MAXIMA and MINIMA).
46. In treatises on the differential calculus much space is usually
Plane devoted to the differential geometry of curves and
cures. surfaces. A few remarks and results relating to the
differential geometry of plane curves are set down here.
(i.) If >' denotes the angle which the radius vector drawn from the origin makes with the tangent to a curve at a point whose polar coordinates are r, 0 and if p denotes the perpendicular from the origin to the tangent, then
cos 4. =dr/ds, sin ,/ =rd6/ds=p/r,
where ds denotes the element of arc. The curve may be determined by an equation connecting p with r.
(ii.) The locus of the 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 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 line, we have
r2 =p2+ (dp/d4,)2.
(iv.) The " 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 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 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 side to which the curve is concave; if the normal is directed towards the left of an observer looking along the tangent in the chosen sense of description the curvature is reckoned positive, in the contrary case negative. The differential d¢ is often called the " angle of contingence." In the 14th century the 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 =iI.
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 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 form f(x) =o, the function f(x) has a factor, (x—xo)3, where xo is the 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 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 constant distance along the normal. The involutes are " orthogonal trajectories " of the tangents to the 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 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 branch of the curve can be expressed (for points sufficiently near the origin) in a power series, which is either
pixllgix2+ .. . , or p2x+ig2x2+ ..
where qi, . . . and q2, . . . are determined without 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 intersection 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 cusp, and in general there is not a series for y in integral powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to 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 formula
u=_1 _ Al 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 definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of classes of " integrable " functions, extensions of the notion lategral of integration to functions which become infinite or in calculus. determinate, and to cases in which the limits of integration become infinite, the 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 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 place of x, by means of an equation giving x in terms of z, we 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 socalled " 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.
lion in braic functions, implicit algebraic functions, exponentials 50• Among the integrals of transcendental functions Newtransterms of and logarithms, trigonometrical and inverse circular which lead to new transcendental functions we may 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 ()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 simple root of the denominator there is a logarithmic 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 resolving a fraction into partial fractions requires a knowledge of the roots of the denominator, but the algebraic 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, 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 radical of the form ' (ax2+bx+c) the radical can be reduced by a linear substitution to one of the forms \I (a2x2), (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 information as to the functional 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 indefinite integrals occupy a considerable space in textbooks of the integral calculus. In regard to the functional character of the integral reference may be made to G. H. Hardy's tract, The Integration of Functions of a Single Variable (Cambridge, 1905), and to the 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 xp=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 Chapter in the Integral Calculus, 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 " binomial integrals," were investigated by Newton (De quadratura curvarum).
(v.) ,fss exlog
d tan, (vi.) rcoxx =log (tan x+sec x).
(vii.) fe" sin (bx+a)dx = (a2+b2)'e"rla sin (bx+a) —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 cos4Ix +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 " errorfunction 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 Gamma functions expressed by the equations
B(l,m)= 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 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 " 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 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 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.) 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(rota,r—cotbr), (o ),
(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 ax2cosbxdxz7r(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 article FUNCTION. In the case of two variables x, y we Thieves. integrate a function f(x,y) over an area; in the case of
three variables x, y, z we integrate a function f(x, y, z) through a 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 process is usually effected by summing first for all the rectangles which lie in a strip between two lines parallel to one axis, say the axis of y, and afterwards for all the strips. This process is 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 identification of a 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 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. dawn the tang enn ants These tangent planes form a 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 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 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 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 expressed either as
zf(xdyydx),
or as
ffdxdy,
the former integral being a line integral taken 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 hand.
53a. We have two theorems of transformation connect Theorems ing volume integrals with surface integrals and surface of Green integrals with line integrals. The first theorem, called and " Green's theorem," is expressed by the equation 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 translation in a righthanded screw. This 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 righthanded system, we have Stokes's theorem in the form 11
J .(ud'+vdy +wdz M l(aya)+meazaz) \ax ay)I
where the integral on the left is taken round the curve s in the chosen sense. When the axes are lefthanded, we may either reverse the sense of 1, m, n and maintain the formula, or retain the sense of 1, m, n and change the sign of the righthand 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 simplyconnected regions of space. The correction for multiplyconnected regions is important in several 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 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:
, tanlbla 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 lineintegral 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 condition may integrals require the replacing of the original integral by the sum of two or more simple integrals.
