In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
line integral (sometimes called a
path integral,
contour integral, or
curve integral; not to be confused with
calculating arc length using integrationDetermining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
) is an
integralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
where the
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
to be integrated is evaluated along a
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
.
The function to be integrated may be a
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
or a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
. 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 lengthDetermining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
or, for a vector field, the
scalar productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on
intervalsIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
. Many simple formulae in physics (for
exampleIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
,
W=
F·
s) have natural continuous analogs in terms of line integrals (
W=∫
C F· d
s). The line integral finds the
workIn physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does 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
fieldIn physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
along a given curve. More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve. This can be visualised 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.
Definition
For some
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
f :
U ⊆
Rn →
R, the line integral along a piecewise smooth
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
C ⊂
U is defined as
where
r: [a, b] →
C is an arbitrary bijective
parametrizationIn mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
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 intuitively interpreted as an elementary
arc lengthDetermining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
. Line integrals of scalar fields over a curve
C do not depend on the chosen parametrization
r of
C.
Derivation
For a
line integral over a scalar field, the integral can be constructed from a
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
using the above definitions of
f,
C and a parametrization
r of
C. This can be done by partitioning the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[
a,
b] into
n sub-intervals [
ti-1,
ti] of length Δ
t = (
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 Δ
si. The product of
f(
r(
ti)) and Δ
si can be associated with the signed area of a rectangle with a height and width of
f(
r(
ti)) and Δ
si respectively. Taking the
limitThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
of the
sumSUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of the terms as the length of the partitions approaches zero gives us Ġ
We note that the distance between subsequent points on the curve, is
Substituting this in to our above Riemann sum yields
which is the Riemann sum for the integral
Basically is the area under the constraint of the scalar function z=f(x,y), where x=u(t) and y=v(t) is the constraint.
Definition
For a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
F :
U ⊆
Rn →
Rn, the line integral along a piecewise smooth
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
C ⊂
U, in the direction of
r, is defined as
where · is the
dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
and
r: [a, b] →
C is a bijective
parametrizationIn mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
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 valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
, but they do depend on its
orientationIn mathematics, a positively oriented curve is a planar simple closed curve such that when traveling on it one always has the curve interior to the left...
. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.
Derivation
The line integral of a vector field can be derived in a very similar manner as in the case of a scalar field. Again using the above definitions of
F,
C and its parametrization
r(
t), we construct the integral from a
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
. Partition the interval [
a,
b] into
n intervals of length Δ
t = (
b −
a)/
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
displacementA displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
vectors, Δ
si. As before, evaluating
F at all the points on the radiation curve and taking the dot product with each displacement vector which gives us the
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
contribution of each partition of
F on
C. Letting the size of the partitions go to zero gives us a sum
We see that the displacement vector between adjacent points on the curve is
Substituting this into our above Riemann sum yields
which is the Riemann sum for the integral defined above.
Path independence
If a vector field
F is the
gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of a
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
G (i.e. if
F is conservative), that is,
then the
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the
compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of
G and
r(
t) is
which happens to be the integrand for the line integral of
F on
r(
t). It follows that, given a path
C , then
In other 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 conservative vector field is called
path independent.
Applications
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 analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. Suppose
U is an open subset of
CA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
, γ : [
a,
b] →
U is a rectifiable curve and
f :
U →
C is a function. Then the line integral
may be defined by subdividing the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[
a,
b] into
a =
t0 <
t1 < ... <
tn =
b and considering the expression
The integral is then the
limitThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
of this sum, as the lengths of the subdivision intervals approach zero.
If
is a continuously differentiable curve, the line integral can be evaluated as an integral of a function of a real variable:
When
is a closed curve, that is, its initial and final points coincide, the notation
is often used for the line integral of
f along
.
The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split into 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
analyticIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
and containing no
singularitiesIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
, then the value of the integral is simply zero; this is a consequence of the Cauchy integral theorem. Because of the
residue theoremThe residue theorem, sometimes called Cauchy's Residue Theorem, in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...
, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable (see
residue theoremThe residue theorem, sometimes called Cauchy's Residue Theorem, in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...
for an example).
Example
Consider the function
f(
z)=1/
z, and let the contour
C be the
unit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
about 0, which can be parametrized by
eit, with
t in [0, 2π]. Substituting, we find
where we use the fact that any complex number
z can be written as
reit where
r is the
modulusIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
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
conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of the corresponding complex function of a complex variable. More specifically, if
and
, then:
provided that both integrals on the right hand side exist, and that the parametrization
of
C has the same orientation as
.
Due to the
Cauchy-Riemann equationsIn mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable...
the curl of the vector field corresponding to the conjugate of a
holomorphic functionIn mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
is zero. This relates through
Stokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
both types of line integral being zero.
Quantum mechanics
The "
path integral formulationThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
" of
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
actually refers not to path integrals in this sense but to
functional integralsFunctional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions...
, 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
probabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
s in quantum
scatteringScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
theory.
See also
- Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
- Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
- Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
- Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....
- Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
- Volume integral
In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain....
- Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
- Nachbin's theorem
In mathematics, in the area of complex analysis, Nachbin's theorem is commonly used to establish a bound on the growth rates for an analytic function. This article will provide a brief review of growth rates, including the idea of a function of exponential type...
- Functional integration
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions...
External links