(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that∂ξ/∂x = ∂η/∂y and ∂ξ/∂y = −∂η/∂x,the Jacobian ∂(ξ, η)/∂(x, v) can be expressed in the form(∂ξ)2+(∂η)2,∂x∂xand in a number of equivalent forms. The area of any portion of the plane is represented by the double integral∫∫ J−1dξ dη,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 ξ = const., or η = 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∫ {1 +(dy)2}1/2dx,dxor, in polar coordinates, by evaluating the integral∫ {r2+(dr)2}1/2dθ.dθIn both cases the integrals are line integrals taken along the curve.(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral∫ J−1/2dηtaken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula∫∫ {1 +(∂z)2+(∂z)2}1/2dx dy.∂x∂yWhen the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula∫∫ [ {∂(y, z)}2+{∂(z, x)}2+{∂(x, y)}2]1/2du dv.∂(u, v)∂(u, v)∂(u, v)When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equationsx = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula∫∫r[ {r2+(∂r)2}sin2θ +(∂r)2]1/2dθ dφ.∂θ∂φThe surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form∫2π0dφ∫π0F (θ,φ) r2sin θ dθ.In every case the domain of integration must be chosen so as to include the whole surface.(ix.) In three-dimensional polar coordinates the Jacobian∂(x, y, z)= r2sin θ.∂(r, θ, φ)The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is∫a0dr∫2π0dφ∫π0F (r, θ, φ) r2sin θ dθ.(x.) Integrations of rational functions through the volume of an ellipsoid x2/a2+ y2/b2+ z2/c2= 1 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(x1)a1+(x2)a2+ ... +(xn)an≤ 1,a1a2anwhere the a’s and α’s are positive, the value of the integral∫∫ ... x1n1−1· x2n2−1... dx1dx2...isa1n1a2n2...Γ (n1/α1) Γ (n2/α2) ....α1α2...Γ (1 + n1/α1+ n2/α2+ ... )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 = aξ, y = bη, z = cζ, 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 sectorsApproximate and Mechanical Integration.of ellipses (1609) was an approximate integration, as also was the method for the quadrature of the hyperbola given by James Gregory in the appendix to hisExercitationes geometricae(1668). In Newton’sMethodus differentialis(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 typey = a0+ a1x + a2x2+ ... + anxnpass through the vertices of (n + 1) 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 hisMathematical Dissertationspublished 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 y1, y2, ... y2n−1, and the extreme ordinates are denoted by y0, y2n. The vertices of the ordinates y0, y1, y2lie 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{ (b − a)/6n } [y0+ y2n+ 2 (y2+ y4+ ... + y2n−2) + 4 (y1+ y3+ ... + y2n−1) ],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 inEncy. d. Math. Wiss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors,Valeur approximative d’une intégrale définie(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 Abdank-Abakanowicz (1882).Bibliography.—For historical questions relating to the subject the chief authority is M. Cantor,Geschichte d. Mathematik(3 Bde., Leipzig, 1894-1901). For particular matters, or special periods, the following may be mentioned: H. G. Zeuthen,Geschichte d. Math. im Altertum u. Mittelalter(Copenhagen, 1896) andGesch. d. Math. im XVI. u. XVII. Jahrhundert(Leipzig, 1903); S. Horsley,Isaaci Newtoni opera quae exstant omnia(5 vols., London, 1779-1785); C. I. Gerhardt,Leibnizens math. Schriften(7 Bde., Leipzig, 1849-1863); Joh. Bernoulli,Opera omnia(4 Bde., Lausanne and Geneva, 1742). Other writings of importance in the history of the subjectare 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) andInstitutiones calculi integralis(3 Bde., Petrop., 1768-1770); J. L. Lagrange,Leçons sur le calcul des fonctions(Paris, 1806,Œuvres, t. x.), andThéorie des fonctions analytiques(Paris, 1797, 2nd ed., 1813,Œuvres, t. ix.); S. F. Lacroix,Traité 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,Résumé des leçons ... sur le calcul infinitésimale(Paris, 1823), andLeçons sur le calcul différentiel(Paris, 1829;Œuvres, sér. 2, t. iv.); F. Minding,Handbuch d. Diff.-u. Int.-Rechnung(Berlin, 1836); F. Moigno,Leçons sur le calcul diff.(4 tt., Paris, 1840-1861); A. de Morgan,Diff. and Int. Calc.(London, 1842); D. Gregory,Examples on the Diff. and Int. Calc.(2 vols., Cambridge, 1841-1846); I. Todhunter,Treatise on the Diff. Calc.andTreatise on the Int. Calc.(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’intégrales définies(Amsterdam, 1858); M. Stegemann,Grundriss d. Diff.- u. Int.-Rechnung(2 Bde., Hanover, 1862) numerous later editions; J. Bertrand,Traité de calc. diff. et int.(2 tt., Paris, 1864-1870); J. A. Serret,Cours de calc. diff. et int.