(ii.) The line integral of a perfect differential of a onevalued 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 understood 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 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 threedimensional 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+ *deck).
The surface integral of a function of (0, 0) over the surface of a sphere r=const. can be expressed in the form
fad$ f o F (8, ¢) r2 sin OdO.
In every case the domain of integration must be chosen so as to include the whole surface.
(ix.) In threedimensional 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 ellipsoid x2/a2+y2/b2+z2/c2 = are often effected by means of a general theorem due to 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 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 early. Kepler's practical measurement of the focal sectors Approxlof ellipses (1609) was an approximate integration, as also mate and was the method for the quadrature of the hyperbola given
by James Gregory in the appendix to his Exercitaliones a~ algeometricae (1668). In Newton's Methodus differentialis cal la' (1711) the subject was taken up systematically. Newton's
object was to effect the approximate quadrature of a given curve by making a curve of the type
y =ao+aix+a2x2+ ... +a,xn
pass through the vertices of (n+I) equidistant ordinates of the given curve, and by taking the area of the new curve so determined as an approximation to the area of the given curve. In 1743 Thomas Simpson in his Mathematical Dissertations published a very convenient rule, obtained by taking the vertices of three consecutive equidistant ordinates to be points on the same parabola. The distance between the extreme ordinates corresponding to the abscissae x=a and x =b is divided into 2n equal segments by ordinates yi, ye, ... yeni, and the extreme ordinates are denoted by yo, Yen. The vertices of the ordinates yo, yi, Y2 lie on a parabola with its axis parallel to the axis of y, so do the vertices of the ordinates y2, y3, y4, and so on. The area is expressed approximately by the formula
{(ba)/6n}[yo+yen+2 (y2 +Y4+ ... +Y2n2)+4(yi+Y3+ • • • +yini)], which is known as Simpson's rule. Since all simple integrals can be represented as areas such rules are applicable to approximate integration in general. For the recent developments reference may be made to the article by A. Voss in Ency. d. Math. bliss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors, Valeta approximative dune integrate definie (Paris, 1905).
Many instruments have been devised for registering mechanically the areas of closed curves and the values of integrals. The best known are perhaps the " planimeter " of J. Amsler (1854) and the " integraph " of AbdankAbakanowicz (1882).
and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral
ffJ1dEdn,
where J denotes the above Jacobian, and the integration is taken through a suitable domain. When the boundary consists of portions of curves for which r: =const., or n =const., the above is generally the simplest way of evaluating it.
(vi.) The problem of " rectifying " a plane curve, or finding its length, is solved by evaluating the integral
f 1 + (a) 2 } Zdx,
or, in polar coordinates, by evaluating the integral
f
J r2 + (d~) 2 } 'do.
In both cases the integrals are line integrals taken along the curve.
(vii.) When we use curvilinear coordinates , n as in (v.) above, the length of any portion of a curve = const. is given by the integral
fJ'idn
taken between appropriate limits for n. There is a similar formula for the arc of a curve n=const.
(viii.) The area of a surface z=f(x, y) can be expressed by the formula
f J l 1+ (az) 2+ (ay) 2 Zdxdy.
