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Please help an ignorant mathematician

  1. Dec 30, 2007 #1
    Hi everyone,

    Does anyone have any recommendations on good textbooks (or websites) that help translate the mathematicians' language of tensors (strictly as multilinear maps to the underlying field) and forms (as alternating tensor fields) to the language used by physicists?

    For example, I want to know what covariance/contravariance is all about, in the context of multilinear algebra and analysis. Also, I still can't figure out how/why tensors can be visualized as an n-dimensional array of numbers.

    I have found many sources that talk about the things above from first principles, but I find these to be un-illuminating. I am looking for a source that places these concepts in context to mathematical theory. (For example, I want to see a proof that shows that tensors can be n-dimensional arrays, and I want to see exactly how covariance/contravariance are related to pullbacks and push-forwards.)

    I am a college undergraduate junior, and I have just finished working through Munkres' Analysis on Manifolds.

    Last edited: Dec 30, 2007
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  3. Dec 30, 2007 #2

    George Jones

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    I'm not sure what you mean by "proof." I tried to talk about this in https://www.physicsforums.com/showpost.php?p=874061&postcount=2", but you might have something else in mind.

    I need to look at pullbacks and push-forwards again before I can comment, but, for a (by definition sloppy) physicist, a covector (covariant) is either an element of a cotangent space or a cross-section of a cotangent bundle and a vector (contravariant) is either an element of a tangent space or a cross-section of a tangent bundle.

    I seem to remember that there is a conflict between the way mathematicians use the terms "covariance" and "contravariance" with respect to mappings, and the way physicists use these terms in differential geometry.

    I'll try and look into this, but it might be a few days before I can look at my books.
    Last edited by a moderator: Apr 23, 2017
  4. Jan 10, 2008 #3
    Lecture Notes on General Relativity, Author: Sean M. Carroll [gr-qc/9712019]

    Before it went to print, you could download this text in PDF format-- and still can according to the site! Just google the keyword above.

    Chapters 1 through 2, out of some 9 chapters total, will explain tensors and forms (antisymmetric tensors with lower indices), covariance and contravariance and such, to the satisfaction of the physicist, and with a minimum of mathematical jargon that is so often intractable to the rest of us.

    It may not have all you are looking for, but I suspect it does,
  5. Jan 12, 2008 #4


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    have you searched this site? there are hundreds, maybe thousands, of words devoted to these questions here.
  6. Jan 13, 2008 #5
    I have been reading Danielson's Vectors and Tensors in Engineering and Physics text voraciously. Definitely makes tensor math more understandable if you are just a lowly engineer (like me).
    I think it is about $25 used. Cheers.

    Upon further inspection of your post, this book may be too basic for your needs, but perhaps someone will find it useful.
    Last edited by a moderator: Apr 23, 2017
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