(i.)This theorem is not true without limitation. The conditions for its validity have been investigated very completely by H. A. Schwarz (see hisGes. math. Abhandlungen, Bd. 2, Berlin, 1890, p. 275). It is a sufficient, though not a necessary, condition that all the differential coefficients concerned should be continuous functions of x, y. In consequence of the relation (i.) the differential coefficients expressed in the two members of this relation are written∂2ƒor∂2ƒ.∂x∂y∂y∂xThe differential coefficient∂nƒ,∂xp∂yq∂zrin which p + g + r = n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms asƒxpyqzror ƒ(p, q, r).x, y, zDifferentialsof higher orders are introduced by the defining equationdnƒ =(dx∂+ dy∂)nƒ∂x∂y= (dx)n∂nƒ+ n(dx)n−1dy∂nƒ+ ...∂xn∂xn−1∂yin which the expression (dx·∂/∂x + dy·∂/∂y)nis developed by the binomial theorem in the same way as if dx·∂/∂x and dy·∂/∂y were numbers, and (∂/∂x)r·(∂/∂y)n−rƒ is replaced by ∂nƒ/∂xr∂yn−r. When there are more than two variables the multinomial theorem must be used instead of the binomial theorem.The problem of forming the second and higher differential coefficients ofimplicit functionscan be solved at once by means of partial differential coefficients, for example, if ƒ(x, y) = 0 is the equation defining y as a function of x, we haved2y=(∂ƒ)−3{ (∂ƒ)2∂2ƒ− 2∂ƒ·∂ƒ·∂2ƒ+(∂ƒ)2∂2ƒ}.dx2∂y∂y∂x2∂x∂y∂x∂y∂x∂y2The differential expression Xdx + Ydy, in which both X and Y are functions of the two variables x and y, is atotal differentialif there exists a function ƒ of x and y which is such that∂ƒ/∂x = X, ∂ƒ/∂y = Y.When this is the case we have the relation∂Y/∂x = ∂X/∂y.(ii.)Conversely, when this equation is satisfied there exists a function ƒ which is such thatdƒ= Xdx + Ydy.The expression Xdx + Ydy in which X and Y are connected by the relation (ii.) is often described as a “perfect differential.” The theory of the perfect differential can be extended to functions of n variables, and in this case there are ½n(n − 1) such relations as (ii.).In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we writep, q, r, s, t for∂z,∂z,∂2z,∂2z,∂2z.∂x∂y∂x2∂x∂y∂y2Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces. When a surface is given by an equation of the form z = ƒ(x, y), the lines of curvature are determined by the equation{ (l + q2)s − pqt} (dy)2+ { (1 + q2)r − (1 + p2)t } dxdy − { (1 + p2)s − pqr} (dx)2= 0,and the principal radii of curvature are the values of R which satisfy the equationR2(rt − s2) − R { (1 + q2)r − 2pqs + (1 + p2)t } √(1 + p2+ q2) + (1 + p2+ q2)2= 0.44.Change of variables.The problem of change of variables was first considered by Brook Taylor in hisMethodus incrementorum. In the case considered by Taylor y is expressed as a function of z, and z as a function of x, and it is desired to express the differential coefficients of y with respect to x without eliminating z. The result can be obtained at once by the rules for differentiating a product and a function of a function. We havedy=dy·dz,dxdzdxd2y=dy·d2z+d2y·(dz)2,dx2dzdx2dz2dxd3y=dy·d3z+ 3d2y·dz·d2z+d3y·(dz)3,dx3dzdx3dz2dxdx2dz3dx. . . . . . . . . . . . . . . . . . . . .The introduction of partial differential coefficients enables us to deal with more general cases of change of variables than that considered above. If u, v are new variables, and x, y are connected with them by equations of the typex = ƒ1(u, v), y = ƒ2(u, v),(i.)while y is either an explicit or an implicit function of x, we have the problem of expressing the differential coefficients of various orders of y with respect to x in terms of the differential coefficients of v with respect to u. We havedy=(∂ƒ2+∂ƒ2dv) / (∂ƒ1+∂ƒ1dv)dx∂u∂vdu∂u∂vduby the rule of the total differential. In the same way, by means of differentials of higher orders, we may express d2y/dx2, and so on.Equations such as (i.) may be interpreted as effecting atransformationby which a point (u, v) is made to correspond to a point (x, y). The whole theory of transformations, and of functions, or differential expressions, which remain invariant under groups of transformations, has been studied exhaustively by Sophus Lie (see, in particular, hisTheorie der Transformationsgruppen, Leipzig, 1888-1893). (See alsoDifferential EquationsandGroups).A more general problem of change of variables is presented when it is desired to express the partial differential coefficients of a function V with respect to x, y, ... in terms of those with respect to u, v, ..., where u, v, ... are connected with x, y, ... by any functional relations. When there are two variables x, y, and u, v are given functions of x, y, we have∂V=∂V∂u+∂V∂v,∂x∂u∂x∂v∂x∂V=∂V∂u+∂V∂v,∂y∂u∂y∂v∂yand the differential coefficients of higher orders are to be formed by repeated applications of the rule for differentiating a product and the rules of the type∂=∂u∂+∂v∂.