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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 .
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.
where is an arbitrary bijectiveparametrization 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 of the curve (i.e., a differential length of ). Line integrals of scalar fields over a curve do not depend on the chosen parametrization r of .[2]
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.
Derivation
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 [ti−1, ti] of length Δt = (b − a)/n, then r(ti) denotes some point, call it a sample point, on the curve C. We can use the set of sample points {r(ti): 1 ≤ i ≤ n} to approximate the curve C as a polygonal path by introducing the straight line piece between each of the sample points r(ti−1) and r(ti). (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 Δsi. The product of f(r(ti)) and Δsi can be associated with the signed area of a rectangle with a height and width of f(r(ti)) and Δsi, 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
where · is the dot product, and r: [a, b] → C is a regular parametrization (i.e: ) 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 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.[2]
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 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 parametert) into n intervals of length Δt = (b − a)/n. Letting ti be the ith point on [a, b], then r(ti) gives us the position of the ith point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their displacement vectors, Δri. As before, evaluating F 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 F on C. Letting the size of the partitions go to zero gives us a sum
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.
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 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.[3]
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).
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 planeC, 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 = t0 < t1 < ... < tn = 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:
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[4]
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