Line integral with respect to arc length

In summary, a line integral with respect to arc length involves using the arc length function s and taking differences in the sums that approach the integral. Ultimately, x and y must be functions of s, even if they are expressed in terms of another parameter. This can be seen through the equation ds^2 = dx^2 + dy^2 and the relationship between s and t as an increasing function. However, if t is an arbitrary parameter, it is not necessary for s to be an increasing function of t.
  • #1
Castilla
241
0
In a line integral with respect to arc length, we have something like f(x, y)ds "inside" the integral sign.

The ds tells us that we are working with the arc length function s, taking diferences (s_K+1 - s_k) in the sums that tend to the line integral.

Question: do we shall understand that x = x(s), y = y(s), or x and y are functions of another parameter? In this last case, this means that we work with two different parameters in the sums that tend to the integral?

Thanks for your help.
 
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  • #2
Ultimately x and y must be functions of s, although they be expressed in terms of another parameter.
Specifically ds2 = dx2 + dy2.
 
  • #3
I think I have understood.

s = s(t), but s is an increasing function, so we have its inverse t = t(s). Then
x = x(t) = x(t(s)), y = y(t) = y(t(s)). Is this ok?
 
  • #4
If t is an arbitrary parameter, it does not necessarily follow that s is an increasing function of t.
 

1. What is a line integral with respect to arc length?

A line integral with respect to arc length is a type of integral that is used to calculate the length of a curve in a given space. It involves integrating a function along a specified curve with respect to the arc length parameter.

2. How is a line integral with respect to arc length different from a regular line integral?

A line integral with respect to arc length is different from a regular line integral in that the former takes into account the actual length of the curve, whereas the latter only considers the coordinates of the curve. This makes it a more accurate measure of length for curved paths.

3. What is the formula for calculating a line integral with respect to arc length?

The formula for calculating a line integral with respect to arc length is ∫C f(x,y) * ds, where C is the curve and ds is the infinitesimal element of arc length along the curve. This can also be written as ∫ab f(x(t), y(t)) * √(x'(t)^2 + y'(t)^2) dt, where a and b are the start and end points of the curve, and x(t) and y(t) are the parametric equations of the curve.

4. What are some real-world applications of line integrals with respect to arc length?

Line integrals with respect to arc length have many real-world applications, such as calculating the work done by a force along a curved path, finding the mass of a wire with varying density, and determining the length of a wire under tension. They are also used in physics and engineering to calculate the displacement of a particle along a curved path and to determine the magnitude and direction of a magnetic field.

5. Are there any limitations to using line integrals with respect to arc length?

Yes, there are some limitations to using line integrals with respect to arc length. One limitation is that it can only be used for smooth curves, meaning that the curve must have a well-defined tangent vector at every point. Additionally, it can be difficult to find the parametric equations of a curve in some cases, making it challenging to compute the line integral with respect to arc length. Lastly, this type of integral may not always give a unique solution, as there may be multiple curves with the same arc length between two points.

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