Area of Curved Surfaces

Area of Curved SurfacesAuthor(s): James PierpontSource: Transactions of the American Mathematical Society, Vol. 7, No. 4 (Oct., 1906), pp. 489-498Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1986240 .

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AREA OF CURVED SURFACES

BY

JAMES PIERPONT

? 1. Introduction.

Let the surface S be defined by the equations

x =f(u, v), y = g(u, v), z = h(u, v),

wheref, g, h are continuous single valued functions in a complete limited region 21. In case their first derivatives are continuous in 2A, i. e., in case S has a continuously turning tangent plane, the area t of S, which we may without ambiguity denote by the same letter, is

(1) S=f1/A2 + j32 + C2 du dv, where

ay a< az ax ax ay au au an au au an .ay a Z' a_ ax' ax ay av vIav -@v vv -a v_

The first attempt to show that the formula (1) still holds in case the derivatives of the coordinates are not continuous was by STOLZ. j He showed the validity of (1) in the case that the six partial derivatives ax/au, ax/av, etc., are limited in 21, and are continuous except at a finite number of points, or on a finite number of ordinary curves.

In the present paper I wish to show that the formula (1) holds for much wider cases.

Before doing this, it is proper to call attention to the fact that the definition of the area of a curved surface varies greatly with different authors. For example, no two of the four authors cited in the second footnote eniploy the same

* Presented to the Society April 28, 1906. Received for publication June 2, 1906. t HOLDER, Beitrdge zur Potentialtheorie, Dissertation, Stuttgart, 1882, p. 29, seq. ; JORDAN,

Cours d'Analyse, Paris, 1893, vol. 1, p. 146, seq. ; STOLZ, Grundziige, Leipzig, 1899, vol. 3, p. 239, seq. ; GOURSAT, Cours d'Analyse, Paris, 1902, vol. 1, p. 306, seq. ; or English trans- lation by HEDRICK, Ginn and Co., p. 272; VALLEE POUSSIN, Cours, vol. 2.

+ These Transactions, vol. 3 (1902), p. 34, seq. 489



490 J. PIERPONT: AREA OF CURVED SURFACES [October

definition, although the four definitions lead to the same result in the case that the surface has a continuously turning tangent plane.

In seeking to remove the restriction of continuity as far as possible, I have found myself forced to adopt a type of definition similar to those employed by HOLDER and STOLZ, which may be described briefly as follows:

Let us effect a rectangular division * of the u, v plane of norm d. By draw- inig the diagonals of these rectangles, we divide the plane into little right angled triangles tK, Such a division we shall call a quasi rectangular division of> norm d. Corresponding to a t,K whose vertices PO,, PI, P2 lie in Af, is a plane triangle r, inscribed in the surface S, and thus to a quasi rectangular divisioln D of 2f corresponds the inscription of a polyhedron in S whose faces are the little trialgiles, TK If 8D denotes the area of this polyhedron, the area of S is defined by STOLZ to be (2) lim '8D.

d = 0

The objection to this definition lies in the restriction of the polyhedra. One caninot help asking: would the limit of SD be the same for a system of divisions which are lnot quasi rectangular ? Let us therefore take a more general system of divisions. Let us divide the u, v plane into little triangles in any way, such however that lno square of finiite size contains an infillite nlumber of trianigles. If each triangle can be enclosed in a square of side d, we shall call such a division a triangular division of norm d. Corresponding to such a division of the u, v plane is a polyhedroni inscribed in the surface S, whose faces are also triangular, as before. If SD again denote the area of this polyhedron we may investigate the limit (2), and if it exists it miay be called the area of S.

But when we take the divisions D in this very general manner, this liimiit does not exist, as SCHWARZ first pointed out.t Some restriction of D is thus necessary. H6LDER, in the paper already referred to, showed that the limit (2) exists for all divisions D such that no angle of any triangle of D is greater thani 7r - , where e (e > 0 ) is taken small at pleasure, and then fixed. As the sines of these angles do not sink below some positive constant, I shall call such triangular divisions positive.

