What is the definition of a line integral?

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Discussion Overview

The discussion revolves around the definition of a line integral, specifically the expression int_aroundc[P(x,y)dx + Q(x,y)dy]. Participants explore the meaning and components of this mathematical concept, touching on aspects of parameterization and limits.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asks for a clear definition of the line integral, presenting the integral notation as a starting point.
  • Another participant questions the clarity of the initial query, suggesting that the notation itself may already constitute a definition.
  • A third participant expresses uncertainty, mentioning that there might be additional definitions involving limits and "zeta," though the context is unclear.
  • One participant proposes that defining the line integral involves choosing a parameterization for the path, which transforms it into a standard integral with hidden limits.

Areas of Agreement / Disagreement

There is no consensus on the definition of a line integral, as participants express differing views on its components and the necessity of additional definitions.

Contextual Notes

Participants reference concepts such as parameterization and limits, but these aspects remain somewhat ambiguous and are not fully resolved in the discussion.

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How would you DEFINE the line integral:

int_aroundc[P(x,y)dx + Q(x,y)dy]
 
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What's that supposed to mean? Isn't it already defined as you have written it?
 
That what I thought but I think there's some definition with limits and zeta and stuff...
 
You would define such an integral by choosing a parameterization for the path. Then it reduces to a regular integral in terms of the parameter. I don't know what "zeta" has to do with it but all the limits are hidden in the definition of the regular integral.
 

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