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. 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, W = F · s) have natural continuous analogs in terms of line integrals (W = ∫C F · ds). The line integral finds the work done on an object moving through an electric or gravitational field, for example. 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 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. For some scalar field f : U ⊆ R n → R {displaystyle f:mathbb {U} subseteq mathbb {R} ^{n} ightarrow mathbb {R} } , the line integral along a piecewise smooth curve C ⊂ U {displaystyle {mathcal {C}}subset mathbb {U} } is defined as where r : [ a , b ] → C {displaystyle mathbf {r} : ightarrow {mathcal {C}}} is an arbitrary bijective parametrization of the curve C {displaystyle {mathcal {C}}} such that r ( a ) {displaystyle mathbf {r} (a)} and r ( b ) {displaystyle mathbf {r} (b)} give the endpoints of C {displaystyle {mathcal {C}}} and a < b {displaystyle 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 {displaystyle f} is called the integrand, the curve C {displaystyle {mathcal {C}}} is the domain of integration, and the symbol d s {displaystyle ds} may be intuitively interpreted as an elementary arc length. Line integrals of scalar fields over a curve C {displaystyle {mathcal {C}}} do not depend on the chosen parametrization r {displaystyle mathbf {r} } of C {displaystyle {mathcal {C}}} . Geometrically, when the scalar field f {displaystyle f} is defined over a plane ( n = 2 ) {displaystyle (n=2)} , its graph is a surface z = f ( x , y ) {displaystyle z=f(x,y)} in space, and the line integral gives the (signed) cross-sectional area bounded by the curve C {displaystyle {mathcal {C}}} and the graph of f {displaystyle f} . See the animation to the right. For a line integral over a scalar field, the integral can be constructed from a Riemann sum using the above definitions of f, C and a parametrization r of C. This can be done by partitioning the interval into n sub-intervals of length Δt = (b − a)/n, then r(ti) denotes some point, call it a sample point, on the curve C. We can use the set of sample points {r(ti) : 1 ≤ i ≤ n} to approximate the curve C by a polygonal path by introducing a straight line piece between each of the sample points r(ti-1) and r(ti). We then label the distance between each of the sample points on the curve as Δsi. The product of f(r(ti)) and Δsi can be associated with the signed area of a rectangle with a height and width of f(r(ti)) and Δsi respectively. Taking the limit of the sum of the terms as the length of the partitions approaches zero gives us We note that, by the mean value theorem, the distance between subsequent points on the curve, is