In the following investigation I have not always been able to adopt as general a definition as does HOLDER, who considers only the case that the first derivatives of the co6rdinates are conltilluous. On the other hand I have striven, whenever possible, to employ triangular divisions much more general thani the quasi rectangular divisions of STOLZ.

*Cf. the author's Lectures on the Theory of Functions of Real Vlariables, vol. 1, Ginn and Co., 1905, i 253; or the author's papers on Multiple Integrals, these Transacti ons, vols. C, 7 (1905, 1906).

t SCHWARZ, Werke, vol. 2, pp. 309 and 369.



1906] J. PIERPONT: AREA OF CURVED SURFACES 491

? 2. Regular surfaces.

Let us begin by considering a class of surfaces having at each point a tangent plane, which however does not necessarily turn continuously. Such surfaces may be called regular.

Let the coordinates x, y, z be one valued totally differentiable * functions in A. Let

Po = (u, V0), P1 = (uo + 8, vo + X') P2 = (UO + s", VO + n")

be the vertices of t. KThe area of the corresponding triangle, 7K , is given by

K K +YK +Z where

Yo z0

2 XK= y, z, 1 =, iY i-Z 'yA

Y2-YO Z2-ZO Y2 Z2 1

while similar expressions hold for YK, ZK. Thus

SD= -1/XK2+ K + ZK2. But

A'x = x -? X= + 3V I

+ aS + 8ax oau. +vf ?a '?I3 ?

at, @ $ la x a x

,, S" wf . A"x= x2 x 0= - 1 n a x7

while similar expressions hold for the other increments. Thus

a)y c,ay, az g Z, au iov au a

2X + 2X+ = ay +ay a,, + az au a?, a,u alv

where XK denotes a sum of several determinants involving the infinitesimals ax, all, /38', /. Similar expressions hold for YK, ZK. We get thus

XyA yX ZJ= Z?~, Z~Ut?~ K K tK + XK s K tK + T-K ? K CK t + ZK

where AK, BK, CK denote the values of the determinants A, B , C at the poinit PO. Thus

SD = (tK AK + K + B + K ) + C + IC

* Lectures, ? 423.

Trans. Am. Math. Soc. 33




Let us suppose that A2 + B2 + C2 remains greater than some positive number, while

/ a \ ~ ~~~~K 1K K (3) <'- 9 tK tK t K

are in general uniformly evanescent in SC; i. e., let them remain numerically less than some positive number arbitrarily small, for any division D of norm d c do except in the vicinity of a discrete set of points in SC. Then

SD tKVAK+BK+ K K+ E K KE where, in general,

I EK I <E/1.*

If now A, B, C are limited and integrable in 9A, we have

lim SD = fVA2 + B2 + C2 dudv.

We ask now under what conditions are the expressions (3) in general uniformly evanescent? The answer is prettv evident from the example given by SCHWARZ. In fact the equation of the tangent plane at the point x0 y0z0 is

AK(x-xo) + B (-O)+CK(-V = 0. AK(X tx X )JBK ( Y YO) Jr CK (7 o )

OIn the other hand, the plane of the triangle TK is

x y z 1

x0 YO z0 1 =0,

K1 Y1 z1 1

X2 Y2 '2 or

XXK + YYK J+ ZZK + UK 0 or finally

x Q 4 )Y ( K+ )+z(GK+ () +

Thus for (3) to converge in general uniformly to zero, it is necessary and suffi- cient that the planes of the faces of the inscribed polyhedra converge in general uniformnly to tangent planes.

Let us call a triangular division such that the faces of the corresponding polyhedra converge in general uniformly to tangent planes a regular division. We have now the following:

THEOREM 1. Let the cobrdinates x , y, z be one valued totally di;erentiable functions qf the parameters u, v in the limited complete region 9A, such that

* A = Cont At denotes upper content of SC. Cf. Lectutres, ? 702.




A2 + B2 + C2 is greater than some positive constant, and is limited and integrable in A. Then

S= limr D f I/A2+ B2 +C2dudv, d=O l

D denoting a regular triangular division of norm d. If we call this limit area, we have at once a number of its properties. W\Ae

mention only the following: THEOREM 2. Let Al * .7n be ubnmixed * partial complete regions of Sf,

whose union is W. Let S1, , S be the pieces of S corresponding to them. Then each SK has an area, and their sum is S.

