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foot of the perpendicular let fall from the origin upon the tangent to a curve at a point is called the pedal of the curve with respect to the origin. The angle >' for the pedal is the same as the angle for the curve. Hence the (p, r) equation of the pedal can be deduced. If the pedal is regarded as the See also:

PRIMARY

primary curve, the curve of which it is the pedal is the " negative pedal " of the primary. We may have pedals of pedals and so on, also negative pedals of negative pedals and so on.

Negative pedals are usually determined as envelopes. (iii.) If 4, denotes the angle which the tangent at any point makes with a fixed See also:

LINE

line, we have See also:

R2(rt-s2)

r2 =p2+ (dp/d4,)2. (iv.) The " See also:

AVERAGE

average curvature " of the arc is of a curve between two points is measured by the quotient Ip~~ As where the upright lines denote, as usual, that the See also:

ABSOLUTE (Lat. absolvere, to loose, set free)

absolute value of the included expression is to be taken, and ¢ is the angle which the tangent makes with a fixed line, so that 0¢ is the angle between the tangents (or normals) at the points. As one of the points moves up to coincidence with the other this average curvature tends to a limit which is the " curvature " of the curve at the point. It is denoted by ds Sometimes the upright lines are omitted and a See also:

RULE

rule of signs is given :—Let the arc s of the curve be measured from some point along the curve in a chosen sense, and let the normal be drawn towards that See also:

side to which the curve is See also:

CONCAVE

concave; if the normal is directed towards the See also:

LEFT

left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned See also:

POSITIVE (or PORTABLE) ORGAN

positive, in the contrary See also:

case negative. The differential d¢ is often called the " angle of contingence." In the 14th See also:

CENTURY (from Lat. centuria, a division of a hundred men)

century the See also:

SIZE

size of the angle between a curve and its tangent seems to have been seriously debated, and the name " angle of contingence " was then given to the supposed an le. (v.) The curvature of a curve at a point is the same as that of a certain circle which touches the curve at the point, and the "radius of curvature" p is the radius of this circle. We have p =i--I. The centre of the circle is called the " centre of curvature "; it is the limiting position of the point of intersection of the normal at the point and the normal at a neighbouring point, when the second point moves up to coincidence with the first. If a circle is described to intersect the curve at the point P and at two other points, and one of these two points is moved up to coincidence with P, the circle touches the curve at the point P and meets it in another point; the centre of the circle is then on the normal. As the third point now moves up to coincidence with P, the centre of the circle moves to the centre of curvature. The circle is then said to " osculate " the curve, or to have " contact of the second See also:

order " with it at P.

(vi.) The following are formulae for the radius of curvature : P =I I + (dx> 2 ~d 22I ' p=lypl -I 41' (vii.) The points at which the curvature vanishes are " points of inflection." If P is a point of inflection and Q a neighbouring point,then, as Q moves up to coincidence with P, the distance from P to the point of intersection of the normals at P and Q becomes greater than any distance that can be assigned. The equation which gives the abscissae of the points in which a straight line meets the curve being expressed in the See also:

FORM (Lat. forma)

form f(x) =o, the See also:

FUNCTION

function f(x) has a See also:

FACTOR (from Lat. facere, to make or do)

factor, (x—xo)3, where xo is the See also:

ABSCISSA (from the Lat. abscissus, cut off)

abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x—xo) occurs (n+I) times in f(x), the curve is said to have " contact of the nth order " with the line. There is an obvious modification when the line is parallel to the See also:

AXIS (Lat. for " axle ")

axis of y. (viii.) The locus of the centres of curvature, or envelope of the normals, of a curve is called the " evolute." A curve which has a given curve as evolute is called an " involute " of the given curve. All the involutes are " parallel " curves, that is to say, they are such that one is derived from another by marking off a See also:

CONSTANT, BENJAMIN (1845-1902)

constant distance along the normal. The involutes are " orthogonal trajectories " of the tangents to the See also:

