Integrals sentence example
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial differential equations, and Jacobi's on analytical mechanics and higher algebra.
He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.
The integrals are then properly functions of the direction in which the light is to be estimated.
We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y.Advertisement
If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture.
To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express the fact that x, x, xs.
The above expressions for the capacity of an ellipsoid of three unequal axes are in general elliptic integrals, but they can be evaluated for the reduced cases when the ellipsoid is one of revolution, and hence in the limit either takes the form of a long rod or of a circular disk.
The following are some examples of cases in which the above methods may be applied to the calculation of areas and integrals.Advertisement
Discussions of the approximate calculation of definite integrals will be found in works on the infinitesimal calculus; see e.g.
His mathematical writings, which account for some forty entries in the Royal Society's catalogue of scientific papers, cover a wide range of subjects, such" s the theory of probabilities, quadratic forms, theory of integrals, gearings, the construction of geographical maps, &c. He also published a Traite de la theorie des nombres.
The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an account of the mode of their construction.
Legendre had pursued the subject which would now be called elliptic integrals alone from 1786 to 1827, the results of his labours having been almost entirely neglected by his contemporaries, but his work had scarcely appeared in 1827 when the discoveries which were independently made by the two young and as yet unknown mathematicians Abel and Jacobi placed the subject on a new basis, and revolutionized it completely.
The remainder of the first volume relates to the Eulerian integrals and to quadratures.Advertisement
The second volume (1817) relates to the Eulerian integrals, and to various integrals and series, developments, mechanical problems, &c., connected with the integral calculus; this volume contains also a numerical table of the values of the gamma function.
The latter portion of the second volume of the Traite des fonctions elliptiques (1826) is also devoted to the Eulerian integrals, the table being reproduced.
In 1788 Legendre published a memoir on double integrals, and in 1809 one on definite integrals.
They were invented by Gauss to facilitate the computation of elliptic integrals.
Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.Advertisement
Investigation soon showed that certain integrals expressing relations between the motions not only of three but of any number of bodies could be found.
It is remarkable that although the seven integrals were found almost from the beginning of the investigation, no others have since been added; and indeed it has recently been shown that no others exist that can be expressed in an algebraic form.
Functions of the variables possessing this property of remaining constant are termed integrals.
Examples of regular functions defined by series or integrals and their analytic continuations.
A facility to allow the punching of both transformed integrals and CI coefficients from the full CI code has been introduced.Advertisement
The MEMORY subdirective sets aside the maximum amount of memory that is not needed for other purposes to hold the 3-center 2-electron integrals.
The angular integrals are carried out using the methods of Racah algebra.
With definite integrals the limits of integration can also change.
The most costly step in the serial 2 nd derivative algorithm is the computation of the 2 nd derivative two-electron integrals.
The kinetic transport coeffs. are computed from explicit collision integrals and compared favorably with detailed simulations.
The theory of Feynman path integrals in such theories is a further active area of research here at King's.
To be able to work with line, surface and volume integrals.
These computationally intractable sums or integrals can be avoided by using approximate Bayesian methods.