Line Integral (redirected from Contour of integration)

line integral

In mathematics, the integral of a function of several variables defined on a line or curve that has been expressed in terms of arc length (see length of a curve). An ordinary definite integral is defined over a line segment, whereas a line integral may use a more general path, such as a parabola or a circle. Line integrals are used extensively in the theory of functions of a complex variable.

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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 overtof the scalar product ofV[x(t)] anddx/dt; this is written ∫V·dx.

For a curve which is defined byx=x(t),y=y(t), and a scalar function ƒ depending onxandy, the line integral of ƒ along the curve is the integral overtof ƒ[x(t),y(t)] · √(dx/dt)²+ (dy/dt)²); this is written ∫ ƒds, whereds= √(dx)²+ (dy)²) is an infinitesimal element of length along the curve.

For a curve in the complex plane defined byz=z(t), and a function ƒ depending onz, the line integral of ƒ along the curve is the integral overtof ƒ[z(t)] (dz/dt); this is written ∫ ƒdz.

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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

∫_C^{f (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.