Can Different Parameterizations Yield the Same Line Integral Value?

In summary, the line integral for the parameterized curve is the same regardless of which parameterization is used. This can be shown by using the chain rule and noting that the second integral in both cases is the same due to a change of variables.
  • #1
dgiroux48
1
0
1. Consider the curve c= (x(t),y(t),z(t)) in space as t varies over [0, T ]. We could also parameterize this curve by
c= x(τ^2 ),y(τ^2 ),z(τ^2) τ ∈ [0, sqrt(T)].
Show that one obtains the same value for the line integral using either parameterization.

The line integral is just the integral for the arc legnth of the parameterized curve. I understand the concept intuitively, I just don't really know how to derive it.

(I don't know how to write out integral signs and all here but the formula can be found at this site Pauls Online Notes : Calculus III - Line Integrals - Part I)Thanks!
 
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  • #2
Looks to me like it is just a matter of using the chain rule.

We are saying that (x(t), y(t), z(t)), for [tex]0\le t\le T[/tex] and [tex](x(\tau^2), y(\tau^2), z(\tau^2))[/tex] are parameterizations of the same curve: [tex]t= \tau^2[/tex] so that [tex]dt= 2\tau d\tau[/tex].
 
  • #3
1- $\int_{C_1} f(x,y,z)ds=\int_0^T f(x(t),y(t),z(t))||r(t)||dt$.

2- On the other hand, $\int_{C_2}f(x,y,z)ds=\int_0^{\sqrt{T}} f(x(t^2),y(t^2),z(t^2))||r(t^2)|| |2t| dt$, where the $|2t|$ comes from the computation of $||r(t^2)||$, since it's a composition of functions.

So now, all you have to do is to note that the second integral in 1 is the same as the second integral in 2, because it is the result of changing the variables.
 

FAQ: Can Different Parameterizations Yield the Same Line Integral Value?

What is line integral parameterization?

Line integral parameterization is a mathematical tool used to calculate the integral of a scalar or vector-valued function over a curve in a multi-dimensional space. It involves defining a parameterized curve and then using this parameterization to evaluate the integral along the curve.

Why is line integral parameterization important?

Line integral parameterization is important because it allows us to calculate the integral of a function over a curve, which is often necessary in physics and engineering applications. It also provides a more systematic approach to solving such integrals compared to other methods.

What is the difference between a line integral and a regular integral?

A line integral involves integrating a function over a curve, while a regular integral involves integrating a function over a single variable. In other words, a line integral takes into account the direction and path of the curve, while a regular integral does not.

How is line integral parameterization related to vector calculus?

Line integral parameterization is a concept in vector calculus, as it involves integrating vector-valued functions over a curve. It is used in many vector calculus applications, such as calculating work done by a force along a curve or finding the center of mass of a lamina.

What are some common applications of line integral parameterization?

Line integral parameterization has many applications in physics and engineering, including calculating work, flux, and circulation of vector fields, finding the mass and center of mass of objects, and solving certain differential equations. It is also used in computer graphics and animation to create smooth curves and surfaces.

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