Line Integrals |
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DefinitionThe process of evaluating a function along a path/curve is line integral as the term named.
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MotivationMuch like double integrals, line integration is a process by which we can integrate multivariable functions. However, with line integral, we integrate the function along a curve. In other words, through line integration, we can find the length of a curve in space. Line integrals are used to calculate work done along a path in a force field, flux thorough a two-dimensional curve . Close line integrals in a physical field gives us insight about the nature (i.e. conservative) of the field.
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Bird's Eye ViewCompared to single variable definite integral, in the line integral, we have a multivariable function integrated along a curvy path. For doing the integral along a line usually, we first have to parameterize the equation of the line/path with a single variable, WHY? From the definition of the line integral ($\int_C f\, ds$) we see that we have a single integral so, we need a single variable which ties up all the information (different coordinate variables) of the path into one, for representing the path of integration! If we have a scalar field $f(x,y,z)$ to be integrated along a curve $C$ parameterized as , \begin{align} x & =g_x(t), y=g_y(t), z=g_z(t) \\\text{or, }\, \vec r & =g_x(t)\hat i+g_y(t)\hat j+g_z(t) \hat k\qquad (a \leq t \leq b)\end{align} then the line integral is- In some cases parametrization may not be needed i.e. if the integration path is along one of the coordinate axes. As an example where the integration path is along $X$ axis, the number of variables reduces to one eliminating the need of further parametrization - In case of vector fields each coordinate is associated with a scalar function. Here we have to put the parametrization information of the curve $C$ into each of that scalar function. And then take dot product of the vector function with the tangent of the curve $\dfrac{d\vec r}{dt}.$ Line integral on a Scalar Field or a Vector field is somewhat similar. Remarks:
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Context of the DefinitionAs we see there are mainly two types of line integrals --
♦ Scalar line integralsHere we have scalar function (let $f(x,y,z)$) and a curve $C$. We will first see the line integral in terms of the sum of values(for insight) then how to evaluate it as an integral. In case of sum method the curve $C$ is chopped up into $n$ small pieces, then the length of that small pieces $\Delta s_i$ multiplied with the value of the function $f(x_i,y_i,z_i)$ at some point($x_i,y_i,z_i$) in that small piece. Then such values of all $n$ small pieces summed up to get the value of the line integral. As we make the pieces very small we get close to the absolute value of line integral of $f(x,y,z)$ along $C$.
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Animation 1: Line in integral in scalar field $f(x,y)$ as sum. |
When the intervals of the arc become infinitesimally small we get the integral form-- Better to see an example -- Evaluate $\int_C (2+x^2y)\, ds$, where $C$ is the upper half ($y\geq 0$) of the unit circle $x^2+y^2=1$.^{[4]} The circle can be parameterized as, One can also do it by directly substituting $y=\sqrt {1-x^2}$ and calculating $ds$ as $\left(\sqrt{1+\dfrac{dy}{dx}}\right)dx$. 0 |
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Animation 2: Line Integration in scalar field $f(x,y)$ as area |
♦ Vector line integralsLet $\vec F(x,y,z)=P(x,y,z)\hat i+Q(x,y,z)\hat j+R(x,y,z)\hat k$ is a continuous vector field in $\mathbb{R}^3$ to be evaluated along a curve $C$. In the spirit of the previous section, $C$ is chopped up into $n$ small subarcs each having a length $\Delta s_i$. In case of vector fields we have to multiply $\Delta s_i$ with the dot product of $\vec F(x_i,y_i,z_i)$ and the unit tangent vector $\vec T(x_i,y_i,z_i)$ at any point $(x_i,y_i,z_i)$ on that subarc. Then all such values for $n$ small subarcs summed up to get the value of the line integral approximately. $$\int_C \vec F\cdot d\vec r \approx \sum_{i=1}^{n} \vec F(x_i,y_i,z_i)\cdot \vec T(x_i,y_i,z_i)\, \Delta s_i$$ 0 |
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Animation 3: Line in integral in vector field $\vec F(x,y)$ as sum. |
When the subarcs become infinitely small the real value of the integral is obtained. if $\vec r(t)$ is the parametric equation of $C$ then $\vec T =\dfrac{\vec r'(t)}{|\vec r'(t)|},\quad \vec r'(t)=\dfrac{d\vec r}{dt}$
Example : Find the line integral in the vector field $\vec F(x,y)=x^2\hat i-xy\hat j$ along a quarter circular path parameterized as $\vec r(t)=\cos t\hat i+\sin t\hat j,\,0\leq t\leq \pi/2$.^{ [4]} Writing $\vec F$ in terms of $t$ :$\vec F(\vec r(t)) \cos^2t\, \hat i-\cos t\sin t\, \hat j$\ So the line integral becomes- 0 |
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Animation 4 : Line integration in vector field of the previous example.(vectors are not to scale) |
♦ Properties of Line Integrals
There is also one type of line integral defined in complex plane named as contour integral. It is has relation with vector line integrals.
Applications1. Finding length or mass of a wire A wire in space is described by the parametric equation- 0 |
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Animation 5 : Parametric equation of helix |
If the mass density in space is $\rho(x,y,z)=1$ then the length and mass values will be same --
2. Calculating work done in a force field Find the work done in a force field $\vec F=(y-x^2)\hat i+(z-y^2)\hat j+(x-z^2)\hat k$ along a path $\vec r(t)=t \hat i+t^2 \hat j+t^3 \hat k,\quad 0\leq t\leq 1$.
If close line integral along any path in a Vector field is zero then the field is a conservative one. In those fields, line integrals depend on the final and initial point, not on the evaluation path of the line integral.
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HistoryLine integrals were invented in the early 19th century to solve problems involving fluid flow, forces, electricity, and magnetism^{[4]}. Pause and PonderOne can see the vector field is composed of scalar functions associated with different directions. More explicitly, in the scalar field, each point in space is associated with some scalar value, but in a vector field, each point in space is associated with some vector. In the case of vector line integral after taking the dot product of the vectors with the tangent of the curve, we get some values, scalar values (similar to the scalar field but along the curve only). Although the initial definitions and context of the application of line integral in Scalar and Vector fields are little different they have similarities in the calculations. With Line Integral you can calculate the work it takes to put a satellite in orbit, the amount of fuel a vehicle ( rovers !) need to go through some specific path over a hilly surface. 0 |
References
Further Reading
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