Line integral with respect to arc length

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Discussion Overview

The discussion revolves around the concept of line integrals with respect to arc length, specifically addressing the relationship between the parameters involved in the integral, such as x, y, and s. Participants explore whether x and y should be understood as functions of the arc length parameter s or if they depend on another parameter entirely.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant notes that in a line integral with respect to arc length, the expression f(x, y)ds indicates that ds relates to the arc length function s, leading to questions about the parameterization of x and y.
  • Another participant asserts that x and y must ultimately be functions of s, although they can be expressed in terms of another parameter, referencing the relationship ds² = dx² + dy².
  • A different participant proposes that if s is defined as an increasing function of t, then x and y can be expressed as functions of t through the inverse relationship t = t(s).
  • However, a later reply challenges the assumption that s must be an increasing function of t, suggesting that t could be arbitrary.

Areas of Agreement / Disagreement

The discussion contains multiple competing views regarding the relationship between the parameters s, t, x, and y. There is no consensus on whether s must be an increasing function of t or if x and y can be defined in other ways.

Contextual Notes

Participants express uncertainty about the nature of the parameterization and the implications of different parameter choices on the line integral, highlighting potential dependencies and assumptions that remain unresolved.

Castilla
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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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Ultimately x and y must be functions of s, although they be expressed in terms of another parameter.
Specifically ds2 = dx2 + dy2.
 
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?
 
If t is an arbitrary parameter, it does not necessarily follow that s is an increasing function of t.
 

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