Line Integral


Also found in: Wikipedia.

line integral

[′līn ¦int·ə·grəl]
(mathematics)
For a curve in a vector space defined byx=x(t), and a vector functionVdefined on this curve, the line integral ofValong the curve is the integral over t of the scalar product ofV[x(t)] and d x/ dt ; this is written ∫V· d x.
For a curve which is defined by x = x (t), y = y (t), and a scalar function ƒ depending on x and y, the line integral of ƒ along the curve is the integral over t of ƒ[x (t), y (t)] · √(dx / dt)2+ (dy / dt)2); this is written ∫ ƒ ds, where ds = √(dx)2+ (dy)2) is an infinitesimal element of length along the curve.
For a curve in the complex plane defined by z = z (t), and a function ƒ depending on z, the line integral of ƒ along the curve is the integral over t of ƒ[z (t)] (dz / dt); this is written ∫ ƒ dz.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Line Integral

 

an integral taken along some curve in the plane or in space. We distinguish line integrals of the first kind and line integrals of the second kind. A line integral of the first kind arises, for example, in problems involving the calculation of the mass of a curve of variable density and is denoted by

Cf (P)ds

where C is the given curve, ds is the differential of its arc, and f(P) is the function of a point on the curve and is the limit of the corresponding integral sums. In the case of a plane curve C given by the equation y = y(x), a line integral of the first kind reduces to an ordinary integral. Specifically,

A line integral of the second kind arises, for example, in connection with the work of a force field. In the case of a plane curve C the integral takes the form

CP(x,y)dx + Q(x,y)dy

and is also the limit of the corresponding integral sums. A line integral of the second kind can be expressed as an ordinary integral. Specifically,

wherex = x(t) and y = y(t) for α ≤ t ≤ β, is the parametric equation of the curve C. Its connection with a line integral of the first kind is given by the equality

CP(x,y)dx + Q (x,y)dy = ∫C {P cosα + Q sin α} ds

Here, α is the angle between the Ox axis and the tangent to the curve pointing in the direction of the increasing arc length.

A line integral of the second kind in space is defined similarly. (SeeVECTOR CALCULUS for a treatment of line integrals of the second kind from the standpoint of vectors.)

Suppose D is some region and C is its boundary. Under certain conditions, the line integral along the curve C and the double integral over the region D (see) are connected by the relation

Similarly, the line integral and the surface integral are connected by the relation

Line integrals are of great importance in the theory of functions of a complex variable. They are widely used in various branches of mechanics, physics, and engineering.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Only the line integrals along paths 1 (Bromwich Contour), 3, and 5 in Figure 2 contribute a nonzero quantity to the closed line integral on the left-hand side of (E.3.3).
By decomposing the integral over the perimeter [C.sub.i] (I ([[theta].sup.(i)])) for each individual segment and circular arch (j) along the perimeter ([C.sub.i]) the line integral I ([[theta].sup.(i)]) can be further simplified into a line integral of a single-variable function F ([[eta].sup.(i)]) as:
where |0> is the physical vacuum state and the line integral appearing in the above expression is along a spacelike path starting at y' and ending at y, on a fixed time slice.
The Line Integral Convolution Method was applied to describe the trajectory of the particles more clearly [24].
where Gauss's theorem over the horizontal plane is used, assuming no contributions from a far field line integral, and [(1 + [([eta]' -- [eta]).sup.2]/[R.sup.2]).sup.-3/2] has been put to unity.
3.1.3 The line integral of the field of a moving charge
The inductive term is calculated as the line integral along the razor-blade of the victim subdomain of a function which is the surface integral of the rooftop function, which extends on the two subpatches on either side of the active subdomain.
The most widely used scheme for producing pencil drawing effects is line integral convolution (LIC) [2], which integrates the noises scattered on the pixels of an image along an integration direction.
Complications due to the simultaneous nature of the price changes are handled with line integral theory.
The branch line integral diminishes as [z.sup.-3/2] become negligible for large z.
Each term of the series corresponds to a successive arrival of a "generalized ray" and each is a definite line integral along a fixed path which can be easily computed numerically.