COMMON

common evolute. (ix.) The equation of an algebraic curve of the nth degree can be expressed in the form uo+ ui+ u2+ ... + un = o, where uo is a constant, and u,. is a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, uo vanishes, and ui =o represents the tangent at the origin. If ul also vanishes, the origin is a See also:

double point and u2 = o represents the tangents at the origin. If u2 has distinct factors, or is of the form a(y—pix)(y—p2x), the value of y on either See also:

BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

branch of the curve can be expressed (for points sufficiently near the origin) in a See also:

POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power See also:

SERIES (a Latin word from serere, to join)

series, which is either pix-l-lgix2+ ..

. , or p2x+ig2x2+ .. where qi, . . . and q2, . . . are determined without See also:

AMBIGUITY (Fr. ambiguite, med. Lat. ambiguitas, from Lat. ambiguus, doubtful; ambi, both ways, agere, to drive)

ambiguity. If pi and p2 are real the two branches have radii of curvature pi, pt determined by the formulae .1. (i+p12)-2giI, P2=(I+p22) IIg2l. When pi and p2 are imaginary the origin is the real point of inter-See also:

section of two imaginary branches. In the real figure of the curve it is an isolated point. If u2 is a square, a(y—px)2, the origin is a See also:

CUSP (Lat. cuspis, a spear, point)

cusp, and in See also:

general there is not a series for y in integral See also:

POWERS, HIRAM (1805-1873)

powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to See also:

ANALYTICAL

analytical geometry and the theory of algebraic functions than to differential calculus. (x.) When the equation of a curve is given in the form uo+ui+ .. . +un_i+un=o where the notation is the same as that in (ix.), the factors of us determine the directions of the asymptotes.

If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un=LIL2 . . . L,., where Li, . . . are linear in x, y, we may resolve un_i/un into partial fractions according to the See also:

formula u=_1 _ Al See also:

A2(z)

A2 nn L+ z+. .+r-, and then Li+AI =o, L2+A2=o, ... are the equations of the asymptotes. When a real factor of us is repeated we may have two parallel asymptotes or we may have a " parabolic asymptote." Sometimes the parallel asymptotes coincide, as in the curve x2(x2+y2—a2) =a*, where x= o is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions. 47. The formal See also:

DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition of an integral, the theorem of the existence of the integral for certain classes of functions, a See also:

list of classes of " integrable " functions, extensions of the notion lategral of integration to functions which become See also:

INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite or in- calculus. determinate, and to cases in which the limits of integration become infinite, the See also:

DEFINITIONS

definitions of multiple integrals, and the possibility of defining functions by means of definite integrals—all these matters have been considered in FUNCTION. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.

48. The See also:

CHIEF (from Fr. chef, head, Lat. caput)

chief methods for the evaluation of indefinite Methods of integrals are the method of integration by parts, and the Integration. introduction of new variables. From the equation d(uv) =udv+vdu we deduce the equation fudvdx=uv— f vddx, or, as it may be written f uwdx = u f wdx —f dx f wdx dx. This is the rule of " integration by parts." As an example we have e°- ens — (x— 1 xe"°dx = x a — a dx — a a2 e . When we introduce a new variable z in See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place of x, by means of an equation giving x in terms of z, we See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express f(x) in terms of z. Let ~(z) denote the function of z into which f(x) is transformed. Then from the equation dx=d;dz we deduce the equation "sin"nxdx (zncos2nxdx = I .3... (zn — I) ( ff(x)dxa f ¢(z)dxdz. (ix.) Jo (2 Joy 2.4...271 naninteger). As an example, in the integral / (x.) f'sin2n+'xdx= f ='cost"+Ixdx=3.5.4. (2n + 1),(n an integer). f (1—x2)dx dx put x=sin z; the integral becomes (x'') f (I } ecosx)n can be reduced by one of the substitutions fcos z. cos zdz=1;(I +cos2z)dz=y(z+I sin 2z) =1(z+sin zcosz). e+cosx -{- 49.