∂x∂x∂u∂x∂vWhen x, y are given functions of u, v, ... we have, instead of the above, such equations as∂V=∂V∂x+∂V∂y;∂u∂x∂u∂y∂uand ∂V/∂x, ∂V/∂y can be found by solving these equations, provided the Jacobian ∂(x, y)/∂(u, v) is not zero. The generalization of this method for the case of more than two variables need not detain us.In cases like that here considered it is sometimes more convenient not to regard the equations connecting x, y with u, v as effecting a point transformation, but to consider the loci u = const., v = const. as two “families” of curves. Then in any region of the plane of (x, y) in which the Jacobian ∂(x, y)/∂(u, v) does not vanish or become infinite, any point (x, y) is uniquely determined by the values of u and v which belong to the curves of the two families that pass through the point. Such variables as u, v are then described as “curvilinear coordinates” of the point. This method is applicable to any number of variables. When the loci u = const., ... intersect each other at right angles, the variables are “orthogonal” curvilinear coordinates. Three-dimensional systems of such coordinates have important applications in mathematical physics. Reference may be made to G. Lamé,Leçons sur les coordonnées curvilignes(Paris, 1859), and to G. Darboux,Leçons sur les coordonnées curvilignes et systèmes orthogonaux(Paris, 1898).When such a coordinate as u is connected with x and y by a functional relation of the form ƒ(x, y, u) = 0 the curves u = const. are a family of curves, and this family may be such that no two curves of the family have a common point. When this is not the case the points in which a curve ƒ(x, y, u) = 0 is intersected by a curve ƒ(x, y, u + Δu) = 0 tend to limiting positions as Δu is diminished indefinitely. The locus of these limiting positions is the “envelope” of the family, and in general it touches all the curves of the family. It is easy to see that, if u, v are the parameters of two families of curves which have envelopes, the Jacobian ∂(x, y)/∂(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ∂(u, v)/∂(x, y) vanishes, a curve of the family u touches a curve of the family v.If three variables x, y, z are connected by a functional relation ƒ(x, y, z) = 0, one of them, z say, may be regarded as animplicit functionof the other two, and the partial differential coefficients of z with respect to x and y can be formed by the rule of the total differential. We have∂z= −∂ƒ/∂ƒ,∂z= −∂ƒ/∂ƒ;∂x∂x∂z∂y∂y∂zand there is no difficulty in proceeding to express the higher differential coefficients. There arises the problem of expressing the partial differential coefficients of x with respect to y and z in terms of those of z with respect to x and y. The problem is known as that of “changing the dependent variable.” It is solved by applying the rule of the total differential. Similar considerations are applicable to all cases in which n variables are connected by fewer than n equations.45.Extension of Taylor’s theorem.Taylor’s theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the formƒ(a + h, b + k) = ƒ(a, b) + dƒ(a, b) +1d2ƒ(a, b) + ...2!+1dn−1ƒ(a, b) +1dnƒ(a + θh, b + θk),(n − 1)!n!in whichdrƒ(a, b) =[ (h∂+ k∂)rƒ(x, y)],∂x∂yx=a, y=banddnƒ(a + θh, b + θk) =[ (h∂+ k∂)nƒ(x, y)].∂x∂yx=a+θh, y=b+θkThe last expression is the remainder after n terms, and in it θ denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (seeMaximaandMinima).46.Plane curves.In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.(i.) If ψ 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, θ and if p denotes the perpendicular from the origin to the tangent, thencos ψ = dr/ds, sin ψ = rdθ/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 thepedalof the curve with respect to the origin. The angle ψ for the pedal is the same as the angle ψ 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 φ denotes the angle which the tangent at any point makes with a fixed line, we haver2= p2+ (dp/dφ)2.(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotientΔφΔswhere the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ 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 bydφdsSometimes 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 dφ 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 angle.(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” ρ is the radius of this circle. We have 1/ρ = |dφ/ds|. 