then d.ypyqi/ D D D 2yi dy1 dy2 dym and ^ ^ ^ Z i 1 2 m i 2y dy dy dy 2y Volm.M /: M ^ ^ ^ D N

4. Generalized tubes in higher dimensions

We are almost ready to prove Theorem 1.1. We must first set up a useful description of a generalized tube. Let W be a generalized tube with guiding curve C and cross-section R as described in the statement of the main theorem. A generalized tube is a fiber-bundle over C with fiber R, that is, a subbundle of the normal bundle over C . Suppose that C is parametrized by arclength by ˛ Œ0; ` ޒn. Suppose n 1 W ! that the cross-section R is a solid in ޒ whose boundary @R is parametrized by an orientation-preserving, differentiable map H ޓn 2 ޒn 1. We also assume W ! that R rotates smoothly about the origin in the normal plane as it is transported along C . For the purpose of the theorem, we also assume that the center of mass of R is the origin in ޒn 1. Define H ޓn 2 ޒn by H .x/ .0; H.x//. W ! E D E The boundary @W of the solid generalized tube consists of the caps at ˛.0/ and ˛.`/ as well as the side surface S, which we can parametrize by

˛.t/ M.t/H .x/ for .t; x/ Œ0; ` ޓn 2; C E E 2 where M Œ0; ` SO.n/ is a differentiable curve of special orthogonal (rotation) W ! matrices in ޒn such that for all t, 011 B0C M.t/ B C M.t/e ˛ .t/: (10) B:C 1 0 @:A D E D 0

1 Note that since M.t/ is a rotation matrix and the unit vector e1 in the y direction E is perpendicular to .0; y2;:::; yn/ yi ޒ , then for all t Œ0; `, the boundary f j 2 g 2 of the cross-section M.t/H .x/, for x ޓn 2, is in a plane perpendicular to the E E 2 tangent vector ˛ .t/. For simplicity later, we write F.t; x/ M.t/H.x/. 0 E D E Recall that since M.t/ is a special orthogonal matrix for all t, then M.t/ 1 D M.t/ , det M.t/ 1, and M .t/ M.t/A.t/, where A.t/ is some antisymmetric > D 0 D matrix for all t. Using the rotation matrix M.t/ provides the following useful fact. GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 781

v 1 v2 E E

v3 E Figure 2. Reversed orientation on a cylinder.

1 Lemma 4.1. The ﬁrst component of the vector M.t/ ˛.t/ is equal to the dot product ˛.t/ ˛ .t/. 0 Proof. By (10), the dot product ˛.t/ ˛ .t/ is 0 ˛.t/ ˛ .t/ ˛.t/ M.t/e1: 0 D > E Taking the transpose of the matrix expression on the right, and since the whole expression is just a real number, we get ˛.t/ ˛ .t/ e M.t/ ˛.t/ .1 0 0/M.t/ 1˛.t/; 0 DE1> > D and the lemma follows. Proof of Theorem 1.1. Case 1: Assume that the guiding curve C is not closed. Let be the .n 1/-form 1 Pn yii. By Corollary 3.3, the volume of the D n i 1 generalized tube is D Z Z Z Z Voln.W/ : (11) D @W D S C capt 0 C capt ` D D We parametrize S by ˛ F but we note that this parametrization is orientation- C reversing. This can be seen by applying our setup to the case of a circular cylinder 3 in ޒ and generalizing to higher dimensions. In Figure 2, v1 is ˛ .t/, v2 is a tangent E 0 E vector to the cross-section boundary in positive orientation, and v3 is the outward E pointing normal vector to the solid M . These three vectors form a left-handed system so the orientation induced from our parametrization is reversed from the boundary orientation on @M induced by the standard orientation of ޒn on M . We can parametrize the caps by G0 and G` where, for each t Œ0; `, we deﬁne n 2 Gt R ޒ with W ! Â0Ã Gt .z / ˛.t/ M.t/ : E D C z Now G0 induces an orientation that is opposite theE boundary orientation on @W, while Gt gives a compatible orientation. Hence, (11) becomes Z Z Z Voln.W/ .˛ F/ G0 G`: (12) D I ޓn 2 C R C R 782 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN

We calculate the integrals on the caps ﬁrst. By the same reasoning as in Proposition 3.4, for each t Œ0; `, 2

1 1 2 n 1 G det.Gt ;Œd.Gt // dz dz dz : t D n ^ ^ ^ Now Â Â Ã Â ÃÃ 0 0> det.Gt ;Œd.Gt // det ˛.t/ M.t/ ; M.t/ E D C z In 1 E Â Â Ã Â ÃÃ 1 0 0> det.M.t// det M.t/ ˛.t/ ; E D C z In 1 E Â Â Ã Â ÃÃ 1 0 0> det M.t/ ˛.t/; ; E ˛.t/ ˛0.t/; D z In 1 D E where 0> .0;:::; 0/, In 1 is the .n 1/ .n 1/ identity matrix and the last E D equality holds by Lemma 4.1. Consequently, Z Z 1 G` G0 Voln 1.R/ ˛.`/ ˛0.`/ ˛.0/ ˛0.0/ : (13) R R D n R Now we must calculate I ޓn 2 .˛ F/. Applying Proposition 3.4, over a n 2 C 1 2 n 2 coordinate patch of ޓ with coordinate system .x ; x ;:::; x /, we have

