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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a line integral is an
integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

integral
where the
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 (geo ...

curve
. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''
contour integral In the mathematical field of complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathemati ...
'' 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

scalar field
or a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

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 length is the distance between two points along a section of 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. In ...

arc length
or, for a vector field, the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
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 * Work (physics), the product of ...

work
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 * Work (physics), the product of ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

scalar field
f\colon U\to\R where U \subseteq \R^n, the line integral along a
piecewise smooth
piecewise smooth
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 (geo ...

curve
\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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Arc length is the distance between two points along a section of 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. In ...

arc length
. 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 Cross-sectional data, or a cross section of a study population, in statistics and econometrics is a type of data set, data collected by observing many subjects (such as individuals, firms, countries, or regions) at the one point or period of time. T ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 by a polygonal path by introducing a straight line piece between each of the sample points and . We then label the distance between each of the 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 (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of the
sum
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 In mathematics, the mean value theorem states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, secant through its endpoints. ...

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)\ \Delta t. 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 In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
F: ''U'' ⊆ R''n'' → R''n'', the line integral along a
piecewise smooth
piecewise smooth
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 (geo ...

curve
''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'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
, and r: 'a'', ''b''→ ''C'' is a
bijective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely cl ...
to the line. Line integrals of vector fields are independent of the parametrization r in
absolute value In , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

absolute value
, 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 design ...
. 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 Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the
musical isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
(which takes the vector field to the corresponding
covector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

parameter
) 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 c ...
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, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 In mathematics, the mean value theorem states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, secant through its endpoints. ...

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 Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

gradient
of a
scalar field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

scalar field
''G'' (i.e. if F is
conservative Conservatism is an aesthetic Aesthetics, or esthetics (), is a branch of philosophy that deals with the nature of beauty and taste (sociology), taste, as well as the philosophy of art (its own area of philosophy that comes out of aest ...
), that is, :\mathbf = \nabla G , then by the multivariable chain rule the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
of the
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
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 * Work (physics), the product of ...

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 curve

For a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

vector field
\mathbf F\colon U\subseteq\R^2\to\R^2, , the line integral across a curve ''C'' ⊂ ''U'', also called the , is defined in terms of a 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
, 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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
and
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...
of complex numbers. Suppose ''U'' is an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
as the lengths of the subdivision intervals approach zero. If the parametrization is
continuously differentiable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 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 in the ...
may be used to equate the line integral of an
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
computes the integral in terms of the singularities.


Example

Consider the function ''f''(''z'') = 1/''z'', and let the contour ''L'' be the counterclockwise
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

unit circle
about 0, parametrized by z(''t'') = ''e''''it'' with ''t'' in , 2πusing the
complex exponential Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

complex exponential
. 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 of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...
.


Relation of complex line integral and line integral of vector field

Viewing complex numbers as 2-dimensional
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
, 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 Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
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 of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis ...
) for any smooth closed curve L. Correspondingly, by
Green's theorem In vector calculus, Green's theorem relates a line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
, the right-hand integrals are zero when \mathbf = \overline is
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between ...
(
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
-free) and
incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among ...

incompressible
(
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

divergence
-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 In vector calculus, Green's theorem relates a line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
, 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 Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic p ...
of
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
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 Quantum mechanics is a fundamental Scientific theory, 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 ...
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 Light or visible light is electromagnetic radiation within the portion of the electromagnetic ...

scattering
theory.


See also

*
Divergence theorem In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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 Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...
*
Methods of contour integration Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to: *Scient ...
* Nachbin's theorem *
Surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Surface integral
*
Volume elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Volume integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...


References


External links

* *
Khan Academy Khan Academy is an American non-profit education Education is the process of facilitating learning, or the acquisition of knowledge, skills, value (ethics), values, morals, beliefs, habits, and personal development. Educational methods ...
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