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A Exact vs Closed forms

  1. Jan 28, 2017 #1
    (I am a mechanical engineer, trying to make up for a poor math education)'

    I understand that:
    1. A CLOSED form is a differential form whose exterior derivative is 0.0.
    2. An EXACT form is the exterior derivative of another form.

    And it stops right there. I am teaching myself differential forms. And as I ratchet up my understanding, I encounter these words--closed and exact--but I am not yet comfortable with their use.

    As a result, I MEMORIZE the two words and their definitions. I do this to get through some rough spots as I continue to learn. But now I am at a point where I am hungering to know why these words matter.

    It would help me, I think, if I knew WHY those words were used. In other words, I just just as easily have written:
    1. A TOMATO form is a differential form whose exterior derivative is 0.0.
    2. A POTATO form is the exterior derivative of another form.
    Please forgive my sarcasm, but I am trying to get BEYOND memorizing the words. Why were those two words chosen?

    And, if you can, answer in terms of pure theoretical math AND, if possible, with a meaningful (perhaps physical for a mechanical engineer) example.

    For example, I THINK I UNDERSTAND that for the case of 1D integration of a form along a line that is CLOSED (like a closed loop or closed circle), that the signed definite integral of the form from "a" to "a-gain" is zero. Does that word CLOSED have anything to do with a the word describing the form. And is this related to the work done by a conservative force in a closed loop? I am almost at the point of seeing that a closed form can represent a conservative force, and an exact form represents a potential function. However, I cannot disambiguate the words CLOSED and EXACT since they all seem to mean the same thing in physics... I just need to see these two words separated.
    Last edited: Jan 28, 2017
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  3. Jan 28, 2017 #2


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  4. Jan 28, 2017 #3


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    It is so bad that it is funny ... :bow:
  5. Jan 28, 2017 #4


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    An exact form can be derived from a potential, so it is related to conservative forces, eg. the circumstances in which E=gradΦ, or in which B=curlA.

    Every exact form is closed, eg. curl(gradΦ)=0 and div(curlA)=0.

    But is every closed form exact? For example, if we see that div(B)=0, can we infer that B=curlA?

    We cannot because we can produce counterexamples, eg. the Dirac magnetic monopole. This is mentioned eg. in Abanov's notes on differential forms http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/phy680.html or in Deschamps's article http://www-liphy.ujf-grenoble.fr/pagesperso/bahram/biblio/Deschamps1981_dif_forms.pdf.
    Last edited: Jan 28, 2017
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