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:—1=| {1 +(dy)2}−3/2d2y|,ρdxdx2ρ =|rdr|=|p +d2p|.dpdφ2(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 ƒ(x) = 0, the function ƒ(x) has a factor (x − x0)3, where x0is the abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x − x0) occurs (n + 1) times in ƒ(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 u0+ u1+ u2+ ... + un= 0, where u0is a constant, and uris a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, u0vanishes, and u1= 0 represents the tangent at the origin. If u1also vanishes, the origin is a double point and u2= o represents the tangents at the origin. If u2has distinct factors, or is of the form a(y − p1x) (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 eitherp1x + ½ q1x2+ ..., or p2x + ½ q2x2+ ...,where q1, ... and q2, ... are determined without ambiguity. If p1and p2are real the two branches have radii of curvature ρ1, ρ2determined by the formulae1=|(1 + p12)−3/2q1|,1=|(1 + p22)−3/2q2|.ρ1ρ2When p1and p2are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is anisolated point. If u2is a square, a(y − px)2, the origin is acusp, 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 u0+ u1+ ... + un−1+ un= 0 where the notation is the same as that in (ix.), the factors of undetermine the directions of theasymptotes. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un= L1L2... Ln, where L1, ... are linear in x, y, we may resolve un−1/uninto partial fractions according to the formulaun−1=A1+A2+ ... +An,unL1L2Lnand then L1+ A1= 0, L2+ A2= 0, ... are the equations of the asymptotes. When a real factor of unis 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) = a4, where x = 0 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 ofIntegral calculus.classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, 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 inFunction. 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.Methods of integration.The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.From the equation d(uv) = u dv + v du we deduce the equation∫udvdx = uv −∫vdudx,dxdxor, as it may be written∫uw dx = u∫w dx −∫du{ ∫w dx}dx.dxThis is the rule of “integration by parts.”As an example we have∫xeaxdx = xeax−∫eaxdx =(x−1)eax.aaaa2When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equationdx =dxdzdzwe deduce the equation∫ƒ(x) dx =∫φ(z)dxdz.dzAs an example, in the integral∫ √(1 − x2)dxput x = sin z; the integral becomes∫ cos z · cos z dz = ∫ ½ (1 + cos 2z)dz = ½ (z + ½ sin 2z) = ½ (z + sin z cos z).49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraicIntegration in terms of elementary functions.functions, implicit algebraic functions, exponentials and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of ex; (v.) all rational integral functions of the variables x, eax, ebx, ... 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 √(a2− x2), √(x2− a2), √(x2+ a2). The substitutions x = a sin θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan ½θ = 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 (ax4+ bx3+ cx2+ dx + e)−1/2the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (seeFunction).Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books 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.)∫ (x2+ a) − ½ dx = log {x + (x2+ a)1/2}.(ii.)∫dx(x − p) √(ax2+ 2bx + c)can be evaluated by the substitution x − p = 1/z, and∫dx(x − p)n√(ax2+ 2bx + c)can be deduced by differentiating (n − 1) times with respect to p.(iii.)∫(Hx + K) dx(αx2+ 2βx + γ) √(ax2+ 2bx + c)can be reduced by the substitution y2= (ax2+ 2bx + c)/(αx2+ 2βx + γ) to the formA∫dy+ B∫dy√(λ1− y2)√(y2− λ2)where A and B are constants, and λ1and λ2are the two values of λ for which (a − λα)x2+ 2(b − λβ)x + c − λγ is a perfect square (see A. G. Greenhill,A Chapter in the Integral Calculus, London, 1888).(iv.) ƒxm(axn+ b)pdx, in which m, n, p are rational, can be reduced, by putting axn= bt, to depend upon ƒtq(1 + t)pdt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1 + t = us. If p + q is an integer and p = r/s we put 1 + t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).(v.)∫dx= log tanx,sin x2(vi.)∫dx= log (tan x + sec x).cos x(vii.) ∫ eaxsin (bx + α) dx = (a2+ b2)−1eax{a sin (bx + α) − b cos (bx + α) }.(viii.) ∫ sinmx cosnx dx can be reduced by differentiating a function of the form sinpx cosqx;e.g.dsin x=1+q sin2x=1 − q+q.dxcosqxcosq−1xcosq+1xcosq−1xcosq+1xHence∫dx=sin x+n − 2∫dx.cosnx(n − 1) cosn−1xn − 1cosn−2x(ix.)∫1/2π0sin2nx dx =∫1/2π0cos2nx dx =1·3 ... (2n − 1)·π, (n an integer).2·4 ... 2n2(x.)∫1/2π0sin2n+1x dx =∫1/2π0cos2n+1x dx =2·4 ... 2n, (n an integer).3·5 ... (2n + 1)(xi.)