Line integral curiosity/confusion

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Discussion Overview

The discussion revolves around the visualization of line integrals in calculus and the implications of a line integral being equal to zero over a curve. Participants explore whether this condition necessitates that the function itself must be zero along the curve.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about how to visualize line integrals graphically, questioning if they can be easily represented.
  • Another participant suggests that visualizing line integrals in terms of work done by a force can aid understanding.
  • A participant argues that a line integral being zero does not imply that the function must be zero along the curve, citing the possibility of positive and negative contributions canceling each other out.
  • It is noted that knowing the integral over a specific curve provides limited information about the function elsewhere.
  • One participant seeks an example function and curve that illustrates the discussed concepts.
  • A later reply advises against relying on visual thinking, suggesting a focus on definitions instead.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the visualization of line integrals or the implications of a zero integral. Multiple competing views remain regarding the interpretation of the integral's value and its relationship to the function.

Contextual Notes

Participants express uncertainty about the visualization of line integrals and the conditions under which a zero integral relates to the function's values along the curve.

optinyx
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Heya!

I was hoping someone could clear this up for me: how would a line integral be represted graphically? I've always liked calculus because it's easy to visualize (almost all the problems have graphs associated with them) - but I don't quite get how to visualize a line integral. Or is it a concept that can't be easily visualized?

any help would, well.. help. :redface:

I've also been pondering something my teacher brought up last week:
Let's say the line integral of f(x,y,z) over a curve C is equal to zero, then would it have to be true that f(x,y,z) = 0 on C?

Thank you! I eagerly await a response. :biggrin:
 
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its easier to visualize it with physics... think of it in terms of Work done under a force. On the other q, you mean for ALL C's?, because if C is closed, then is not necesarily true, think of it in terms of the work done :)
 
No, the fact that the line integral of f over a curve is 0 does NOT mean that the function must be identically 0 for two reasons:

First it is possible that f is positive on part of the curve and negative on another part so that the two parts cancel.
Also, knowing the integral on a particular curve only gives you information about f ON THAT CURVE. Even if f were 0 on the curve itself, it might be non-zero elsewhere.
 
okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

If I understand though, then that's what hallsofivy was saying right? - about it being positive on one part and negative on the other? I think I get it.

Could anyone give me an example function, f(x,y,z), and a curve where this would be the case?

Thank you for all the help.
 
optinyx said:
okay, I think I understand now. thinking of it in terms of work done makes it easier, like in physics.. when you push a block ten feet in one direction and then ten feet back to where you started, they say the total work done was zero because the total distance was zero. Or am I completely off my rocker? (it's been a while since general physics).

displacement*
 
Last edited:
try to stop thinking visually, it doesn't help in the real world, or the world of mathematics, just work with the definitions.
 

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