Is the ordinary integral a special case of the line integral?

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Discussion Overview

The discussion revolves around whether the ordinary integral over the real line can be considered a special case of the line integral, specifically when the line is straight and the field is defined only along that line. The scope includes theoretical considerations and conceptual clarifications regarding the definitions and properties of fields and integrals.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that the ordinary integral can be viewed as a special case of the line integral under certain conditions.
  • Another participant agrees with this perspective but questions the validity of calling it a field if it is only defined along a 1D line.
  • A further contribution states that a field defined on a line must satisfy specific requirements to be considered a field, providing an example involving the real and imaginary axes.
  • Concerns are raised about the inherent differences between ordinary integrals and line integrals, particularly regarding the measure of lines in higher dimensions.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the ordinary integral as a special case of the line integral, with some agreeing while others raise questions about the definitions and properties involved. The discussion remains unresolved regarding the fundamental differences between the two types of integrals.

Contextual Notes

There are limitations related to the definitions of fields and the measure of lines in higher dimensions that are not fully explored, leaving some assumptions unaddressed.

LucasGB
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Can I consider the ordinary integral over the real line a special case of the line integral, where the line is straight and the field is defined only along the line?
 
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You certainly can do it that way, if you want to.
 
But can it be called a field if it is only defined along a 1D line?
 
If the field "defined" on the line actually satisfies the requirements to be a field, yes. Say, for example, if you're in C and choose your line to be R, then you're good. But if you choose the imaginary axis to be your line and do not change the definition of multiplication then you do not have a field.

But as for your question, I'm not sure. I think the two may be inherently different. If we're in R^2 the regular integral of any function along a line will be 0 because lines have measure 0, whereas it would be silly to have line integrals be 0 on all lines.
 

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