∫dxcan be reduced by one of the substitutions(1 + e cos x)ncos φ =e + cos x, cosh u =e + cos x,1 + e cos x1 + e cos xof which the first or the second is to be employed according as e < or > 1.50.New transcendents.Among the integrals of transcendental functions which lead to new transcendental functions we may notice∫x0dx, or∫log x−∞ezdz,log xzcalled the “logarithmic integral,” and denoted by “Li x,” also the integrals∫x0sin xdx and∫x∞cos xdx,xxcalled the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral∫x0e−x2dxcalled the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (seeTables, Mathematical).51.Eulerian integrals.New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equationsB(l, m) =∫10xl−1(1 − x)m−1dx,Γ(n) =∫∞0e−ttn−1dt.When n is a positive integer Γ(n + 1) = 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 formulaB(l, m) · Γ(l + m) = Γ(l) · Γ(m).The Gamma function satisfies the difference equationΓ(x + 1) = x Γ(x),and also the equationΓ(x) · Γ(1 − x) = π/sin (xπ),with the particular resultΓ(½)= √π.The number−[d{ log Γ (1 + x) }]x=0, or −Γ′(1),dxis called “Euler’s constant,” and is equal to the limitlim.n=∞[ (1 + ½ +1⁄3+ ... +1)− log n];nits value to 15 decimal places is 0.577 215 664 901 532.The function log Γ(1 + x) can be expanded in the serieslog Γ (1 + x) = ½ log(xπ)− ½ log1 + x+ { 1 + Γ′;(1) } xsin xπ1 − x−1⁄3(S3− 1) x3−1⁄5(S5− 1) x5− ...,whereS2r+1= 1 +1+1+ ...,22r+132r+1and the series for log Γ(1 + x) converges when x lies between −1 and 1.52.Definite integrals.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∫10(1 − x2)−1/2log x dx = −½ π log 2,although the indefinite integral of (1 − x2)−1/2log 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.(ii.)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 ofintegration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (seeFunction). The method of contour integration involves the introduction of complex variables (seeFunction: §Complex Variables).A few results are added(i.)∫∞0xa−1dx =π, (1 > a > 0),1 + xsin aπ(ii.)∫∞0xa−1− xb−1dx = π (cot aπ − cot bπ), (0 < a or b < 1),1 − x(iii.)∫∞0xa−1log xdx =π2, (a > 1),x − 1sin2aπ(iv.)∫∞0x2· cos 2x · e−x2dx = −1⁄4e−1√π,(v.)∫101 − x2dx= log tanπ,1 + x4log x8(vi.)∫∞0sin mxdx = ½(1−1+1),e2πx− 1emm2(vii.)∫π0log (1 − 2α cos x + α2) dx = 0 or 2π log α according as α < or > 1,(viii.)∫∞0sin xdx = ½ π,x(ix.)∫∞0cos axdx = ½ πb−1e−ab,x2+ b2(x.)∫∞0cos ax − cos bxdx = ½ π (b − a),x2(xi.)∫∞0cos ax − cos bxdx = logb,xa(xii.)∫∞0cos x − e−mxdx = log m,x(xiii.)∫∞−∞e−x2+2axdx = √π · ea2,(xiv.)∫∞0x−1/2sin x dx =∫∞0x−1/2cos x dx = √(½ π),53.Multiple Integrals.The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the articleFunction. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z) through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by∫∫ ƒ(x, y) dx dy.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 of ƒ 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 ƒ(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∫badx{ ∫ƒ2 (x)ƒ(x, y)dy},ƒ1 (x)and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(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∫∫∫ ƒ(x, y, z) dx dy dz.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 first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, andSurface Integrals.draw the tangent planes at all these points. 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 some arbitrarily 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 ƒ 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∫∫∫ ƒ dS.TheLine Integrals.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∫ ƒ dswhere 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∫ (udx+ vdy+ wdz)ds or ∫s(u dx + v dy + w dz).dsdsdsIn like manner surface integrals usually present themselves in the form∫∫ (lξ + mη + nζ) dSwhere l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.The area of a bounded portion of the plane of (x, y) may be expressed either as½ ∫ (x dy − y dx),or as∫∫ dx dy,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.Theorems of Green and Stokes.We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation∫∫∫ (∂ξ+∂η+∂ζ)dx dy dz = ∫∫ (lξ + mη + nζ) dS,∂x∂y∂zwhere 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 l, 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.,∫∫ (∂ξ+∂η)dx dy =∫ (ξdy− ηdx)ds,∂x∂ydsdsthe 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 right-handed 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 right-handed system, we have Stokes’s theorem in the form∫s(u dx + v dy + w dz) =∫∫ {l(∂w−∂v)+ m(∂u−∂w)+ n(∂v−∂u) }dS,∂y∂z∂z∂x∂x∂ywhere the integral on the left is taken round the curve s in the chosen sense. When the axes are left-handed, we may either reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and 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 ∂ξ/∂x, ∂η/∂y, ∂ζ/∂z 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 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. ItChange of Variables in a Multiple Integral.is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the second set of variables. It is necessary in the second place to determine the limits of integration which must be employed 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 x1, x2, ..., xn, and those of the other set by u1, u2, ..., un, we have the relationdx1dx2... dxn=∂ (x1, x2, ..., xn)du1du2... dun.