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 W = F ⋅ s {\displaystyle W=\mathbf {F} \cdot \mathbf {s} } , have natural continuous analogues in terms of line integrals, in this case W = ∫ L F ( s ) ⋅ d s {\textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} } , which computes the work done on an object moving through an electric or gravitational field F along a path L {\displaystyle L} .
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 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.
Line integral of a scalar field
Definition For some scalar field f : U → R {\displaystyle f\colon U\to \mathbb {R} } where U ⊆ R n {\displaystyle U\subseteq \mathbb {R} ^{n}} , the line integral along a piecewise smooth curve C ⊂ U {\displaystyle {\mathcal {C}}\subset U} is defined as
∫ C f d s = ∫ a b f ( r ( t ) ) | r ′ ( t ) | d t , {\displaystyle \int _{\mathcal {C}}f\,ds=\int _{a}^{b}f\left(\mathbf {r} (t)\right)\left|\mathbf {r} '(t)\right|\,dt,}
where r : [ a , b ] → C {\displaystyle \mathbf {r} \colon [a,b]\to {\mathcal {C}}} is an arbitrary bijective parametrization of the curve C {\displaystyle {\mathcal {C}}} such that r(a) and r(b) give the endpoints of C {\displaystyle {\mathcal {C}}} 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 C {\displaystyle {\mathcal {C}}} is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length of the curve C {\displaystyle {\mathcal {C}}} (i.e., a differential length of C {\displaystyle {\mathcal {C}}} ). Line integrals of scalar fields over a curve C {\displaystyle {\mathcal {C}}} do not depend on the chosen parametrization r of C {\displaystyle {\mathcal {C}}} . Geometrically, when the scalar field f is defined over a plane (n = 2), its graph is a surface 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. See the animation to the right.
Derivation 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 [a, b] into n sub-intervals [ti−1, ti] 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 as a polygonal path by introducing the straight line piece between each of the sample points r(ti−1) and r(ti). (The approximation of a curve to a polygonal path is called rectification of a curve, see here for more details.) We then label the distance of the line segment between adjacent 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
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