The indefinite integrals of certain classes of functions can be cos 4 = cosh u = expressed by means of a finite number of operations of addition or 1+e cos x' I e-{-eeoscosxx' multiplication in terms of the so-called " elementary " of which the first or the second is to be employed according as Integra- functions. The elementary functions are rational alge- e< or > I. See also:

lion in braic functions, implicit algebraic functions, exponentials 50• Among the integrals of transcendental functions Newtransterms of and logarithms, trigonometrical and inverse circular which See also:

lead to new transcendental functions we may See also:

NOTICE

notice cendents. element- functions. The following are among the classes of i -= z ary func- functions whose integrals involve the elementary functions dx or , dz, Lions, only: (i.) all rational functions; (ii.) all irrational functions ()See also:

LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log x' _ xz of the form f(x, y), where f denotes a rational algebraic function called the " logarithmic integral," and of x and y, and y is connected with x by an algebraic equation of the the integrals second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of es; (v.) all rational integral functions of the variables x, e", ... sin mx, cos mx, sin nx, cos nx, . in which a, b, . and m, n, ... are any constants. The integration of a rational function is generally effected by resolving the function into partial fractions, the function being first expressed as the quotient of two rational integral functions. Corresponding to any See also:

SIMPLE

simple See also:

root of the denominator there is a logarithmic See also:

TERM

term in the integral. If any of the roots of the denominator are repeated there are rational algebraic terms in the integral. The operation of re-solving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic See also:

PART

part of the integral can always be found without obtaining all the roots of the denominator. Reference may be made to C. Hermite, Cours d'analyse, See also:

Paris, 1873. The integration of other functions, which can be integrated in terms of the elementary functions, can usually be effected by transforming the functions into rational functions, 'possibly after preliminary integrations by parts.

In the case of rational functions of x and a See also:

RADICAL (Lat. radix, a root)

radical of the form ' (ax2+bx+c) the radical can be reduced by a linear substitution to one of the forms \I (a2-x2), (x2—a2), x1 (x2+a2). The substitutions x =a sin D, x =a sec 0, x =a tan 0 are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin 0 and cos 0, and it can be reduced to a rational function of t by the substitution tan 40=t. There are many other substitutions by which such integrals can be determined. Sometimes we may have See also:

INFORMATION (from Lat. informare, to give shape or form to, to represent, describe)

information as to the functional See also:

CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character of the integral without being able to determine it. For example, when the subject of integration is of the form (ax"+bx3+cx2+dx+e)-I the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (see FUNCTION). Methods of reduction and substitution for the evaluation of in-definite integrals occupy a considerable space in See also:

TEXT (Lat. textum, woven fabric, from texere, to weave)

text-books of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. See also:

Hardy's See also:

TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

tract, The Integration of Functions of a Single Variable (See also:

Cambridge, 1905), and to the See also:

MEMOIRS

memoirs there quoted. A few results are added here (i.) f (x2+a)-Idx = log lx+(x2+a)I }.

denoted by " Li x," also (ii.) f (x _ p) (Q x+zbx+c) can be evaluated by the substitution x-p=1/z, andf (.r—p)nv/ (daxx2+zbx+c) can be deduced by differentiating (n—I) times with respect to p. (iii.) f (a (Hx+K)dx can be reduced by the sub- stitution (ax +2bx+c) stitution y2 = (ax2+2bx+c)/(ax2+2px+y) to the form Af dy +B dy (Ti—Y2) JAI (Y2—),2) where A and B are constants, and Ti and X2 are the two values of X for which (a—Ta)x +2(b—TQ)x+c—Ty is a perfect square (see A. G. Greenhill, A See also:

CHAPTER (a shortened form of chapiter, a word still used in architecture for a capital; derived from O. Fr. chapitre, Lat. capitellum, diminutive of caput, head)

Chapter in the Integral Calculus, See also:

LONDON

London, 1888). (iv.) Jx"'(ax"+b)gdx, in which m, n, p are rational, can be reduced, by putting ax" = bt, to depend upon f t4(I +t)gdt. If p is an integer and q a fraction re's, we putt =u'. If q is an integer and p = r/s we put 1+1 =us. If p+q is an integer and p = r/s we put 1+t=tu'. These integrals, called " See also:

BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

binomial integrals," were investigated by See also:

Newton (De quadratura curvarum). (v.) ,fss ex-log d tan-, (vi.) rcoxx =log (tan x+sec x). (vii.) fe" sin (bx+a)dx = (a2+b2)-'e"rla sin (bx+a) —See also:

BOOS, MARTIN (1762–1825)

boos (bx+a) }. (viii.) f x cos" x dx can be reduced by differentiating a function of the form sing x cos4 x; d sin x1 gsin2x= 1—q q e.g. dx cos4 x cos4-Ix +cosy+Ix cos4-'x+cos4+Ix• ,J 1( dx sin x n—2 dx cos' x (n — 1) cos"—' x+n— 1— I,f cos"—2x' Hence x f sin x f cos x o x dx and x dx, called the " sine integral " and the " cosine integral," and denoted by " Si x" and " Ci x," also the integral f e o~dx called the " See also:

ERROR (Lat. error, from errare, to wander, to err)

error-function integral," and denoted by " Erf x." All these functions havebeen tabulated (seeTABLES,MATHEMATICAL).

51. New functions can be introduced also by means of the definite integrals of functions of two or more variables with re- Eulerian spect to one of the variables, the limits of integration integrals. being fixed. Prominent among such functions are the Beta and See also:

GAMMA

Gamma functions expressed by the equations B(l,m)= See also:

FIX, THEODORE (180o-1846)

fix''(I—x)''''dx, F(n) =f 0 When n is a positive integer r(n+I) =n ! . The Beta function (or " Eulerian integral of the first See also:

KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind ") is expressible in terms of Gamma functions (or " Eulerian integrals of the second kind ") by the formula B(l, m) . P(l+m)=1'(l) . I'(m). The Gamma function satisfies the difference equation r(x+1) =xr(x), and also the equation r(x) . P(1 —x) =7r/ sin (xir), with the particular result r(2)=e.( r. The number —(dx d llog 1'0 +x)}js=o or—1''(1), is called " See also:

EULER, LEONHARD (1707-1783)

Euler's constant," and is equal to the limit lim.,,=,,o (1+2+ +...+n) —log n] ; its value to 15 decimal places is 0.577 215 664 901 532. The function log P(I+x) can be See also:

EXPANDED

expanded in the series log P(1+x)=2 log(sin x2r) -2 log I-+11-}-1''(1)}x 1(S3—1)70—1; (S6—1)x6—..., where I I S2r+I = I +22r+I+32r+I+ .. , and the series for log P(1 +x) converges when x lies between — i and 1. 52.

Definite integrals can sometimes be evaluated when the limits of integration are some particular See also:

numbers, although the corresponding indefinite integrals cannot be found. For example, we have the result f o I (I —x2)-I logxdx= - 2irlog2, although the indefinite integral of (I —x2)-I log x cannot be found. Numbers of definite integrals are expressible in terms of the transcendental functions mentioned in § 50 or in terms of Gamma functions. For the calculation of definite integrals we have the following methods (i.) Differentiation with respect to a parameter. (ri.) Integration with respect to a parameter. (iii.) Expansion in infinite series and integration term by term. (iv.) See also:

CONTOUR

Contour integration. The first three methods involve an interchange of the order of two limiting operations, and they are valid only when the functions satisfy certain conditions of continuity, or, in case the limits of Definite integrals. integration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (see FUNCTION). The method of contour integration involves the introduction of complex variables (see FUNCTION: § Complex Variables). A few results are added (i.)f o I+xdx=sinaa' (1>a>o), (ii.) f o x° I —~idx=~r(See also:

ROTA, COURT OF

rota,r—cotbr), (o <a or b<I) (..,) f x— x"_i loIg xdx _ ,r2 ui sinzaa' (a>), (iv.) J x2.cos 2x.e yzdx= 0 (v) I +—x2x"dxtog fi tan'r, of vi ) si: ( n mx dx = , I I o;_ m+1) , (vii.) f log(I—2acosx+a2)dx =o or 2ir log a according as a < or> 1, :log( viii.) sin ( x ~ dx o x (ix.) J f x cos ax _ o x2+bzdx—2nb ie nn (x.) f cos ax-2cosbxdxz7r(b—a), (xi) cos ax—cos bxdx =logb J x be (xii.) f cos x —e '"xdx = log in, o x (xiii.) f e z +2a=dx = „Ia. ea2, (xiv.) f x'^ sin xdx =f :e lcosxdx=I(2n). 0 0 53.

The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the M uhiple See also:

ARTICLE (from Lat. articulus, a joint)

article FUNCTION. In the case of two variables x, y we Thieves. integrate a function f(x,y) over an See also:

AREA

area; in the case of three variables x, y, z we integrate a function f(x, y, z) through a See also:

VOLUME

volume. The integral of a function f(x, y) over an area in the plane of (x, y) is denoted by ff f(x, y)dxdy. The notation refers to a method of evaluating the integral. We may suppose the area divided into a very large number of very small rectangles by lines parallel to the axes. Then we multiply the value off at any point within a rectangle by the measure of the area of the rectangle, sum for all the rectangles, and pass to a limit by increasing the number of rectangles indefinitely and diminishing all their sides indefinitely. The See also:

PROCESS

process is usually effected by summing first for all the rectangles which See also:

lie in a See also:

STRIP

strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is See also:

EQUIVALENT

equivalent to integrating f(x, y) with respect to y, keeping x constant, and taking certain functions of x as the limits of integration for y, and then integrating the result with respect to x between constant limits. The integral obtained in this way may be written in such a form as 12c=) adx f nc=)f (x, y)dy and is called a " repeated integral.” The See also:

IDENTIFICATION (Lat. idem, the same)

identification of a See also:

SURFACE

surface integral, such as f[f(x, y)dxdy, with a repeated integral cannot always be made, but implies that the function satisfies certain conditions of continuity. In the same way volume integrals are usually evaluated by regarding them as repeated integrals, and a volume integral is written in the form fff f(x, y, z)dxdydz. Integrals such as surface and volume integrals are usually called " multiple integrals." Thus we have " double " integrals, " triple " integrals, and so on. In contradistinction to multiple integrals the See also:

ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary integral of a function of one variable with respect to that variable is called a " simple integral.

A more general type of surface integral may be defined by taking an arbitrary surface, with or without an edge. We suppose in the Surface first place that the surface is closed, or has no edge. We rge er of Integrals. See also:

DAWN (the 16th-century form of the earlier " Jawing " or " dawning," from an old verb " daw," O. Eng. dagian, to become day; cf. Dutch dagen, and Ger. tagen)

dawn the tang enn ants These tangent planes form a See also:

POLYHEDRON (Gr. rain, many, ESpa, a base)

polyhedron having a large number of faces, one to each marked point; and we may choose the marked points so that all the linear dimensions of any See also:

FACE (from Lat. fades, derived either from facere, to make, or from a root fa-, meaning " appear "; cf. Gr. cbatvstv)

face are less than somearbitrarily chosen length. We may devise a rule for increasing the number of marked points indefinitely and decreasing the lengths of all the edges of the polyhedra indefinitely. If the sum of the areas of the faces tends to a limit, this limit is the area of the surface. If we multiply the value of a function f at a point of the surface by the measure of the area of the corresponding face of the polyhedron, sum for all the faces, and pass to a limit as before, the result is a surface integral, and is written fffds. The See also:

EXTENSION (Lat. ex, out ; tendere, to stretch)

extension to the case of an open surface bounded by an edge presents no difficulty. A line integral taken along a curve is defined in a similar way, and is written ffds where ds is the element of arc of the curve (§ 33). The direction cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line integrals usually See also:

PRESENT

present themselves in the form f lugs } vds+wds) ds or f,,(udx+vdy+wdz). In like manner surface integrals usually present themselves in the form Jf(1E+mgt+ni )dS where 1, m, n are the direction cosines of the normal to the surface drawn in a specified sense. The area of a hounded portion of the plane of (x, y) may be ex-pressed either as zf(xdy-ydx), or as ffdxdy, the former integral being a line integral taken See also:

ROUND (O. Fr. rond, Lat. rotundus, the Fr. is the source also of Du. rond; Ger., Swed., Dan. and Nor. rond)

round the boundary of the portion, and the latter a surface integral taken over the area within this boundary. In forming the line integral the boundary is supposed to be described in the positive sense, so that the included area is on the left See also:

hand.

53a. We have two theorems of transformation connect- Theorems See also:

ing volume integrals with surface integrals and surface of See also:

Green integrals with line integrals. The first theorem, called and " Green's theorem," is expressed by the equation See also:

Stokes. fff az+ay+az) dxdydz=ff(lE+mn+n3')dS, where the volume integral on the left is taken through the volume within a closed surface S, and the surface integral on the right is taken over S, and 1, m, n denote the direction cosines of the normal to S drawn outwards. There is a corresponding theorem for a closed curve in two dimensions, viz., (ax+ay) dxdy= f ( as—ids) ds, if a(xi, xz, .. , x„) the sense of description of s being the positive sense. This theorem is a particular case of a more general theorem called " Stokes's theorem." Let s denote the edge of an open surface S, and let S be covered with a network of curves so that the meshes of the network are nearly plane, then we can choose a sense of description of the edge of any mesh, and a corresponding sense for the normal to S at any point within the mesh, so that these senses are related like the directions of rotation and See also:

TRANSLATION

translation in a right-handed See also:

SCREW (O.E. scrue, from O. Fr. escroue, mod. ecrou; ultimate origin uncertain; the word, or a similar one, appears in Teutonic languages, cf. Ger. Schraube, Dan. skrue, but Skeat, following Diaz, finds the origin in Lat. scrobs, a ditch, hole, particularl

screw. This See also:

convention fixes the sense of the normal (l, m, n) at any point on S when the sense of description of s is chosen. If the axes of x, y, z are a right-handed See also:

SYSTEM

system, we have Stokes's theorem in the form 11 J .(ud'+vdy +wdz M l(ay-a)+meaz-az) \ax ay)I where the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either See also:

REVERSE

reverse the sense of 1, m, n and maintain the formula, or retain the sense of 1, m, n and See also:

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change the sign of the right-hand member of the equation. For the validity of the theorems of Green and Stokes it is in general necessary that the functions involved should satisfy certain conditions of continuity. For example, in Green's theorem the differential coefficients OElax, an/ay, N-/Oz must be continuous within S.

Further, there are restrictions upon the nature of the curves or surfaces involved. For example, Green's theorem, as here stated, applies only to simply-connected regions of space. The correction for multiply-connected regions is important in several See also:

PHYSICAL

physical theories. 54. The process of changing the variables in a multiple integral, such as a surface or volume integral, is divisible into two stages.'" It is necessary in the first place to determine the differential Chan of element expressed by the product of the differentials of the change es first set of variables in terms of the differentials of the in a second set of variables. It is necessary in the second place Multiple to determine the limits of integration which must be em- Integral. ployed when the integral in terms of the new variables is evaluated as a repeated integral. The first part of the problem is solved at once by the introduction of the Jacobian. If the variables of one set are denoted by xi, x2, ..., x,,, and those of the other set by ui, u2, .. , u,,,, we have the relation dxidx2 ... dxa =a (7ii, nz, ... duiduz ... dug. Line Integrals. 14 In regard to the second See also:

STAGE (Fr. 6/age; from Lat. stare, to stand)

stage of the process the limits of integration must be determined by the rule that the integration with respect to the second set of variables is to be taken through the same domain as the integration with respect to the first set. For example, when we have to integrate a function f (x, y) over the area within a circle given by x2+y2=a2, and we introduce polar coordinates so that x =r cos 8, y =r sin 0, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a r o, 2,r> 0 o, and we have faadx f~({s z~)f(x,y)dy= fodr fof(rcos0,rsin8)rd0.

