Line integral

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).

, 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.

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.

Here, and in the rest of the article, the absolute value bars denote the standard (Euclidean) norm of a vector.

is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length of the curve

[2] 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

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).

We then label the distance of the line segment between adjacent sample points on the curve as Δsi.

Taking the limit of the sum of the terms as the length of the partitions approaches zero gives us

Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation.

Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.

As before, evaluating F at all the points on the curve and taking the dot product with each displacement vector gives us the infinitesimal contribution of each partition of F on C. Letting the size of the partitions go to zero gives us a sum

By the mean value theorem, we see that the displacement vector between adjacent points on the curve is

In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them.

For this reason, a line integral of a conservative vector field is called path independent.

The flow is computed in an oriented sense: the curve C has a specified forward direction from r(a) to r(b), and the flow is counted as positive when F(r(t)) is on the clockwise side of the forward velocity vector r'(t).

The integral is then the limit of this Riemann sum as the lengths of the subdivision intervals approach zero.

When L is a closed curve (initial and final points coincide), the line integral is often denoted

To establish a complete analogy with the line integral of a vector field, one must go back to the definition of differentiability in multivariable calculus.

The line integrals of complex functions can be evaluated using a number of techniques.

The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals.

This also implies the path independence of complex line integral for analytic functions.

Consider the function f(z) = 1/z, and let the contour L be the counterclockwise unit circle about 0, parametrized by z(t) = eit with t in [0, 2π] using the complex exponential.

This is a typical result of Cauchy's integral formula and the residue theorem.

Viewing complex numbers as 2-dimensional vectors, the line integral of a complex-valued function

is analytic (satisfying the Cauchy–Riemann equations) for any smooth closed curve L. Correspondingly, by Green's theorem, the right-hand integrals are zero when

are identical to the vanishing of curl and divergence for F. By Green's theorem, the area of a region enclosed by a smooth, closed, positively oriented curve

However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.