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Scalar line integrals can be calculated using Equation \ref{eq12a}; vector line integrals can be calculated using Equation \ref{lineintformula}. Two key concepts expressed in terms of line integrals are flux and circulation.
With line integrals we will start with integrating the function \(f\left( {x,y} \right)\), a function of two variables, and the values of \(x\) and \(y\) that we’re going to use will be the points, \(\left( {x,y} \right)\), that lie on a curve \(C\).
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.
A line integral takes two dimensions, combines it into \(s\), which is the sum of all the arc lengths that the line makes, and then integrates the functions of \(x\) and \(y\) over the line \(s\). Definition of a Line Integral
There are two types of line integrals: scalar line integrals and vector line integrals. Scalar line integrals are integrals of a scalar function over a curve in a plane or in space. Vector line integrals are integrals of a vector field over a curve in a plane or in space.
Line integrals (also referred to as path or curvilinear integrals) extend the concept of simple integrals (used to find areas of flat, two-dimensional surfaces) to integrals that can be used to find areas of surfaces that "curve out" into three dimensions, as a curtain does.
Use a line integral to compute the work done in moving an object along a curve in a vector field. Describe the flux and circulation of a vector field. We are familiar with single-variable integrals of the form ∫b af(x)dx, where the domain of integration is an interval [a, b].
The line integral of \(f\) with respect to \(x\) is, \[\int\limits_{C}{{f\left( {x,y} \right)\,dx}} = \int_{{\,a}}^{{\,b}}{{f\left( {x\left( t \right),y\left( t \right)} \right)x'\left( t \right)\,dt}}\]
With Line Integrals we will be integrating functions of two or more variables where the independent variables now are defined by curves rather than regions as with double and triple integrals. We will also investigate conservative vector fields and discuss Green’s Theorem in this chapter.
The integral Z C F dr= Z b a F(r(t)) r0(t) dt is called theR line integral of F along C. We think of F(r(t)) r0(t) as power and C F dras the work. Even so F and rare column vectors, we write in this lecture [F 1(x);:::;F n(x)] and r0= [x0 1;:::;x 0 n] to avoid clutter. Mathematically, F: Rn!Rn