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Visualising the exterior derivative ?

  1. May 28, 2013 #1
    Is there a geometric interpretation/visualization of the exterior derivative, at least in the case of three dimensions ?

    Suppose we have a 1-form on a 3-dimensional basis {dx1, dx2,dx3} :

    [tex]\displaystyle{\omega =f_{i}dx^{i}}[/tex]

    with a set of real-valued coefficients f. The exterior derivative is then, by definition, the 2-form

    [tex]\displaystyle{d\omega =\sum_{i,j}\frac{\partial f_{j}}{\partial x^{i}}dx^{i}\wedge dx^{j}}[/tex]

    Intuitively, a 1-form in three dimensions is an oriented line segment, a 2-form an oriented surface element. So, is there an intuitive way to "visualize" the exterior derivative operation, in a geometric sense ? Can it be visualised roughly as "wedging" with an orthogonal basis element in a way that orientation is preserved, thereby increasing the degree by one ? This would explain how, for example, the exterior derivative turns an oriented line segment into an oriented surface element.

    I am fine with the abstract definitions of the operators, and its connections to the usual div/grad/curl, but it would be very helpful to have some way to intuitively visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations; my problem is that, just by looking at dF=0 and d*F=uJ it is very hard to visualise what this actually implies in a geometric sense.
     
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  3. May 28, 2013 #2

    WannabeNewton

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  4. May 28, 2013 #3
    That is excellent, thank you very much WannabeNewton. I actually have Gravitation (MTW) here, but haven't read it yet. I am only doing this as a hobby, so I'm not a physics student and thus have to first nail the maths :)
     
  5. May 28, 2013 #4

    WannabeNewton

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    Ah I see. What GR text are you using now then, if not MTW (if you don't mind me asking)?
     
  6. May 28, 2013 #5
    The GR text I studied from is not in English ( I grew up bilingually ). It is "Allgemeine Relativitätstheorie" by Prof Thorsten Fliessbach of the University of Siegen in Germany. Unlike MTW it is very much focused on coordinate-dependent calculations, and does not use differential forms. I also own a copy of Wald's textbook, as well as Carroll's "Spacetime and Geometry". I have yet to study these; working two full-time jobs leaves me with little spare time.

    Currently I am labouring through Darlings "Differential Forms and Connections", which is not easy for me as a layman since it is more abstract than I am used to. But it does present things in a very rigorous way, and I do get the main concepts if not all the mathematical details.
     
  7. May 28, 2013 #6

    WannabeNewton

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    Ah ok. Well I have personally not gone through much of MTW either; I have almost exclusively worked through only Wald and Carroll (so I am now trying to go through different texts now like Padmanabhan). I have never heard of Darling's text but you might try Spivak's "Calculus on Manifolds" as it is concise and has tons of diagrams. I wish you the best of luck with regards to your self-studies, in the face of your two jobs.
     
  8. May 28, 2013 #7
    Thanks for the prompt reply, btw. I am quite new here; I have been very active on a number of other forums, one of which I am moderating, but to be honest as of late, and since my own knowledge and understanding is gradually building up, I feel as if I am starting to "outgrow" those a little. I am sick of having to deal with cranks and getting every serious discussion derailed; it is too much of a distraction from making real progress. This forum seems much more rigorous and focused.

    My own area of personal interest is geometrodynamics / differential geometry in general, and GR in particular.
     
  9. May 28, 2013 #8
    Yes, I have heard of it, thank you for the recommendation.
    Between Wald and Carroll, which one would you think is better suited to a layman such as myself ? Just be leaving through them I would think probably Carroll.
     
  10. May 28, 2013 #9

    WannabeNewton

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    If you can get access to both, I would say try to use them in conjunction since you can benefit greatly from doing end of chapter problems in both texts. If you can only get access to one then yeah start out with Carroll.
     
  11. May 28, 2013 #10
    Yes, I own both texts, so I can use them in conjunction.
     
  12. May 28, 2013 #11

    WannabeNewton

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    Sounds like a plan then!
     
  13. May 28, 2013 #12
    Thanks again for your help. I might need to pick your brains with a few other bits and pieces; like I said, this looks like a good forum to me.
     
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