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In mathematics, a **line integral** is an integral where the function to be integrated is evaluated along a curve.^{ [1] } The terms *path integral*, *curve integral*, and *curvilinear integral* are also used; * contour integral * is used as well, although that is typically reserved for line integrals in the complex plane.

- Vector calculus
- Line integral of a scalar field
- Line integral of a vector field
- Path independence
- Applications
- Flow across a curve
- Complex line integral
- Example
- Relation of complex line integral and line integral of vector field
- Quantum mechanics
- See also
- References
- External links

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as , have natural continuous analogues in terms of line integrals, in this case , which computes the work done on an object moving through an electric or gravitational field **F** along a path .

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.

For some scalar field where , the line integral along a piecewise smooth curve is defined as^{ [2] }

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

The function f is called the integrand, the curve is the domain of integration, and the symbol *ds* may be intuitively interpreted as an elementary arc length. Line integrals of scalar fields over a curve do not depend on the chosen parametrization **r** of .^{ [3] }

Geometrically, when the scalar field f is defined over a plane (*n* = 2), its graph is a surface *z* = *f*(*x*, *y*) in space, and the line integral gives the (signed) cross-sectional area bounded by the curve and the graph of f. See the animation to the right.

For a line integral over a scalar field, the integral can be constructed from a Riemann sum using the above definitions of f, C and a parametrization **r** of C. This can be done by partitioning the interval [*a*, *b*] into n sub-intervals [*t*_{i−1}, *t*_{i}] of length Δ*t* = (*b* − *a*)/*n*, then **r**(*t*_{i}) denotes some point, call it a sample point, on the curve C. We can use the set of sample points {**r**(*t*_{i}) : 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**(*t*_{i−1}) and **r**(*t*_{i}). We then label the distance between each of the sample points on the curve as Δ*s*_{i}. The product of *f*(**r**(*t*_{i})) and Δ*s*_{i} can be associated with the signed area of a rectangle with a height and width of *f*(**r**(*t*_{i})) and Δ*s*_{i}, respectively. Taking the limit of the sum of the terms as the length of the partitions approaches zero gives us

By the mean value theorem, the distance between subsequent points on the curve, is

Substituting this in the above Riemann sum yields

which is the Riemann sum for the integral

For a vector field **F** : *U* ⊆ **R**^{n} → **R**^{n}, the line integral along a piecewise smooth curve *C* ⊂ *U*, in the direction of **r**, is defined as^{ [2] }

where · is the dot product, and **r**: [*a*, *b*] → *C* is a bijective parametrization of the curve *C* such that **r**(*a*) and **r**(*b*) give the endpoints of *C*.

A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line.

Line integrals of vector fields are independent of the parametrization **r** in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.^{ [3] }

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.

The line integral of a vector field can be derived in a manner very similar to the case of a scalar field, but this time with the inclusion of a dot product. Again using the above definitions of **F**, C and its parametrization **r**(*t*), we construct the integral from a Riemann sum. We partition the interval [*a*, *b*] (which is the range of the values of the parameter t) into n intervals of length Δ*t* = (*b* − *a*)/*n*. Letting *t _{i}* be the ith point on [

By the mean value theorem, we see that the displacement vector between adjacent points on the curve is

Substituting this in the above Riemann sum yields

which is the Riemann sum for the integral defined above.

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

then by the multivariable chain rule the derivative of the composition of *G* and **r**(*t*) is

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

In other words, the integral of **F** over *C* depends solely on the values of *G* at the points **r**(*b*) and **r**(*a*), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called *path independent*.

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*.^{ [4] }

For a vector field , **F**(*x*, *y*) = (*P*(*x*, *y*), *Q*(*x*, *y*)), the **line integral across a curve***C* ⊂ *U*, also called the *flux integral*, is defined in terms of a piecewise smooth parametrization **r**: [*a*,*b*] → *C*, **r**(*t*) = (*x*(*t*), *y*(*t*)), as:

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

The flow is computed in an oriented sense: the curve C has a specified forward direction from **r**(*a*) to **r**(*b*), and the flow is counted as positive when **F**(**r**(*t*)) is on the clockwise side of the forward velocity vector **r'**(*t*).

In complex analysis, the line integral is defined in terms of multiplication and addition of complex numbers. Suppose *U* is an open subset of the complex plane **C**, *f* : *U* → **C** is a function, and is a curve of finite length, parametrized by *γ* : [*a*,*b*] → *L*, where *γ*(*t*) = *x*(*t*) + *iy*(*t*). The line integral

may be defined by subdividing the interval [*a*, *b*] into *a* = *t*_{0} < *t*_{1} < ... < *t*_{n} = *b* and considering the expression

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

If the parametrization γ is continuously differentiable, the line integral can be evaluated as an integral of a function of a real variable:^{ [2] }

When L 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^{ [5] } to be

The line integrals of complex functions can be evaluated using a number of techniques. The most direct is to split into real and imaginary parts, reducing the problem to evaluating two real-valued line integrals. The Cauchy integral theorem may be used to equate the line integral of an analytic function to the same integral over a more convenient curve. It also implies that over a closed curve enclosing a region where *f*(*z*) is analytic without singularities, the value of the integral is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities.

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 [0, 2π] using the complex exponential. Substituting, we find:

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

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) for any smooth closed curve L. Correspondingly, by Green's theorem, the right-hand integrals are zero when is irrotational (curl-free) and incompressible (divergence-free). In fact, the Cauchy-Riemann equations for are identical to the vanishing of curl and divergence for **F**.

By Green's theorem, the area of a region enclosed by a smooth, closed, positively oriented curve is given by the integral This fact is used, for example, in the proof of the area theorem.

The path integral formulation of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function *of* a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.

In physics, **potential energy** is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ*f*(*p*) of a function *f* at a point *p* measures by how much the average value of *f* over small spheres or balls centered at *p* deviates from *f*(*p*).

In vector calculus, **Green's theorem** relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.

**Scalar potential**, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In the mathematical field of complex analysis, **contour integration** is a method of evaluating certain integrals along paths in the complex plane.

In geometry and algebra, the **triple product** is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued **scalar triple product** and, less often, the vector-valued **vector triple product**.

**Arc length** is the distance between two points along a section of a curve.

In calculus, the **Leibniz integral rule** for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

The following are important identities involving derivatives and integrals in vector calculus.

A **parametric surface** is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

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 a generalization of the fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

In differential calculus, there is no single uniform **notation for differentiation**. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.

**Two-dimensional space** is a geometric setting in which two values are required to determine the position of an element. The set ℝ^{2} of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.

In differential geometry, the **radius of curvature**, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

**Stokes' theorem**, also known as **Kelvin–Stokes theorem** after Lord Kelvin and George Stokes, the **fundamental theorem for curls** or simply the **curl theorem**, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the *flux of its curl* through the enclosed surface.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Kwong-Tin Tang (30 November 2006).
*Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms*. Springer Science & Business Media. ISBN 978-3-540-30268-1. - 1 2 3 "List of Calculus and Analysis Symbols".
*Math Vault*. 2020-05-11. Retrieved 2020-09-18. - 1 2 Nykamp, Duane. "Line integrals are independent of parametrization".
*Math Insight*. Retrieved September 18, 2020. - ↑ "16.2 Line Integrals".
*www.whitman.edu*. Retrieved 2020-09-18. - ↑ Ahlfors, Lars (1966).
*Complex Analysis*(2nd ed.). New York: McGraw-Hill. p. 103.

- "Integral over trajectories",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Khan Academy modules:
- Path integral at PlanetMath .
- Line integral of a vector field – Interactive

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