Can Circulation be Computed in a System with an Open Path of Integration?

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Discussion Overview

The discussion revolves around the concept of circulation in vector fields, specifically whether circulation can be computed along an open path of integration. Participants explore the definitions and implications of circulation and line integrals in both open and closed contexts.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that circulation is typically defined only for closed paths, questioning whether it is appropriate to refer to line integrals over open paths as circulation.
  • Others propose that while line integrals over open paths are well-defined, they should not be labeled as circulation.
  • There is a suggestion that the term "flux" is used for surface integrals over open and closed surfaces, while no specific name exists for line integrals over open curves.
  • A participant raises a question about the relevance of calling a straight line integral "circulation" if it does not pertain to circular motion.

Areas of Agreement / Disagreement

Participants generally agree that circulation is defined for closed paths, but there is disagreement on the implications of this definition and whether open paths can be associated with the concept of circulation.

Contextual Notes

The discussion highlights the dependence on definitions and the potential confusion surrounding the terminology used in vector calculus, particularly regarding circulation and line integrals.

Jhenrique
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Exist circulation in a system/circuit/vector field/anything where the path of circulation is open?

Is possible to compute the circulation of a system/circuit/vector field/anything with an open path of integration?
 
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Do you have a specific mathematical definition of circulation in mind? Usually circulation is defined only for closed paths. If paths are open, one would no longer call it circulation.
 
If that's the definition, then "circulation" the word should only mean for closed paths. Of course you can take a line integral over an open path, that's perfectly well defined, but you just don't call it "circulation".

Are you having trouble with the definition of the word "circulation"? Or is there a math problem you're confused about?
 
Matterwave said:
If that's the definition, then "circulation" the word should only mean for closed paths. Of course you can take a line integral over an open path, that's perfectly well defined, but you just don't call it "circulation".

Are you having trouble with the definition of the word "circulation"? Or is there a math problem you're confused about?

The surface integral of a vector field over a open surface has name: "flux"
The surface integral of a vector field over a closed surface has name: "flux"

The line integral of a vector field over a closed curve has name: "circulation"
But the line integral of a vector field over a open curve hasn't name...
 
Is that a fundamental problem you think? Say you have a straight line integral, would you still like to call this integral "circulation" even though it has nothing to do with anything circular?
 

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