THEOREM 3. Let WA be an unmixed partial complete region of SC depend- ing on a parameter X _0 , such that lim 9fA = At. Then

lim S- S. A=0

THEOREM 4. The area of S remains utnaltered when S is subjected to a displacement or when the parameters u, v are subjected to the transformation of theorenzs 31, 32 in the author's paper t on Improper Integrals.

?3. Irregular suraces.

We turn now to certain surfaces which do not have tangent planes at every point, i. e., surfaces for which one or more of the first partial differential coefficients of the co6rdinates do not exist at certain points, and which may be styled irregular surfaces.

THEOREM 5. Let the coordinates x, y, z be one valued functions, having limited total difference quotients in the limited complete region 2f. Let D be a positive division of norm some number do. Then

max SD is finite, and evanescent with RI.

For, let the difference quotients remain = ,u. We have

BIut SD XK + EY + EK

I KI 2 I /| 2{ + IA I IA Y I} ~/2 P 2.OP /~t~cc w ih e b s P 0= P2p.p2 AsD esa 0os

where 0 K is the angle made by the sides PO P1, POP,. As D is a positive

* Lectures, ? 711. tThese Transactions, vol. 7 (1906), p. 172. t Lectures, ? 708.




division, one of the angles of tK is such that csc OK is numerically less than some positive number Mf. Thus

XK < 1I2ft,

where p, Mare independent of I and d. Similar relations hold for I ,K

I Z; and the truth of the theorem is now obvious. THEOREM 6. In the limited complete region 2I, let the cobrdinates x, y, z

be as in theorem 1, except that at certain points forming a discrete aggregate a, the first partial differential coefficients do not exist. Let their total difference quotients be limited in Wf. Then

lim SD f 1A2 + B2 + C2 du dv, d=Oa

where D denotes a positive regular division of norm d. Let us first show the limit on the left exists. We may choose an unmixed

partial region 3 of 9X, containinig a as inner* points, such that 3 is as small as we please. Let G = 2-3 , while Sc may denote the area of the surface corresponding to d;. The triangles tK fall into two groups; first a group G, containing points of 03, and second a group G2, containing only points of S. Then

SDEZI/ X2 + Y, + ZKEZZE G1 G2

But 3 may be chosen so small that the first sum is < c/4 for any d < do. Moreover, by taking do still smaller if necessary, we have

I EG2 Sc I < e/4. Hence

(4) I SD- ScI< </2 (d do).

Similarly for any other division D' of norm d',

I SD, SC| <e/2, (d' <do),

decreasing do still farther if necessary. Thus

I SDI-SDI <e, (d d' < do).

Hence lim 5D exists; call it S. Since S exists, we may take do so small that

IS-SDI<e/2, (d<do).

* Le(tures, ? 704, 4.




This with (4) gives JS- S01<E;

that is,

S=limSc=limf X 2 A2+B2+ C2dudv,

by Lectures, ? 724. The preceding theorem takes care of a large class of irregular surfaces whose

total difference quotients are limited, either with the original set of parameters or with a new set. When the total difference quotients are not limited, and it is not practical to transform to a niew system, whose total difference quotients are limited, the following theorems may be useful. They rest on a new system of divisions which we define as follows. Let us effect a quadrate division of the u, v plane of norm d, and take the triangles tK so that for any triangular division D associated with d, no square contains more than n triangles, and no triangle lies in more than v squares; n, v being arbitrarily large constants independent of d. Such a division we call a quasi quadrate division of norm d.

We shall also need to introduce a new classification of functions according to their variation in 2A, or along lines parallel to the u, v axes.