When the coordinates of the points of a surface are expressed as
is
are cited in the course of the article. A list of some of the more important treatises on the differential and integral calculus is appended. The list has no pretensions to completeness; in particular, most of the recent books in which the subject is presented in an elementary way for beginners or engineers are omitted.—L. Euler, Institutiones calculi differentialis (Petrop., 1755) and Institutiones calculi integralis (3 Bde., Petrop., 1768–1770) ; J. L. Lagrange, Lecons sur le calcul des fonctions (Paris, 18o6, Euvres, t. x.), and, Theorie des fonctions analytiques (Paris, 1797, 2nd ed., 1813, fEuvres, t. ix.); S. F. Lacroix, Traite de calcul diff. et de calcul int. (3 tt., Paris, 1808–1819). There have been numerous later editions; a translation by Herschel, Peacock and Babbage of an abbreviated edition of Lacroix's treatise was published at Cambridge in 1816. G. Peacock, Examples of the Differential and Integral Calculus (Cambridge, 1820) ; A. L. Cauchy, Resume des lecons . . sur le calcul infinitesimale (Paris, 1823), and Lecons sur le calcul differentiel (Paris, 1829; fEuvres, set.. 2, t. iv.); F. Minding, Handbuchd. Diff.u. Int.Rechnung (Berlin, 1836) ; F. Moigno, Lecons sur le calcul cliff. (4 tt., Paris, 184o–1861) ; A. de Morgan, Diff. and Int. Cale. (London, 1842) ; D. Gregory, Examples on: the Diff. and Int. Cale. (2 vols., Cambridge, 1841–1846); I. Todhunter, Treatise on the Diff. Cale. and Treatise on the Int. Cale. (London, 1852), numerous later editions; B. Price, Treatise on the Infinitesimal Calculus (2 vols., Oxford, 1854), numerous later editions; D. Bierens de Haan, Tables d'integrales difinies (Amsterdam, 1858) ; M. Stegemann, Grundriss d. Diff. u. Int.Rechnung (2 Bde., Hanover, 1862) numerous later editions; J. Bertrand, Traite de calc. diff. et int. (2 tt., Paris, 1864–187o) ; J. A. Serret, Cours de calc. diff. et int. (2 tt., Paris, 1868, 2nd ed., 188o, German edition by Harnack, Leipzig, 1884–1886, later German editions by Bohlmann, 1896, and Scheffers, 1906,1 incomplete) ; B. Williamson, Treatise on the Diff. Cale. (Dublin, 1872), and Treatise on the Int. Cale. (Dublin, 1874) numerous later editions of both; also the article " Infinitesimal Calculus " in the 9th ed. of the Ency. Brit. ; C. Hermite, Cours d'analyse (Paris, 1873) ; O. Schlomilch, Compendium d. hoheren Analysis (2 Bde., Leipzig, 1874) numerous later editions; J. Thomae, Einleitung in d. Theorie d. bestimmten Integrale (Halle, 1875) ; R. Lipschitz, Lehrbuch d. Analysis (2 Bde., Bonn, 1877, 188o); A. Harnack, Elemente d. Diff.u. Int.Rechnung (Leipzig, 1882, Eng. trans. by Cathcart, London, 1891); M. Pasch, Einleitung in d. Diff.u. Int.Rechnung (Leipzig, 1882) ; Genocchi and Peano, Calcolo differenziale (Turin, 1884, German edition by Bohlmann and Schepp, Leipzig, 1898, 1899); H. Laurent, Traite d'analyse (7 tt., Paris, 1885–1891) ; J. Edwards, Elementary Treatise on the Diff. Cale. (London, 1886), several later editions; A. G. Greenhill, Diff. and Int. Cale. (London, 1886, 2nd ed., 1891); E. Picard, Traite d'analyse (3 tt., Paris, 1891–1896); O. Stolz, Grundziege d. Diff. u. Int.Rechnung (3 Bde., Leipzig, 1893–1899) ; C. Jordan, Cours d'analyse (3 tt., Paris, 1893–1896) ; L. Kronecker, Vorlesungen ii. d. Theorie d. einfachen u. vielfachen Integrate (Leipzig, 1894) ; J. Perry, The Calculus for Engineers (London, 1897) ; H. Lamb, An Elementary Course of Infinitesimal Calculus (Cambridge, 1897) ; G. A. Gibson, An Elementary Treatise on the Calculus (London, 1901) ; E. Goursat, Cours d'analyse mathematique (2 tt., Paris, 1902–1905) ; C.J. de la Vallee Poussin, Cours d'analyse infenitesimale (2 tt., Louvain and Paris, 1903–1906) ; A. E. H. Love, Elements of the Diff. and Int. Cale. (Cambridge, 1909) ; W. H. Young, The Fundamental Theorems of the Diff. Cale. (Cambridge, 1910). A resume of the infinitesimal calculus is given in the articles "Diff.u. IntRechnung " by A. Voss, and " Bestimmte Integrale " by G. Brunel in Ency. d. math. Wiss. (Bde. ii. A. 2, and ii. A. 3, Leipzig, 1899, 1900). Many questions of principle are discussed exhaustively by E. W. Hobson, The Theory of Functions of a Real Variable (Cambridge,
1907). (A. E. H. L.)
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