∂ (u1, u2, ..., un)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 ƒ(x, y) over the area within a circle given by x2+ y2= a2, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have∫a−adx∫√(a2−x2)ƒ(x, y) dy =∫a0dr∫2π0ƒ(r cos θ, r sin θ) r dθ.−√(a2−x2)If we have to integrate over the area of a rectangle a ≥ x ≥ 0, b ≥ y ≥ 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—∫a0dx∫b0ƒ(x, y) dy =∫0tan−1b/adθ∫0a sec θƒ(r cos θ, r sin θ) r dr+∫1/2πtan−1b/adθ∫0b cosec θƒ(r cos θ, r sin θ) r dr.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 along a portion of the axis of x. When a change ofLine Integrals and Multiple Integrals.variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may 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 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∫ ½ r2dθ or ∫ ½ p ds,where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be 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 formulaπ ∫ y2dx,and the area of the surface is expressed by the formula2π ∫ y ds,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.
(i.)
This theorem is not true without limitation. The conditions for its validity have been investigated very completely by H. A. Schwarz (see hisGes. math. Abhandlungen, Bd. 2, Berlin, 1890, p. 275). It is a sufficient, though not a necessary, condition that all the differential coefficients concerned should be continuous functions of x, y. In consequence of the relation (i.) the differential coefficients expressed in the two members of this relation are written
The differential coefficient
in which p + g + r = n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms as
Differentialsof higher orders are introduced by the defining equation
in which the expression (dx·∂/∂x + dy·∂/∂y)nis developed by the binomial theorem in the same way as if dx·∂/∂x and dy·∂/∂y were numbers, and (∂/∂x)r·(∂/∂y)n−rƒ is replaced by ∂nƒ/∂xr∂yn−r. When there are more than two variables the multinomial theorem must be used instead of the binomial theorem.
The problem of forming the second and higher differential coefficients ofimplicit functionscan be solved at once by means of partial differential coefficients, for example, if ƒ(x, y) = 0 is the equation defining y as a function of x, we have
The differential expression Xdx + Ydy, in which both X and Y are functions of the two variables x and y, is atotal differentialif there exists a function ƒ of x and y which is such that
∂ƒ/∂x = X, ∂ƒ/∂y = Y.
When this is the case we have the relation
∂Y/∂x = ∂X/∂y.
(ii.)
Conversely, when this equation is satisfied there exists a function ƒ which is such that
dƒ= Xdx + Ydy.
The expression Xdx + Ydy in which X and Y are connected by the relation (ii.) is often described as a “perfect differential.” The theory of the perfect differential can be extended to functions of n variables, and in this case there are ½n(n − 1) such relations as (ii.).
In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we write
Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces. When a surface is given by an equation of the form z = ƒ(x, y), the lines of curvature are determined by the equation
{ (l + q2)s − pqt} (dy)2+ { (1 + q2)r − (1 + p2)t } dxdy − { (1 + p2)s − pqr} (dx)2= 0,
and the principal radii of curvature are the values of R which satisfy the equation
R2(rt − s2) − R { (1 + q2)r − 2pqs + (1 + p2)t } √(1 + p2+ q2) + (1 + p2+ q2)2= 0.
44.Change of variables.The problem of change of variables was first considered by Brook Taylor in hisMethodus incrementorum. In the case considered by Taylor y is expressed as a function of z, and z as a function of x, and it is desired to express the differential coefficients of y with respect to x without eliminating z. The result can be obtained at once by the rules for differentiating a product and a function of a function. We have
The introduction of partial differential coefficients enables us to deal with more general cases of change of variables than that considered above. If u, v are new variables, and x, y are connected with them by equations of the type
x = ƒ1(u, v), y = ƒ2(u, v),
(i.)
while y is either an explicit or an implicit function of x, we have the problem of expressing the differential coefficients of various orders of y with respect to x in terms of the differential coefficients of v with respect to u. We have
by the rule of the total differential. In the same way, by means of differentials of higher orders, we may express d2y/dx2, and so on.
