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Thread: Levi-Civita Density

  1. #1 Levi-Civita Density 
    Administrator Markus Hanke's Avatar
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    I am currently studying Classical Dynamics of Particles and Systems by Marion/Thornton. In the first chapter matrices, vectors and vector calculus are discussed, and in the context of that the vector product in three dimensions is introduced like so :



    with the justification that



    The symbol is herein called the Levi-Civita density. Purely algebraically this all checks out ( I have done the sums myself ), but I was wondering the following :

    1. Why is called a density ? Is there any significance to that ?
    2. While this definition of the vector product works algebraically, it is not at all intuitive due to the presence of aforementioned density. Can someone explain the deeper significance of the Levi-Civita symbol ? Is there any way to visualise this geometrically ? I get that this is a definition in terms of coordinate basis ( unit ) vectors, and that makes sense. I just don't really get the presence of the Levi-Civita symbol, and its true meaning.

    I also don't really understand this bit :



    Again, algebraically there is no problem, but I don't really get the deeper reasoning and significance behind this.
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  2. #2  
    mvb
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    I have never run into any significance in physics to the use of the word "density" in referring to the Levi-Civita symbol. If there is a deeper significance to ijk , it would be that is a fully antisymmetric tensor. It changes sign if you interchange any two adjacent subscripts or if you reverse the order of the subscripts, and it is therefore zero if any two subscripts are the same.

    Caveat: I am not very familiar with the details of general relativity, and there could be something there that I am not aware of. There could be other things in physics I am not aware of that would involve using the symbol as a density, of course. I doubt that there is an especially widespread use however.
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  3. #3  
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    Actually, the very last relation ( cross product between unit vectors ) does make sense - in this instance all the Levi-Civita symbol does is provide the correct signs ( +,-,0 ) in the sum, so that we get the third unit vector, as required.

    I still don't get why the authors call this a density, though.
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  4. #4  
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    Quote Originally Posted by Markus Hanke View Post
    I am currently studying Classical Dynamics of Particles and Systems by Marion/Thornton. In the first chapter matrices, vectors and vector calculus are discussed, and in the context of that the vector product in three dimensions is introduced like so :



    with the justification that



    The symbol is herein called the Levi-Civita density. Purely algebraically this all checks out ( I have done the sums myself ), but I was wondering the following :

    1. Why is called a density ? Is there any significance to that ?
    2. While this definition of the vector product works algebraically, it is not at all intuitive due to the presence of aforementioned density. Can someone explain the deeper significance of the Levi-Civita symbol ? Is there any way to visualise this geometrically ? I get that this is a definition in terms of coordinate basis ( unit ) vectors, and that makes sense. I just don't really get the presence of the Levi-Civita symbol, and its true meaning.

    I also don't really understand this bit :



    Again, algebraically there is no problem, but I don't really get the deeper reasoning and significance behind this.
    I have no idea why it is called a density, and have never heard it called that before. I have always just treated it as a useful device in recalling how to compute cross products, essentially a menumonic device. It is equivalent to a formal determinant and I find it easiest to remember in that form.

    Generally "Levi-Civita" is applied to the connection that occurs in the context Riemannian geometry -- the key point is that a torsion-free manifold admits a unique connection that gives rise to a Riemannian metric and conversely. The cross product is unique to three-space, although there is a generalization of a sort in higher dimensions to a "product" that requires more than just two factors. This is not terribly useful. The more general and useful construction is the Grassman algebra, which gives one the "wedge product" of differential forms, and that turns out to be very useful indeed.

    You might want to look at the treatment of vector calculus in Mike Spivak's little book Calculus on Manifolds. It gives a nice simple treatment of vector calculus and an introduction to differential forms in the context of 3-space. There you will find out that the fundamental theorem of calculus, the "divergence theorem" and "Stokes theorem" and really the same thing (and mathematicians tend to call the generalization Stokes Theorem and let it go at that.
    Last edited by DrRocket; 04-01-2013 at 02:14 AM.
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  5. #5  
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    I have no idea why it is called a density, and have never heard it called that before. I have always just treated it as a useful device in recalling how to compute cross products, essentially a menumonic device. It is equivalent to a formal determinant and I find it easiest to remember in that form.
    Thanks...that sounds reasonable.
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    The expression is only valid in , where is orthonormal basis in ... . However, in , with , the cross product of two vector , that is , doesn’t yield a vector in , but a pseudo-vector ... . Thus, in this case, the cross product is outer product ... .

    For example ... ,
    • for , then ,
    • for , then ,
    • for , then ,
    • for , then ,
    • etc. ... .

    CMIIW ... . Thanks ... .
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