Definite integral of a scalar or vector field along a path
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Line integral" – news · newspapers · books · scholar · JSTOR(June 2023) (Learn how and when to remove this message)
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.[1] The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as , have natural continuous analogues in terms of line integrals, in this case , which computes the work done on an object moving through an electric or gravitational field F along a path .
^Kwong-Tin Tang (30 November 2006). Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms. Springer Science & Business Media. ISBN 978-3-540-30268-1.
mathematics, a lineintegral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear...
definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally...
In scientific visualization, lineintegral convolution (LIC) is a method to visualize a vector field, such as fluid motion. global method integration-based...
also known as the fundamental theorem of calculus for lineintegrals, says that a lineintegral through a gradient field can be evaluated by evaluating...
surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral...
\limits _{D}f(x,y)\,dx\,dy.} The fundamental theorem of lineintegrals says that a lineintegral through a gradient field can be evaluated by evaluating...
infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes...
In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form ∫ a ( x ) b ( x ) f ( x , t...
property that its lineintegral is path independent; the choice of path between two points does not change the value of the lineintegral. Path independence...
method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real...
f(x)=e^{-x^{2}}} over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is ∫ − ∞ ∞ e − x 2 d x = π . {\displaystyle...
of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the lineintegral of a vector field represents...
positive real line: ∫ 0 ∞ sin x x d x = π 2 . {\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x}}\,dx={\frac {\pi }{2}}.} This integral is not absolutely...
to define a suitable integral. Also, one might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the necessary...
trigonometry, side lengths (Heron's formula), vectors, coordinates, lineintegrals, Pick's theorem, or other properties. Heron of Alexandria found what...
Path integral may refer to: Lineintegral, the integral of a function along a curve Contour integral, the integral of a complex function along a curve...
as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It...
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...
calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially...
powerful tool to evaluate lineintegrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as...
Mathematically, this is expressed as the lineintegral of the electric field along that path. In electrostatics, this lineintegral is independent of the path taken...
three-space to the lineintegral of the vector field over the surface boundary. The second fundamental theorem of calculus states that the integral of a function...
improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context...
Global Information
