In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a line integral is an
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
where the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
to be integrated is evaluated along a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
.
The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''
contour integral
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 method of complex analysis.
...
'' 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
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
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
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
or, for a vector field, the
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of the vector field with a
differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval est ...
. Many simple formulae in physics, such as the definition of
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
as
, have natural continuous analogues in terms of line integrals, in this case
, which computes the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
done on an object moving through an electric or gravitational field F along a path
.
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
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
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
![Line integral of scalar field](https://upload.wikimedia.org/wikipedia/commons/4/42/Line_integral_of_scalar_field.gif)
Definition
For some
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
where
, the line integral along a
piecewise smooth
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
is defined as
:
where
is an arbitrary
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
of the curve
(i.e., a differential length of
). 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
Cross-sectional data, or a cross section of a study population, in statistics and econometrics, is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. The analy ...
area bounded by the curve
and the graph of . See the animation to the right.
Derivation
For a line integral over a scalar field, the integral can be constructed from a
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
using the above definitions of , and a parametrization of . This can be done by partitioning the
interval into sub-intervals of length , then denotes some point, call it a sample point, on the curve . We can use the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of sample points to approximate the curve as a
polygonal path
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
by introducing the straight line piece between each of the sample points and . (The approximation of a curve to a polygonal path is called ''rectification of a curve,'' see
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
for more details.) We then label the distance of the line segment between adjacent sample points on the curve as . The product of and can be associated with the signed area of a rectangle with a height and width of and , respectively. Taking the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
of the
sum of the terms as the length of the partitions approaches zero gives us
:
By the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, 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 field
Definition
For a
vector field F: ''U'' ⊆ R
''n'' → R
''n'', the line integral along a
piecewise smooth
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
''C'' ⊂ ''U'', in the direction of r, is defined as
:
where · is the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
, and r:
'a'', ''b''→ ''C'' is a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the line of the integration.
Line integrals of vector fields are independent of the parametrization r in
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, but they do depend on its
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.
From the viewpoint of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the
musical isomorphism
In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced by ...
(which takes the vector field to the corresponding
covector
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
field), over the curve considered as an
immersed 1-manifold.
Derivation
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 , and its parametrization , we construct the integral from a
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
. We partition the
interval (which is the range of the values of the
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
) into intervals of length . Letting be the th point on , then gives us the position of the th point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their
displacement
Displacement may refer to:
Physical sciences
Mathematics and Physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
vectors, . As before, evaluating at all the points on the curve and taking the dot product with each displacement vector gives us the
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
contribution of each partition of on . Letting the size of the partitions go to zero gives us a sum
:
By the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, 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 independence
If a vector field F is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
of a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
''G'' (i.e. if F is
conservative
Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
), that is,
:
then by the
multivariable chain rule the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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''.
Applications
The line integral has many uses in physics. For example, the
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
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 curve
For a
vector field , , the line integral across a curve ''C'' ⊂ ''U'', also called the
''flux integral'', is defined in terms of a
piecewise smooth
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
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 has a specified forward direction from to , and the flow is counted as positive when is on the clockwise side of the forward velocity vector .
Complex line integral
In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the line integral is defined in terms of
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
and
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
of complex numbers. Suppose ''U'' is an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
C, 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 and considering the expression
:
The integral is then the limit of this
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
as the lengths of the subdivision intervals approach zero.
If the parametrization is
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, 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
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
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions ...
may be used to equate the line integral of an
analytic function
In 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. Functions of each type are infinitely differentiable, but complex an ...
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
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
computes the integral in terms of the singularities. This also implies the path independence of complex line integral for analytic functions.
Example
Consider the function ''f''(''z'') = 1/''z'', and let the contour ''L'' be the counterclockwise
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
about 0, parametrized by z(''t'') = ''e''
''it'' with ''t'' in
, 2π
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
using the
complex exponential
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
. Substituting, we find:
:
This is a typical result of
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
and the
residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
.
Relation of complex line integral and line integral of vector field
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
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
) for any smooth closed curve L. Correspondingly, by
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orient ...
, the right-hand integrals are zero when
is
irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
(
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
-free) and
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
(
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
-free). In fact, the Cauchy-Riemann equations for
are identical to the vanishing of curl and divergence for F.
By
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orient ...
, 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 mechanics
The
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...
of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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 amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the quan ...
s in quantum
scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
theory.
See also
*
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
*
Gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
*
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 method of complex analysis.
...
*
Nachbin's theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including ...
*
Surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
*
Volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = \ ...
*
Volume integral
In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
References
External links
*
*
Khan Academy
Khan Academy is an American non-profit educational organization created in 2008 by Sal Khan. Its goal is creating a set of online tools that help educate students. The organization produces short lessons in the form of videos. Its website also in ...
modules:
*
"Introduction to the Line Integral"*
"Line Integral Example 1"*
"Line Integral Example 2 (part 1)"*
"Line Integral Example 2 (part 2)"*
Line integral of a vector field – Interactive
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Complex analysis
Vector calculus