1 1 n 2 .˛ F/ det ˛.t/ F.x/; ˛ .t/ Ft .t; x/; M.t/ŒdH dt dx dx ; C D n C E 0 C E ^ ^ ^ where here Ft @F=@t. This can be broken down by multilinearity of the determi- D nant as follows: 1 .˛ F/ det˛.t/;˛ .t/;M.t/ŒdH C D n 0 det F.x/;Ft .t;x/;M.t/ŒdH det F.x/;˛ .t/;M.t/ŒdH C E E C E 0 Á 1 n 2 det ˛.t/;Ft .t;x/;M.t/ŒdH dt dx dx : (14) C E ^ ^^

We now consider the integration over Œ0; ` ޓn 2 of the four forms in (14). For the ﬁrst determinant in (14), det ˛.t/; ˛ .t/; M.t/ŒdH det ˛.t/; M.t/e1; M.t/ŒdH 0 D E 1 det.M.t// det M.t/ ˛.t/; e1;ŒdH D E 1 det e1; M.t/ ˛.t/; ŒdH : D E GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 783

Doing Laplace expansion down the ﬁrst column, we obtain an integral of the form in Proposition 3.7, with a vector a that depends on t. Hence, by Proposition 3.7, E Z Z ` 1 n 2 det ˛.t/; ˛0.t/; M.t/ŒdH dt dx dx ޓn 2 t 0 ^ ^ ^ D Z ` Z n 2 1 n 2 . 1/ det ˛.t/; ˛0.t/; M.t/ŒdH dx dx dt 0: D t 0 ޓn 2 ^ ^ ^ D D For the second determinant in (14),

det F.t; x/; Ft .t; x/; M.t/ŒdH det M.t/H .x/; M.t/A.t/H .x/; M.t/ŒdH E E D E E det.M.t// detH .x/; A.t/H .x/; ŒdH D E E detH .x/; A.t/H .x/; ŒdH : D E E Performing a Laplace expansion of the determinant using the ﬁrst two columns of this last determinant produces terms similar to the forms described in Proposition 3.8. Since the centroid of R is assumed to be at the origin, then for all t, integrating all n 2 these terms over ޓ gives 0. For the third determinant in (14), we have

det F.t; x/; ˛ .t/; M.t/ŒdH det M.t/H .x/; M.t/e1; M.t/ŒdH E 0 D E E det.M.t// det H .x/; e1;ŒdH D E E det e1; H .x/; ŒdH D E E det.H.x/; ŒdH/; D E where the last equality follows by Laplace expansion of the determinant on the ﬁrst row .1 0 0/. By Proposition 3.4, Z Z ` 1 n 2 det F.x/; ˛0.t/; Œd.F/ dt dx dx ޓn 2 t 0 E ^ ^ ^ Z D 1 n 2 ` det H.x/; Œd.H / dx dx D n 2 E ^ ^ ޓ .n 1/` Voln 1.R/: D As with previous determinants, the fourth determinant becomes

1 det ˛.t/; Ft .t; x/; M.t/ŒdH det M.t/ ˛.t/; A.t/H .x/; ŒdH : E D E 784 COLE ADAMS, STEPHEN LOVETT AND MATTHEW MCMILLAN

Since there are zeros in the ﬁrst rows of H and d.H /, and because M 1 M , D > another Laplace expansion gives

1 det M ˛; AH ;ŒdH n n i X j j q X j i 1 q ˛iM . 1/ A H dj H ˛i . 1/ M A H dj H D 1 q j j j q j j j 2 j 2 n D D X q j i j i 1 ˛iH dj H . 1/ .M A M A /; D j j 1 q j q j 2 D where we use the Einstein summation convention over the repeated indices appearing in superscript and subscript, namely i and q. By Proposition 3.5, after integration on ޓn 2, all terms will reduce to 0 except those for which q j , which will give D the volume of the cross-section, Voln 1.R/. Set I Œ0; `. So, D Z Z 1 1 n 2 det M ˛; AH ;Œd.H / dt dx dx n 2 ^ ^ ^ ޓ I Z Z n X q j i j i 1 1 n 2 ˛iH dj H . 1/ .M1Aq Mj Aq/ dt dx dx D ޓn 2 I j j ^ ^ ^ j 2 D Z Z n X j j i j i 1 1 n 2 ˛iH dj H . 1/ .M1Aj Mj Aj / dt dx dx D ޓn 2 I j j ^ ^ ^ j 2 D Z Z n X j 1 j i j 1 n 2 ˛iH dj H . 1/ Mj A1 dt dx dx D ޓn 2 I j j ^ ^ ^ j 2 D Z n Z X i j j j 1 1 n 2 ˛i Mj A1 dt . 1/ H dj 1H dx dx : D I ޓn 2 j j ^ ^ j 2 D i j But ˛ .t/ M.t/e1, so ˛ .t/ M .t/e1 M.t/A.t/e1. Hence M A are the com- 0 D E 00 D 0 E D E j 1 ponents of the covariant vector ˛00.t/>. Note that since A.t/ is an antisymmetric 1 Pn i j matrix, A1 0. Thus ˛i j 2 Mj A1 ˛.t/ ˛00.t/. Hence, we get D D D Z Z 1 1 n 2 det M ˛; AH ;Œd.H / dt dx dx ޓn 2 I ^ ^ ^ Z Voln 1.R/ ˛i˛i00 dt D I Â Z Ã ˇ` Voln 1.R/ ˛.t/ ˛0.t/ˇ0 ˛0 ˛0 dt D I ˇ` Voln 1.R/ ˛.t/ ˛0.t/ˇ0 ` : D GENERALIZATIONSOFPAPPUS’CENTROIDTHEOREMVIASTOKES’THEOREM 785