Let D be a quadrate division of the u, v plane, of norm d _ do. Let OK osc b (u., v) denote the oscillation of 4 in one of the squares of D. The extreme

max E cox d, D

for all divisions D is called * the variation of 4 in Af. If the variation is not only limited, but evanescent with X, we shall say 4 has limited fluctuations in 2t. Let a denote the points of 2Y lying on a line parallel to the u axis. Let us effect a division D' of this line into intervals of length d. Let

=K OSC u( u v) in one of these intervals. Then

= max 1:)1 DI

denotes the variation of 4) in a. Let us consider now all the aggregates a lying on lines parallel to the u axis,

and suppose ii : a-, where a- _0 . If now for each there exists a constant G, such that c65 < a-G; that is, if c. are uniformly evanescent with cr, we shall say that 4(u, v) has limited fluctuation with respect to Mb. Similar remarks apply to v.

With the aid of these notions we may prove the following theorems. THEOREM 7. Let the coordinates x, y, z be one valued limitedfunctions in

the limited complete region A. Let x, y have limited total diference quotients,

* Lectur es, ? 709.




while z has limited variation in S. Let D denote a quasi quadrate division of norm, d - do. Then

max SD D

is finite. For, as before

2 1X. `5IA' I " + IrA1I. Y K | / | + y I z|

But ,u denoting a sufficiently large constant,

Let a), = osc z in the square s,. If the triangle tIK lies in the squares s,, , r

A' KZ t?? + + r

Hence A' i\ + I i\z 1\ | ( t1 + ..+ cor

Thus, n, v, 3, 3' denoting sufficiently large constants

IJX. <,tnEdd(w)l? + + co J9

the summation extending over all squares s, containing points of 9C. But z having limited variation (5) L d < X. Hence

E I XK,, Y" I < /unPX. Finally as in theorem 5,

El ZKj <31 The theorem is thus established.

THEOREM 8. The coordinates x, y, z being as in the preceding theorem, except that z has limited fluctuation in A, and D denoting a quasi quadrate division of norm d cdo

niax SD D

isfinite, and evanescent with RY. The reasoning is the same as in theorem 8, except that now 3 and X' are

evanescent with RC. THEOREM 9. Let the coordinates x, y, z have limited total diference

quotients in Xf, while the variation of z along any line parallel to the u or v aaxes is < M. Let A lie in a square of side s _ 0. Then

max 8D <Gs;




where G is some constant independent of s, and D is a quasi rectangular division of norm c do.

For, here

2E IX. ElAy I 'y A + E I A'Y I/y Z A Mz< 'F cu d + X'c,,du;

where X' denotes a sufficiently large constant; du, dv denote the length of sides of one of the triangles t,K parallel respectively to the u, v axes, and co , v the oscillation of z along these sides. Since the variation is < X in both directions,

Z it dV= Edv wu < XEdv < Ms. v

Similarly,

ENVdu < Ms.

The rest of the proof follows as before. THEOREM 10. The symbols having the same meaning as before, except

that z has limited fluctuation with respect to u, v,

Max SD < GS2. D

The demonstration is siinilar to the foregoing. Following the line of proof used in establishing theorem 6, and employing

the theorems just given, we readily prove the following theorems: THEOREM 11. Let 9I be a limited complete region, containing the discrete

aggregate a. Let b be a partial region qf f containing the points (i, such that 3 - - 1 b is exterior to ai, moreover let 1 _ 0. Let the coardinates x, y, z be one valued totally differentiable functions in X, such that A2 + B2 + C(2 iever sinks below a positive constant in any ~3, is properly integrabie in any Q-, and improperly integrable in Wt. Let x, y have limited total difference qutotients and z lim'titedfluctuation in b. Thena

lim SD f/A 2 + B2 + C2dudv, d=O a

where D is any quasi quadrate division qf norm d. THEORE.M 12. Let the symbols have the same meaning as in the preceding

theorem except that: 1) a reduces to afinite aggregate; 2) z has limited variation along any line parallel to the t, v axes; 3) D denotes a regular quasi rectangular division. TIhen

limSD=f. d=O )



498 J. PIERPONT: AREA OF CURVED SURFACES

THEOREM 13. Let the symbols have the same meaning as in theorem 11 except that: 1) z has limited fluctuation with respect to u, v in b; 2) D denotes a regular quasi rectangular division. Then

limSD 'f d=O

THEOREM 14. If we call the limits in theorems 11, 12, 13 area, the theorems 2, 3, 4 still hold.

ROCK BOUND, SACHEM'S HEAD, April, 1906.



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