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line integral
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 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 (for example, ) have natural continuous analogs in terms of line integrals (). The line integral finds the work done on an object moving through an electric or gravitational field, for example.
== Vector calculus ==
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve. More specifically, the line integral over a scalar field 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 ''x''-''y'' 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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