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Concept# Line integral

Summary

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. 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 .
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by z = f(x,y) and a curve C in the xy plane. The line integral of f would be the area of the "curtain" created—when the points of the surface that are directly over C are carved out.
For some scalar field where , the line integral along a piecewise smooth curve is defined as
where is an arbitrary bijective parametrization of the curve such that r(a) and r(b) give the endpoints of and a < b. Here, and in the rest of the article, the absolute value bars denote the standard (Euclidean) norm of a vector.
The function f is called the integrand, the curve is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length of the curve (i.e., a differential length of ).

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Stokes' theorem

Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.

Curve orientation

In mathematics, an orientation of a curve is the choice of one of the two possible directions for travelling on the curve. For example, for Cartesian coordinates, the x-axis is traditionally oriented toward the right, and the y-axis is upward oriented. In the case of a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections), the curve is said to be positively oriented or counterclockwise oriented, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when traveling on it.

Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a 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 variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.

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