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In mathematics, a line integral is an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
. 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 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 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 alge ...
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 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 t ...
as W=\mathbf\cdot\mathbf, have natural continuous analogues in terms of line integrals, in this case \textstyle W = \int_L \mathbf(\mathbf)\cdot d\mathbf, 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 t ...
done on an object moving through an electric or gravitational field F along a path L.


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 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


Definition

For some scalar field f\colon U\to\R where U \subseteq \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. P ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
\mathcal \subset U is defined as :\int_ f(\mathbf)\, ds = \int_a^b f\left(\mathbf(t)\right) \left, \mathbf'(t)\ \, dt. where \mathbf r\colon ,bto\mathcal 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 ...
parametrization of the curve \mathcal such that and give the endpoints of \mathcal 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 \mathcal is the domain of integration, and the symbol may be intuitively interpreted as an elementary arc length of the curve \mathcal (i.e., a differential length of \mathcal). Line integrals of scalar fields over a curve \mathcal do not depend on the chosen parametrization of \mathcal. 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 \mathcal 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 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 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 of the sum of the terms as the length of the partitions approaches zero gives us :I = \lim_ \sum_^n f(\mathbf(t_i)) \, \Delta s_i. By the mean value theorem, the distance between subsequent points on the curve, is :\Delta s_i = \left, \mathbf(t_i+\Delta t)-\mathbf(t_i)\ \approx \left, \mathbf(t_i)\ Substituting this in the above Riemann sum yields :I = \lim_ \sum_^n f(\mathbf(t_i)) \left, \mathbf'(t_i)\ \Delta t which is the Riemann sum for the integral :I = \int_a^b f(\mathbf(t)) \left, \mathbf'(t)\ dt.


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. P ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
''C'' ⊂ ''U'', in the direction of r, is defined as :\int_C \mathbf(\mathbf)\cdot\,d\mathbf = \int_a^b \mathbf(\mathbf(t))\cdot\mathbf'(t)\,dt 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 alge ...
, 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 ...
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, 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.


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 vectors, . As before, evaluating 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 on . Letting the size of the partitions go to zero gives us a sum :I = \lim_ \sum_^n \mathbf(\mathbf(t_i)) \cdot \Delta\mathbf_i By the mean value theorem, we see that the displacement vector between adjacent points on the curve is :\Delta\mathbf_i = \mathbf(t_i + \Delta t)-\mathbf(t_i) \approx \mathbf'(t_i) \,\Delta t. Substituting this in the above Riemann sum yields :I = \lim_ \sum_^n \mathbf(\mathbf(t_i)) \cdot \mathbf'(t_i)\,\Delta t, 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 gr ...
of a scalar field ''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 in ...
), that is, :\mathbf = \nabla G , 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. ...
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 :\frac = \nabla G(\mathbf(t)) \cdot \mathbf'(t) = \mathbf(\mathbf(t)) \cdot \mathbf'(t) which happens to be the integrand for the line integral of F on r(''t''). It follows, given a path ''C '', that :\int_C \mathbf(\mathbf)\cdot\,d\mathbf = \int_a^b \mathbf(\mathbf(t))\cdot\mathbf'(t)\,dt = \int_a^b \frac\,dt = G(\mathbf(b)) - G(\mathbf(a)). 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 t ...
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 \mathbf F\colon U\subseteq\R^2\to\R^2, , 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. P ...
parametrization , , as: :\int_C \mathbf F(\mathbf r)\cdot d\mathbf r^\perp = \int_a^b \begin P\big(x(t),y(t)\big) \\ Q\big(x(t),y(t)\big) \end \cdot \begin y'(t) \\ -x'(t) \end ~dt = \int_a^b -Q~dx + P~dy. Here ⋅ is the dot product, and \mathbf'(t)^\perp = (y'(t),-x'(t)) is the clockwise perpendicular of the velocity vector \mathbf'(t)=(x'(t),y'(t)). 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, 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, is a function, and L\subset U is a curve of finite length, parametrized by , where . The line integral :\int_L f(z)\,dz may be defined by subdividing the interval 'a'', ''b''into and considering the expression :\sum_^ f(\gamma(t_k)) \, \gamma(t_k) - \gamma(t_) = \sum_^n f(\gamma_k) \,\Delta\gamma_k. 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, the line integral can be evaluated as an integral of a function of a real variable: :\int_L f(z)\,dz = \int_a^b f(\gamma(t)) \gamma'(t)\,dt. When is a closed curve (initial and final points coincide), the line integral is often denoted \oint_L f(z)\,dz, sometimes referred to in engineering as a ''cyclic integral''. The line integral with respect to the conjugate complex differential \overline is defined to be :\int_L f(z) \overline := \overline = \int_a^b f(\gamma(t)) \overline\,dt. 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 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 wel ...
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 about 0, parametrized by z(''t'') = ''e''''it'' with ''t'' in , 2πusing the
complex exponential The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
. Substituting, we find: :\begin \oint_L \frac \,dz &= \int_0^ \frac ie^ \,dt = i\int_0^ e^e^\,dt \\ &= i \int_0^ dt = i(2\pi-0)= 2\pi i. \end 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 ...
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 wel ...
.


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 f(z) has real and complex parts equal to the line integral and the flux integral of the vector field corresponding to the conjugate function \overline. Specifically, if \mathbf (t) = (x(t), y(t)) parametrizes ''L'', and f(z)=u(z)+iv(z) corresponds to the vector field \mathbf(x,y) = \overline = (u(x + iy), -v(x + iy)), then: :\begin \int_L f(z)\,dz &= \int_L (u+iv)(dx+i\,dy) \\ &= \int_L (u,-v)\cdot (dx,dy) + i\int_L (u,-v)\cdot (dy,-dx) \\ &= \int_L \mathbf(\mathbf)\cdot d\mathbf + i\int_L \mathbf(\mathbf)\cdot d\mathbf^\perp. \end By Cauchy's theorem, the left-hand integral is zero when f(z) 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 differenti ...
) for any smooth closed curve L. Correspondingly, by Green's theorem, the right-hand integrals are zero when \mathbf = \overline is irrotational ( curl-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 eq ...
(
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 t ...
-free). In fact, the Cauchy-Riemann equations for f(z) 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 L is given by the integral \textstyle\frac \int_L \overline \, dz. 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 i ...
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 chemistr ...
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 qu ...
s in quantum scattering theory.


See also

* Divergence theorem *
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 * Surface integral *
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 ...


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
{{Authority control Complex analysis Vector calculus