(2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack, Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and Scheffers, 1906, incomplete); B. Williamson,Treatise on the Diff. Calc.(Dublin, 1872), andTreatise on the Int. Calc.(Dublin, 1874) numerous later editions of both; also the article “Infinitesimal Calculus” in the 9th ed. of theEncy. Brit.; C. Hermite,Cours d’analyse(Paris, 1873); O. Schlömilch,Compendium d. höheren 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, 1880); 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,Traité d’analyse(7 tt., Paris, 1885-1891); J. Edwards,Elementary Treatise on the Diff. Calc.(London, 1886), several later editions; A. G. Greenhill,Diff. and Int. Calc.(London, 1886, 2nd ed., 1891); É. Picard,Traité d’analyse(3 tt., Paris, 1891-1896); O. Stolz,Grundzüge d. Diff.- u. Int.-Rechnung(3 Bde., Leipzig, 1893-1899); C. Jordan,Cours d’analyse(3 tt., Paris, 1893-1896); L. Kronecker,Vorlesungen ü. d. Theorie d. einfachen u. vielfachen Integrale(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); É. Goursat,Cours d’analyse mathématique(2 tt., Paris, 1902-1905); C.-J. de la Vallée Poussin,Cours d’analyse infinitésimale(2 tt., Louvain and Paris, 1903-1906); A. E. H. Love,Elements of the Diff. and Int. Calc.(Cambridge, 1909); W. H. Young,The Fundamental Theorems of the Diff. Calc.(Cambridge, 1910). A résumé of the infinitesimal calculus is given in the articles “Diff.- u. Int-Rechnung” by A. Voss, and “Bestimmte Integrale” by G. Brunel inEncy. 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).
(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that
∂ξ/∂x = ∂η/∂y and ∂ξ/∂y = −∂η/∂x,
the Jacobian ∂(ξ, η)/∂(x, v) can be expressed in the form
and in a number of equivalent forms. The area of any portion of the plane is represented by the double integral
∫∫ J−1dξ dη,
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 ξ = const., or η = 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
or, in polar coordinates, by evaluating the integral
In both cases the integrals are line integrals taken along the curve.
(vii.) When we use curvilinear coordinates ξ, η as in (v.) above, the length of any portion of a curve ξ = const. is given by the integral
∫ J−1/2dη
taken between appropriate limits for η. There is a similar formula for the arc of a curve η = const.
(viii.) The area of a surface z = ƒ(x, y) can be expressed by the formula
When the coordinates of the points of a surface are expressed as functions of two parameters u, v, the area is expressed by the formula
When the surface is referred to three-dimensional polar coordinates r, θ, φ given by the equations
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,
and the equation of the surface is of the form r = ƒ(θ, φ), the area is expressed by the formula
The surface integral of a function of (θ, φ) over the surface of a sphere r = const. can be expressed in the form
∫2π0dφ∫π0F (θ,φ) r2sin θ dθ.
In every case the domain of integration must be chosen so as to include the whole surface.
(ix.) In three-dimensional polar coordinates the Jacobian
The volume integral of a function F (r, θ, φ) through the volume of a sphere r = a is
∫a0dr∫2π0dφ∫π0F (r, θ, φ) r2sin θ dθ.
(x.) Integrations of rational functions through the volume of an ellipsoid x2/a2+ y2/b2+ z2/c2= 1 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
where the a’s and α’s are positive, the value of the integral
∫∫ ... x1n1−1· x2n2−1... dx1dx2...
is
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 = aξ, y = bη, z = cζ, 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 sectorsApproximate and Mechanical Integration.of ellipses (1609) was an approximate integration, as also was the method for the quadrature of the hyperbola given by James Gregory in the appendix to hisExercitationes geometricae(1668). In Newton’sMethodus differentialis(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 = a0+ a1x + a2x2+ ... + anxn
pass through the vertices of (n + 1) 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 hisMathematical Dissertationspublished 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 y1, y2, ... y2n−1, and the extreme ordinates are denoted by y0, y2n. The vertices of the ordinates y0, y1, y2lie 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
{ (b − a)/6n } [y0+ y2n+ 2 (y2+ y4+ ... + y2n−2) + 4 (y1+ y3+ ... + y2n−1) ],
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 inEncy. d. Math. Wiss., Bd. II., A. 2 (1899), and to a monograph by B. P. Moors,Valeur approximative d’une intégrale définie(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 Abdank-Abakanowicz (1882).