Equations such as (i.) may be interpreted as effecting atransformationby which a point (u, v) is made to correspond to a point (x, y). The whole theory of transformations, and of functions, or differential expressions, which remain invariant under groups of transformations, has been studied exhaustively by Sophus Lie (see, in particular, hisTheorie der Transformationsgruppen, Leipzig, 1888-1893). (See alsoDifferential EquationsandGroups).
A more general problem of change of variables is presented when it is desired to express the partial differential coefficients of a function V with respect to x, y, ... in terms of those with respect to u, v, ..., where u, v, ... are connected with x, y, ... by any functional relations. When there are two variables x, y, and u, v are given functions of x, y, we have
and the differential coefficients of higher orders are to be formed by repeated applications of the rule for differentiating a product and the rules of the type
When x, y are given functions of u, v, ... we have, instead of the above, such equations as
and ∂V/∂x, ∂V/∂y can be found by solving these equations, provided the Jacobian ∂(x, y)/∂(u, v) is not zero. The generalization of this method for the case of more than two variables need not detain us.
In cases like that here considered it is sometimes more convenient not to regard the equations connecting x, y with u, v as effecting a point transformation, but to consider the loci u = const., v = const. as two “families” of curves. Then in any region of the plane of (x, y) in which the Jacobian ∂(x, y)/∂(u, v) does not vanish or become infinite, any point (x, y) is uniquely determined by the values of u and v which belong to the curves of the two families that pass through the point. Such variables as u, v are then described as “curvilinear coordinates” of the point. This method is applicable to any number of variables. When the loci u = const., ... intersect each other at right angles, the variables are “orthogonal” curvilinear coordinates. Three-dimensional systems of such coordinates have important applications in mathematical physics. Reference may be made to G. Lamé,Leçons sur les coordonnées curvilignes(Paris, 1859), and to G. Darboux,Leçons sur les coordonnées curvilignes et systèmes orthogonaux(Paris, 1898).
When such a coordinate as u is connected with x and y by a functional relation of the form ƒ(x, y, u) = 0 the curves u = const. are a family of curves, and this family may be such that no two curves of the family have a common point. When this is not the case the points in which a curve ƒ(x, y, u) = 0 is intersected by a curve ƒ(x, y, u + Δu) = 0 tend to limiting positions as Δu is diminished indefinitely. The locus of these limiting positions is the “envelope” of the family, and in general it touches all the curves of the family. It is easy to see that, if u, v are the parameters of two families of curves which have envelopes, the Jacobian ∂(x, y)/∂(u, v) vanishes at all points on these envelopes. It is easy to see also that at any point where the reciprocal Jacobian ∂(u, v)/∂(x, y) vanishes, a curve of the family u touches a curve of the family v.
If three variables x, y, z are connected by a functional relation ƒ(x, y, z) = 0, one of them, z say, may be regarded as animplicit functionof the other two, and the partial differential coefficients of z with respect to x and y can be formed by the rule of the total differential. We have
and there is no difficulty in proceeding to express the higher differential coefficients. There arises the problem of expressing the partial differential coefficients of x with respect to y and z in terms of those of z with respect to x and y. The problem is known as that of “changing the dependent variable.” It is solved by applying the rule of the total differential. Similar considerations are applicable to all cases in which n variables are connected by fewer than n equations.
45.Extension of Taylor’s theorem.Taylor’s theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the form
in which
and
The last expression is the remainder after n terms, and in it θ denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (seeMaximaandMinima).
46.Plane curves.In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.
(i.) If ψ 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, θ and if p denotes the perpendicular from the origin to the tangent, then
cos ψ = dr/ds, sin ψ = rdθ/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 thepedalof the curve with respect to the origin. The angle ψ for the pedal is the same as the angle ψ 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 φ denotes the angle which the tangent at any point makes with a fixed line, we have
r2= p2+ (dp/dφ)2.
(iv.) The “average curvature” of the arc Δs of a curve between two points is measured by the quotient
where the upright lines denote, as usual, that the absolute value of the included expression is to be taken, and φ is the angle which the tangent makes with a fixed line, so that Δφ 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
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 dφ 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 angle.
