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Appealing to Mathematical Intuition

  1. Jul 5, 2009 #1
    I'm new. Now you know.

    I'm also in the second year of an undergraduate nanotechnology engineering program at a Canadian university, taking a first course in quantum mechanics and simultaneously an advanced (sic) calculus course. The overlap between Hilbert Spaces and Fourier Series / Transforms drove me to Wikipedia and textbooks, where I journeyed from inner product spaces all the way back through vector spaces to set theory (not at ZFC) in an effort to answer my questions. I had a lot of them. It was all so beautiful. Then my calculus class moved on to vector calculus, and I tried thinking about a change from Cartesian to spherical coordinates in terms of vector spaces, which was not a simple concept (for me) (see [Aside 1]). Now, as I read about Stokes' Theorem, I've gotten stuck on topology and differential forms. *Sigh...

    In all of this, some concepts are incredibly appealing (see [Aside 2] below) once they are understood. However, I find most of the explanations lacking an intuitive sense. Mathematics is, even if not on the face of it, almost entirely intuitive, but many notions are made befuddling by the reliance on notation.

    My question is this: what resources are you aware of that consistently appeal to a sense of visual intuition (it doesn't have to be geometric)? I don't mind verbosity or redundancy, since different explanations of the same phenomenon can often be useful.

    Also, I have no idea if this is the right section...

    [Aside 1]

    My reasoning follows:

    - Intuitively, in three dimensional Euclidian space (call it E3), for each 3-tuple corresponding to Cartesian coordinates (x, y, z), there is (at least) one corresponding 3-tuple that represents the same point in spherical coordinates (r, qa, qb). Call two corresponding 3-tuples "pairs" (ignoring degeneracy and strangeness at (0,0)).

    - Because of the way the pairs are related (e.g. x0 = r0*cos(qa0)), they cannot be in the same vector space, in the sense that the characteristic operations would not preserve the relationship between vectors. e.g. that v1 + v2 in Cartesian coordinates = (x1+x2, y1+y2, z1+z1), is not the same vector as v1 + v2 (r1+r2, qa1+qa2, qb1+qb2) in spherical coordinates.

    - Given the familiar E3 space, with addition defined as the addition of components, it is possible to define the unfamiliar S3 space (I made it up in my brain; it's the three-dimensional spherical space), with vector addition defined in whichever super-awkward relationship preserves vectors as though they were added in E3, and scalar multiplication defined as (exercise). Given E3 and S3, there is a way to transform each vector in one space to a vector in the other that will preserve the operations. Homo...morphism? I'm not entirely certain that S3 could satisfy all requirements of a vector space, but let's pretend it does.

    This reasoning took me to hell (category theory) and back, and I have no idea if it's right.

    [End Aside 1]

    [Aside 2]

    I am always motivated while studying an abstract algebraic structure by the idea that the emergent theorems could be applied to any objects known to behave as this structure does. This is particularly applicable to me, as I view nanotechnology engineering as nothing more than emergence engineering.

    If, for example, we consider an algebraic set as a set of molecular entities and algebraic operations as some form of interaction between these entities (be it a reaction, or otherwise), then it would be possible to "perform algebra" in any molecular system that can be related to an algebraic system. Even more awesome is that any algorithm that can be implemented in the associated algebraic structure can be implemented in the molecular system. Imagine approximating the behaviour of an "ideal" molecule as a Taylor Polynomial. Imagine finding mappings between systems that preserve structure!
  2. jcsd
  3. Jul 5, 2009 #2
    You might consider Roger Penrose's "Road to Reality" (Knopf 2004). It's over 1000 pages, but it treats a broad range of mathematics and its application to physical theory in an intuitive way (with illustrations) while at the same time maintaining rigor. It's not a textbook, nor is it a substitute for one, but it might be what you're looking for. I'm starting my second reading, and I don't pretend to fully follow all of it, but can you select the topics you're most interested in. It lists for 40 USD, but I found a copy in factory outlet store for 10 USD.
    Last edited: Jul 5, 2009
  4. Jul 5, 2009 #3
    I have no idea what you are talking about in your Aside 1. Sorry. Haha.

    You say that you have recently become interested in topology and differential forms. There are a few books that I know of that are very intuitive books for this subject matter.

    https://www.amazon.com/Advanced-Cal...sr_1_3?ie=UTF8&s=books&qid=1246852193&sr=8-3" by Harold Edwards
    This book is a great intuitive approach to differential forms and integration. It spends the first three chapters on intuition alone, and then it goes into the theory in the later chapters.

    https://www.amazon.com/Geometric-Ap...sr_1_1?ie=UTF8&s=books&qid=1246852172&sr=8-1" by David Bachman
    I haven't read this book, but it seems to be a good recommendation with good reviews.

    https://www.amazon.com/Introduction...sr_1_1?ie=UTF8&s=books&qid=1246852384&sr=1-1" by Loring Tu
    This is, in my opinion, by far the best intuitive, but rigorous and complete, introduction to differential forms, smooth manifolds, and integration on manifolds. The other option is https://www.amazon.com/Introduction...sr_1_3?ie=UTF8&s=books&qid=1246852517&sr=1-3" by John M. Lee.

    Depending on how much mathematics you know, I would start with Edwards' book and then move on to Loring Tu's manifolds book.
    Last edited by a moderator: May 4, 2017
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