Bibliography.—For historical questions relating to the subject the chief authority is M. Cantor,Geschichte d. Mathematik(3 Bde., Leipzig, 1894-1901). For particular matters, or special periods, the following may be mentioned: H. G. Zeuthen,Geschichte d. Math. im Altertum u. Mittelalter(Copenhagen, 1896) andGesch. d. Math. im XVI. u. XVII. Jahrhundert(Leipzig, 1903); S. Horsley,Isaaci Newtoni opera quae exstant omnia(5 vols., London, 1779-1785); C. I. Gerhardt,Leibnizens math. Schriften(7 Bde., Leipzig, 1849-1863); Joh. Bernoulli,Opera omnia(4 Bde., Lausanne and Geneva, 1742). Other writings of importance in the history of the subjectare 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) andInstitutiones calculi integralis(3 Bde., Petrop., 1768-1770); J. L. Lagrange,Leçons sur le calcul des fonctions(Paris, 1806,Œuvres, t. x.), andThéorie des fonctions analytiques(Paris, 1797, 2nd ed., 1813,Œuvres, t. ix.); S. F. Lacroix,Traité 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,Résumé des leçons ... sur le calcul infinitésimale(Paris, 1823), andLeçons sur le calcul différentiel(Paris, 1829;Œuvres, sér. 2, t. iv.); F. Minding,Handbuch d. Diff.-u. Int.-Rechnung(Berlin, 1836); F. Moigno,Leçons sur le calcul diff.(4 tt., Paris, 1840-1861); A. de Morgan,Diff. and Int. Calc.(London, 1842); D. Gregory,Examples on the Diff. and Int. Calc.(2 vols., Cambridge, 1841-1846); I. Todhunter,Treatise on the Diff. Calc.andTreatise on the Int. Calc.(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’intégrales définies(Amsterdam, 1858); M. Stegemann,Grundriss d. Diff.- u. Int.-Rechnung(2 Bde., Hanover, 1862) numerous later editions; J. Bertrand,Traité de calc. diff. et int.(2 tt., Paris, 1864-1870); J. A. Serret,Cours de calc. diff. et int.(2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack, Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and Scheffers, 1906, incomplete); B. Williamson,Treatise on the Diff. Calc.(Dublin, 1872), andTreatise on the Int. Calc.(Dublin, 1874) numerous later editions of both; also the article “Infinitesimal Calculus” in the 9th ed. of theEncy. Brit.; C. Hermite,Cours d’analyse(Paris, 1873); O. Schlömilch,Compendium d. höheren 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, 1880); 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,Traité d’analyse(7 tt., Paris, 1885-1891); J. Edwards,Elementary Treatise on the Diff. Calc.(London, 1886), several later editions; A. G. Greenhill,Diff. and Int. Calc.(London, 1886, 2nd ed., 1891); É. Picard,Traité d’analyse(3 tt., Paris, 1891-1896); O. Stolz,Grundzüge d. Diff.- u. Int.-Rechnung(3 Bde., Leipzig, 1893-1899); C. Jordan,Cours d’analyse(3 tt., Paris, 1893-1896); L. Kronecker,Vorlesungen ü. d. Theorie d. einfachen u. vielfachen Integrale(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); É. Goursat,Cours d’analyse mathématique(2 tt., Paris, 1902-1905); C.-J. de la Vallée Poussin,Cours d’analyse infinitésimale(2 tt., Louvain and Paris, 1903-1906); A. E. H. Love,Elements of the Diff. and Int. Calc.(Cambridge, 1909); W. H. Young,The Fundamental Theorems of the Diff. Calc.(Cambridge, 1910). A résumé of the infinitesimal calculus is given in the articles “Diff.- u. Int-Rechnung” by A. Voss, and “Bestimmte Integrale” by G. Brunel inEncy. 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).