(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” ρ is the radius of this circle. We have 1/ρ = |dφ/ds|. 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:—
(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 ƒ(x) = 0, the function ƒ(x) has a factor (x − x0)3, where x0is the abscissa of the point of inflection P, and the line is the tangent at P. When the factor (x − x0) occurs (n + 1) times in ƒ(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 u0+ u1+ u2+ ... + un= 0, where u0is a constant, and uris a homogeneous rational integral function of x, y of the rth degree. When the origin is on the curve, u0vanishes, and u1= 0 represents the tangent at the origin. If u1also vanishes, the origin is a double point and u2= o represents the tangents at the origin. If u2has distinct factors, or is of the form a(y − p1x) (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
p1x + ½ q1x2+ ..., or p2x + ½ q2x2+ ...,
where q1, ... and q2, ... are determined without ambiguity. If p1and p2are real the two branches have radii of curvature ρ1, ρ2determined by the formulae
When p1and p2are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is anisolated point. If u2is a square, a(y − px)2, the origin is acusp, 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 u0+ u1+ ... + un−1+ un= 0 where the notation is the same as that in (ix.), the factors of undetermine the directions of theasymptotes. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un= L1L2... Ln, where L1, ... are linear in x, y, we may resolve un−1/uninto partial fractions according to the formula
and then L1+ A1= 0, L2+ A2= 0, ... are the equations of the asymptotes. When a real factor of unis 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) = a4, where x = 0 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 ofIntegral calculus.classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, 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 inFunction. 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.Methods of integration.The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.
From the equation d(uv) = u dv + v du we deduce the equation
or, as it may be written
This is the rule of “integration by parts.”
As an example we have
When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equation
we deduce the equation
As an example, in the integral
∫ √(1 − x2)dx
put x = sin z; the integral becomes
∫ cos z · cos z dz = ∫ ½ (1 + cos 2z)dz = ½ (z + ½ sin 2z) = ½ (z + sin z cos z).
49. The indefinite integrals of certain classes of functions can be expressed by means of a finite number of operations of addition or multiplication in terms of the so-called “elementary” functions. The elementary functions are rational algebraicIntegration in terms of elementary functions.functions, implicit algebraic functions, exponentials and logarithms, trigonometrical and inverse circular functions. The following are among the classes of functions whose integrals involve the elementary functions only: (i.) all rational functions; (ii.) all irrational functions of the form ƒ(x, y), where ƒ denotes a rational algebraic function of x and y, and y is connected with x by an algebraic equation of the second degree; (iii.) all rational functions of sin x and cos x; (iv.) all rational functions of ex; (v.) all rational integral functions of the variables x, eax, ebx, ... 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 √(a2− x2), √(x2− a2), √(x2+ a2). The substitutions x = a sin θ, x = a sec θ, x = a tan θ are then effective in the three cases. By these substitutions the subject of integration becomes a rational function of sin θ and cos θ, and it can be reduced to a rational function of t by the substitution tan ½θ = 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 (ax4+ bx3+ cx2+ dx + e)−1/2the integral cannot be expressed explicitly in terms of elementary functions. Such integrals lead to new functions (seeFunction).
Methods of reduction and substitution for the evaluation of indefinite integrals occupy a considerable space in text-books 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.)
∫ (x2+ a) − ½ dx = log {x + (x2+ a)1/2}.
(ii.)
can be evaluated by the substitution x − p = 1/z, and
can be deduced by differentiating (n − 1) times with respect to p.
(iii.)
can be reduced by the substitution y2= (ax2+ 2bx + c)/(αx2+ 2βx + γ) to the form
where A and B are constants, and λ1and λ2are the two values of λ for which (a − λα)x2+ 2(b − λβ)x + c − λγ is a perfect square (see A. G. Greenhill,A Chapter in the Integral Calculus, London, 1888).
(iv.) ƒxm(axn+ b)pdx, in which m, n, p are rational, can be reduced, by putting axn= bt, to depend upon ƒtq(1 + t)pdt. If p is an integer and q a fraction r/s, we put t = us. If q is an integer and p = r/s we put 1 + t = us. If p + q is an integer and p = r/s we put 1 + t = tus. These integrals, called “binomial integrals,” were investigated by Newton (De quadratura curvarum).
(v.)
(vi.)
(vii.) ∫ eaxsin (bx + α) dx = (a2+ b2)−1eax{a sin (bx + α) − b cos (bx + α) }.
(viii.) ∫ sinmx cosnx dx can be reduced by differentiating a function of the form sinpx cosqx;
Hence
(ix.)
(x.)
(xi.)
of which the first or the second is to be employed according as e < or > 1.
