IntegralIn mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.
Elliptic integralIn integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
Gaussian integralThe Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution.
Differential formIn mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S: The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms.
Direct method in the calculus of variationsIn mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy. The calculus of variations deals with functionals , where is some function space and .
Riemann–Stieltjes integralIn mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
Leibniz integral ruleIn calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form where and the integrands are functions dependent on the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative. It is named after Gottfried Leibniz.
Calculus of variationsThe calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
Closed and exact differential formsIn mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the of d, and a closed form is in the kernel of d. For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α.
Quadratic formIn mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.