Now putting into (12) the integrals of the four determinants in (14) and the integrals for the caps (13), we get ` n Voln.W/ .n 1/ Voln 1.R/` Voln 1.R/ ˛.t/ ˛0.t/ 0 ` D j ` Voln 1.R/ ˛.t/ ˛0.t/ 0 : (15) C j Hence Voln.W/ Voln 1.R/`; D which establishes the main theorem when the guiding curve of the generalized tube is not closed. Case 2: If C is a closed curve, then in (12) we do not have integrals for the caps. Then (15) becomes ˇ` n Voln.W/ .n 1/ Voln 1.R/` Voln 1.R/ ˛.t/ ˛0.t/ˇ0 ` ; D and since ˛.0/ ˛ .0/ ˛.`/ ˛ .`/, the result of the main theorem follows for this 0 D 0 case as well.

References

[Domingo-Juan and Miquel 2004] M. C. Domingo-Juan and V. Miquel, “Pappus type theorems for motions along a submanifold”, Differential Geom. Appl. 21:2 (2004), 229–251. MR 2005e:53041 Zbl 1061.53038 [Goodman and Goodman 1969] A. W. Goodman and G. Goodman, “Generalizations of the theorems of Pappus”, Amer. Math. Monthly 76 (1969), 355–366. MR 39 #2047 Zbl 0172.45902 [Gray and Miquel 2000] A. Gray and V. Miquel, “On Pappus-type theorems on the volume in space forms”, Ann. Global Anal. Geom. 18:3-4 (2000), 241–254. MR 2001i:53046 Zbl 1009.53027 [Lovett 2010] S. Lovett, Differential geometry of manifolds, A K Peters, Natick, MA, 2010. MR 2011k: 53001 Zbl 1205.53001

Received: 2014-03-18 Accepted: 2014-11-22 [email protected] School of Engineering, Vanderbilt University, Nashville, TN 37235-1826, United States [email protected] Department of Mathematics and Computer Science, Wheaton College, 501 College Avenue, Wheaton, IL 60187, United States [email protected] St Catherine’s College, Oxford OX1 3UJ, United Kingdom

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involve 2015 2015 vol. 8 no. 5 inv lve

o.8 o 5 no. 8, vol. a journal of mathematics

A simplification of grid equivalence 721 NANCY SCHERICH A permutation test for three-dimensional rotation data 735 DANIEL BEROAND MELISSA BINGHAM Power values of the product of the Euler function and the sum of divisors function 745 LUIS ELESBAN SANTOS CRUZAND FLORIAN LUCA On the cardinality of infinite symmetric groups 749 MATT GETZEN Adjacency matrices of zero-divisor graphs of integers modulo n 753 MATTHEW YOUNG Expected maximum vertex valence in pairs of polygonal triangulations 763 TIMOTHY CHUAND SEAN CLEARY Generalizations of Pappus’ centroid theorem via Stokes’ theorem 771 COLE ADAMS,STEPHEN LOVETTAND MATTHEW MCMILLAN A numerical investigation of level sets of extremal Sobolev functions 787 STEFAN JUHNKEAND JESSE RATZKIN Coalitions and cliques in the school choice problem 801 SINAN AKSOY,ADAM AZZAM,CHAYA COPPERSMITH,JULIE GLASS, GIZEM KARAALI,XUEYING ZHAOAND XINJING ZHU The chromatic polynomials of signed Petersen graphs 825 MATTHIAS BECK,ERIKA MEZA,BRYAN NEVAREZ,ALANA SHINEAND MICHAEL YOUNG Domino tilings of Aztec diamonds, Baxter permutations, and snow leopard 833 permutations BENJAMIN CAFFREY,ERIC S.EGGE,GREGORY MICHEL,KAILEE RUBIN AND JONATHAN VER STEEGH The Weibull distribution and Benford’s law 859 VICTORIA CUFF,ALLISON LEWISAND STEVEN J.MILLER Differentiation properties of the perimeter-to-area ratio for finitely many 875 overlapped unit squares PAUL D.HUMKE,CAMERON MARCOTT,BJORN MELLEMAND COLE STIEGLER On the Levi graph of point-line configurations 893 JESSICA HAUSCHILD,JAZMIN ORTIZ AND OSCAR VEGA