50.New transcendents.Among the integrals of transcendental functions which lead to new transcendental functions we may notice
called the “logarithmic integral,” and denoted by “Li x,” also the integrals
called the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral
∫x0e−x2dx
called the “error-function integral,” and denoted by “Erf x.” All these functions have been tabulated (seeTables, Mathematical).
51.Eulerian integrals.New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed. Prominent among such functions are the Beta and Gamma functions expressed by the equations
B(l, m) =∫10xl−1(1 − x)m−1dx,
Γ(n) =∫∞0e−ttn−1dt.
When n is a positive integer Γ(n + 1) = 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) · Γ(l + m) = Γ(l) · Γ(m).
The Gamma function satisfies the difference equation
Γ(x + 1) = x Γ(x),
and also the equation
Γ(x) · Γ(1 − x) = π/sin (xπ),
with the particular result
Γ(½)= √π.
The number
is called “Euler’s constant,” and is equal to the limit
its value to 15 decimal places is 0.577 215 664 901 532.
The function log Γ(1 + x) can be expanded in the series
−1⁄3(S3− 1) x3−1⁄5(S5− 1) x5− ...,
where
and the series for log Γ(1 + x) converges when x lies between −1 and 1.
52.Definite integrals.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
∫10(1 − x2)−1/2log x dx = −½ π log 2,
although the indefinite integral of (1 − x2)−1/2log 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:—
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 ofintegration are infinite, when the functions tend to zero at infinite distances in a sufficiently high order (seeFunction). The method of contour integration involves the introduction of complex variables (seeFunction: §Complex Variables).
A few results are added
(i.)
(ii.)
(iii.)
(iv.)
∫∞0x2· cos 2x · e−x2dx = −1⁄4e−1√π,
(v.)
(vi.)
(vii.)
∫π0log (1 − 2α cos x + α2) dx = 0 or 2π log α according as α < or > 1,
(viii.)
(ix.)
(x.)
(xi.)
(xii.)
(xiii.)
∫∞−∞e−x2+2axdx = √π · ea2,
(xiv.)
∫∞0x−1/2sin x dx =∫∞0x−1/2cos x dx = √(½ π),
53.Multiple Integrals.The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the articleFunction. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z) through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by
∫∫ ƒ(x, y) dx dy.
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 of ƒ 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 ƒ(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
and is called a “repeated integral.” The identification of a surface integral, such as ∫∫ ƒ(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
∫∫∫ ƒ(x, y, z) dx dy dz.
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 first place that the surface is closed, or has no edge. We may mark a large number of points on the surface, andSurface Integrals.draw the tangent planes at all these points. 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 some arbitrarily 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 ƒ 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
∫∫∫ ƒ dS.
TheLine Integrals.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
∫ ƒ ds
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
In like manner surface integrals usually present themselves in the form
∫∫ (lξ + mη + nζ) dS
where l, m, n are the direction cosines of the normal to the surface drawn in a specified sense.
The area of a bounded portion of the plane of (x, y) may be expressed either as
½ ∫ (x dy − y dx),
or as
∫∫ dx dy,
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.Theorems of Green and Stokes.We have two theorems of transformation connecting volume integrals with surface integrals and surface integrals with line integrals. The first theorem, called “Green’s theorem,” is expressed by the equation
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 l, 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.,
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 right-handed 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 right-handed system, we have Stokes’s theorem in the form
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 reverse the sense of l, m, n and maintain the formula, or retain the sense of l, m, n and 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 ∂ξ/∂x, ∂η/∂y, ∂ζ/∂z 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 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. ItChange of Variables in a Multiple Integral.is necessary in the first place to determine the differential element expressed by the product of the differentials of the first set of variables in terms of the differentials of the second set of variables. It is necessary in the second place to determine the limits of integration which must be employed 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 x1, x2, ..., xn, and those of the other set by u1, u2, ..., un, we have the relation
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 ƒ(x, y) over the area within a circle given by x2+ y2= a2, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have
If we have to integrate over the area of a rectangle a ≥ x ≥ 0, b ≥ y ≥ 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—
∫a0dx∫b0ƒ(x, y) dy =∫0tan−1b/adθ∫0a sec θƒ(r cos θ, r sin θ) r dr
+∫1/2πtan−1b/adθ∫0b cosec θƒ(r cos θ, r sin θ) r dr.
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 along a portion of the axis of x. When a change ofLine Integrals and Multiple Integrals.variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may 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 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
∫ ½ r2dθ or ∫ ½ p ds,
where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be 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
π ∫ y2dx,
and the area of the surface is expressed by the formula
2π ∫ y ds,
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.