If we have to integrate over the area of a rectangle 0, b~ y o, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows: , tan-lbla a sec 0 f odx f of(x,y)dy= f o dO f o f(rcosO, rsin0)rdr i ' +f dO b cosec o tan ib/a 0 f (r cose, rsin o)rdr. 55. A few additional results in relation to line integrals and multiple integrals are set down here. (i.) Any simple integral can be regarded as a line-integral taken Line along a portion of the axis of x. When a change of Integrals variables is made, the limits of integration with respect to the new variable must be such that the domain of and Multiple integration is the same as before. This See also:

CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition may integrals require the replacing of the See also:

ORIGINAL

original integral by the sum of two or more simple integrals. (ii.) The line integral of a perfect differential of a one-valued function, taken along any closed curve, is zero. (iii.) The area within any plane closed curve can be expressed by either of the formulae f 2r2do or f Z pds, where r, 0 are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to he under-stood as line integrals taken along the curve. When the same integrals are taken between limits which correspond to two points. of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores. (iv.) The volume enclosed by a surface which is generated by the revolution of a curve about the axis of x is expressed by the formula of y2dx, and the area of the surface is expressed by the formula 2afyds, where ds is the differential element of arc of the curve. When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles.

The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x. (v.) When we use See also:

CURVILINEAR

curvilinear coordinates n which are conjugate functions of x, y, that is to say are such that 6E/ax=On/Oy and 6E/6y=-On/ax, the Jacobian a(, n)/a(x, y) can be expressed in the form \ax/ 2+ \ax) ifunctions of two parameters u, v, the area is expressed by the formula ff 5 8(Y, z) '+ a('' x) 2 a(x, Y) 2 z a(u,v) + 8(u,v) +)8(u,v) ] dude. When the surface is referred to three-dimensional polar coordinates r, 0, ¢ given by the equations x = r sin 0 cos 0, y =r sin 0 sin di, z =r cos 8, .and the equation of the surface is of the form r=f(8,4,), the area is expressed by the formula f f r L r2+ ( a B ) 2 sin 28+ *See also:

DECK

deck). The surface integral of a function of (0, 0) over the surface of a See also:

SPHERE (Gr. vclsa?pa, a ball or globe)

sphere r=const. can be expressed in the form fad$ f o F (8, ¢) r2 sin See also:

OdO. In every case the domain of integration must be chosen so as to include the whole surface. (ix.) In three-dimensional polar coordinates the Jacobian a(x,y, z)=r2 sin 0. a(r, 8, 0) The volume integral of a function F (r, 0, el)) through the volume of a sphere r =a is f adr f o"d4 f o F(r, B, 4,)r2sin OdO. (x.) Integrations of rational functions through the volume of an See also:

ELLIPSOID

ellipsoid x2/a2+y2/b2+z2/c2 = are often effected by means of a general theorem due to See also:

LEJEUNE, LOUIS FRANCOIS, BARON (1776-1848)

Lejeune Dirichlet (1839), which is as follows: when the domain of integration is that given by the inequality (L.)ai + (a2)a2+...+(an)and1 where the a's and a's are positive, the value of the integral ff... x2'2'`1... dxidxa... P (1) ai/ r \ai . aia2 . rl+al+L+...\) If, however, the See also:

OBJECT

object aimed at is an integration through the volume of an ellipsoid it is simpler to reduce the domain of integration to that within a sphere of' radius unity by the transformation x=aE, y=bn, z=ci-, and then to perform the integration through the sphere by transforming to polar coordinates as in (ix). 56. Methods of approximate integration began to be devised very See also:

early.

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