Everything about Line Integral totally explained
In
mathematics, a
line integral (sometimes called a
path integral or
curve integral) is an
integral where the
function to be integrated is evaluated along a
curve. Various different line integrals are in use. In the case of a closed curve it's also called a
contour integral.
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 formulas in physics (for
example,
) have natural continuous analogs in terms of line integrals (
). The line integral finds the
work done on an object moving through an electric or gravitational field, for example.
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
field along a given curve.
Line integral of a scalar field
For some
scalar field f :
U ⊆
Rn R, the line integral along a
curve C ⊂
U is defined as
»
where we use the fact that any complex number
z can be written as
reit where
r is the
modulus of
z. On the unit circle this is fixed to 1, so the only variable left is the angle, which is denoted by
t.
This answer can be also verified by the
Cauchy integral formula.
Quantum mechanics
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.
Further Information
Get more info on 'Line Integral'.
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