line integral

In mathematics, a line integral (sometimes called a path integral or curve integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve in two dimensions or the complex plane it is also called a contour integral.

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 formulas in physics (for example, W=vec Fcdotvec s) have natural continuous analogs in terms of line integrals (W=int_C vec Fcdot dvec s). 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.

Line integral of a scalar field

For some scalar field f : URn to R, the line integral along a curve CU is defined as

int_C f, ds = int_a^b f(mathbf{r}(t)) |mathbf{r}'(t)|, dt.
where r: [a, b] to C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.

The function f is called the integrand, the curve C is the domain of integration, and the symbol ds may be heuristically interpreted as an elementary arc length. Line integrals of scalar fields do not depend on the chosen parametrization r.

Line integral of a vector field

For a vector field F : URn to Rn, the line integral along a curve CU, in the direction of r, is defined as

int_C mathbf{F}(mathbf{r})cdot,dmathbf{r} = int_a^b mathbf{F}(mathbf{r}(t))cdotmathbf{r}'(t),dt.

where cdot is the dot product and r: [a, b] to C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.

A line integral of a scalar field is thus a line integral of a vector field where the vectors are always tangential to the line.

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.

Path independence

If a vector field F is the gradient of a scalar field G, that is,

nabla G = mathbf{F},

then the derivative of the composition of G and r(t) is

frac{dG(mathbf{r}(t))}{dt} = nabla G(mathbf{r}(t)) cdot mathbf{r}'(t) = mathbf{F}(mathbf{r}(t)) cdot mathbf{r}'(t)

which happens to be the integrand for the line integral of F on r(t). It follows that, given a path C , then

int_C mathbf{F}(mathbf{r})cdot,dmathbf{r} = int_a^b mathbf{F}(mathbf{r}(t))cdotmathbf{r}'(t),dt = int_a^b frac{dG(mathbf{r}(t))}{dt},dt = G(mathbf{r}(b)) - G(mathbf{r}(a)).

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

For this reason, a line integral of a vector field which is the gradient of a scalar field is called path independent.


The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.

Complex line integral

The line integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, gamma : [a, b] to U is a rectifiable curve and f : U to C is a function. Then the line integral

int_gamma f(z),dz

may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression

sum_{1 le k le n} f(gamma(t_k)) (gamma(t_k) - gamma(t_{k-1}) ).

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

If gamma is a continuously differentiable curve, the line integral can be evaluated as an integral of a function of a real variable:

int_gamma f(z),dz
=int_a^b f(gamma(t)),gamma,'(t),dt.

When gamma is a closed curve, that is, its initial and final points coincide, the notation

oint_gamma f(z),dz

is often used for the line integral of f along gamma.

The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split in to real and imaginary parts reducing the problem to that of evaluating two real-valued line integrals, the Cauchy integral formula may be used in other circumstances. If the line integral is a closed curve in a region where the function is analytic and containing no singularities, then the value of the integral is simply zero, this is a consequence of the Cauchy integral theorem. Because of the residue theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable (see residue theorem for an example).


Consider the function f(z)=1/z, and let the contour C be the unit circle about 0, which can be parametrized by eit, with t in [0, 2π]. Substituting, we find

oint_C f(z),dz = int_0^{2pi} {1over e^{it}} ie^{it},dt = iint_0^{2pi} e^{-it}e^{it},dt
=iint_0^{2pi},dt = i(2pi-0)=2pi i
where we use the fact that any complex number z can be written as reit where r is the modulus of z. On the unit circle this is fixed to 1, so the only variable left is the angle, which is denoted by t. This answer can be also verified by the Cauchy integral formula.

Relation between the line integral of a vector field and the complex line integral

Viewing complex numbers as 2-dimensional vectors, the line integral of a 2-dimensional vector field corresponds to the real part of the line integral of the conjugate of the corresponding complex function of a complex variable. More specifically, if mathbf{r} (t)=x(t)mathbf{i}+y(t)mathbf{j} and f(z)=u(z)+iv(z), then:

int_C overline{f(z)},dz = int_C (u-iv),dz = int_C (umathbf{i}+vmathbf{j})cdot dmathbf{r} - iint_C (umathbf{i}-vmathbf{j})cdot dmathbf{r},

provided that both integrals on the right hand side exist, and that the parametrization z(t) of C has the same orientation as mathbf{r}(t).

Due to the Cauchy-Riemann equations the curl of the vector field corresponding to the conjugate of a holomorphic function is zero. This relates through Stokes' theorem both types of line integral being zero.

Also, the line integral can be evaluated using the change of variables.

Quantum mechanics

The "path integral formulation" of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path. 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.

See also

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