Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.[1] 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 , have natural continuous analogs 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 s.

Vector calculusEdit

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.

Line integral of a scalar fieldEdit

 
The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

DefinitionEdit

For some scalar field  , the line integral along a piecewise smooth curve   is defined as

 

where   is an arbitrary bijective parametrization of the curve   such that   and   give the endpoints of   and  . Here, and in the rest of the article, the absolute value bars denote the standard (euclidean) norm of a vector.

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

Geometrically, when the scalar field   is defined over a plane  , its graph is a surface   in space, and the line integral gives the (signed) cross-sectional area bounded by the curve   and the graph of  . See the animation to the right.

DerivationEdit

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 = (ba)/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 ≤ in} 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

 

Substituting this in the above Riemann sum yields

 

which is the Riemann sum for the integral

 

Line integral of a vector fieldEdit

DefinitionEdit

For a vector field F : URnRn, the line integral along a piecewise smooth curve CU, in the direction of r, is defined as

 

where · is the dot product and r: [a, b] → 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.

From the viewpoint of differential geometry, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the musical isomorphism (which takes the vector field to the corresponding covector field) over the curve considered as an immersed 1-manifold.

DerivationEdit

 
The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its tangent vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral. (Click on image for a detailed description.)

The line integral of a vector field can be derived in a manner very similar to the case of a scalar field, but this time with the inclusion of a dot product. Again using the above definitions of F, C and its parametrization r(t), we construct the integral from a Riemann sum. We partition the interval [a,b] (which is the range of the values of the parameter t) into n intervals of length Δt = (ba)/n. Letting ti be the ith point on [a,b], then r(ti) gives us the position of the ith point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their displacement vectors, Δri. 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

 

Substituting this in the above Riemann sum yields

 

which is the Riemann sum for the integral defined above.

Path independenceEdit

If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is,

 

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

 

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

 

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.

ApplicationsEdit

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.

Flow across a curveEdit

For a vector field  ,  , the line integral across a curve CU, also called the flux integral, is defined in terms of a piecewise smooth parametrization  ,  , as:

 

Here • is the dot product, and   is the clockwise perpendicular of the velocity vector  .

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   is on the clockwise side of the forward velocity vector  .

Complex line integralEdit

In complex analysis, the line integral is defined in terms of multiplication and addition of complex numbers. Suppose U is an open subset of the complex plane C, f : UC is a function, and   is a curve of finite length, parametrized by  , where   The line integral

 

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

 

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

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

 

When   is a closed curve (initial and final points coincide), the line integral is often denoted  , sometimes referred to in engineering as a cyclic integral.

The line integral with respect to the conjugate complex differential   is defined[2] to be

 

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. The Cauchy integral theorem may be used to equate the line integral of an analytic function to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where   is analytic without singularities, the value of the integral is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities.

ExampleEdit

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. Substituting, we find:

 

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

Relation of complex line integral and line integral of vector fieldEdit

Viewing complex numbers as 2-dimensional vectors, the line integral of a complex-valued function   has real and complex parts equal to the line integral and the flux integral of the vector field corresponding to the conjugate function  . Specifically, if   parametrizes L, and   corresponds to the vector field   , then:

 

By Cauchy's theorem, the left-hand integral is zero when   is analytic (satisfying the Cauchy–Riemann equations). Correspondingly, by Green's theorem, the right-hand integrals are zero when   is irrotational (curl-free) and incompressible (divergence-free). In fact, the Cauchy-Riemann equations for   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   is given by the integral  . This fact is used, for example, in the proof of the area theorem.

Quantum mechanicsEdit

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 alsoEdit

ReferencesEdit

  1. ^ Kwong-Tin Tang (30 November 2006). Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms. Springer Science & Business Media. ISBN 978-3-540-30268-1.
  2. ^ Ahlfors, Lars (1966). Complex Analysis (2nd ed.). New York: McGraw-Hill. p. 103.

External linksEdit