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Thread: Geometric Interpretation of Exterior Derivative ?

  1. #1 Geometric Interpretation of Exterior Derivative ? 
    Administrator Markus Hanke's Avatar
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    Is there a geometric interpretation/visualization of the exterior derivative, at least in three dimensions ?

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



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



    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 ? For example, I can roughly visualise the Hodge Star as giving us the elements which are orthogonal to whatever the operator acts on in the particular coordinate basis, at least up to a sign. I know this isn't precise or rigorous, but it does help me to grasp the intuitive meaning, and it even works in more than three dimensions. Does something similar exist for the exterior derivative ? Can it be visualised roughly as "wedging" with a linearly independent 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 visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations. Guitarist has shown us the algebraic part, which I am fine with; 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.
    Last edited by Markus Hanke; 05-26-2013 at 10:56 AM.
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    Quote Originally Posted by Markus Hanke View Post
    Is there a geometric interpretation/visualization of the exterior derivative, at least in three dimensions ?

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



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



    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 ? For example, I can roughly visualise the Hodge Star as giving us the elements which are orthogonal to whatever the operator acts on in the particular coordinate basis, at least up to a sign. I know this isn't precise or rigorous, but it does help me to grasp the intuitive meaning, and it even works in more than three dimensions. Does something similar exist for the exterior derivative ? Can it be visualised roughly as "wedging" with a linearly independent 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 visualise it as well. Ultimately I am trying to gain an intuitive understanding of the differential forms notation for the Maxwell equations. Guitarist has shown us the algebraic part, which I am fine with; 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.
    The first thing one must do is interpret the derivative as a linear operator. What it is, at a given point in the domain of a function f is the best linear operator that approximates f itself. So if f is the function, then at a fixed point x you have a linear operator D_xf. So now, viewing f as a function on a manifold, a 0-form, we have the induced map on the tangent bundle, df. Now if the manifold happens to be just n-space, then the tangent space at a point x is also that n-space and df acting on that tangent space acts as D_xf. So the exterior derivative is really just the obvious generalization of the ordinary derivative (of a function of several variables) to the setting of a manifold. "d" applied to a 0-form (a smooth function) is then just the derivative in this sense, which is a 1-form. Note that df is a linear operator on the tangent space, and hence is itself just an element of the cotangent space. Both the tangent space and cotangent space are of the same dimension as the underlying manifold. So in dimension 3 both are 3-dimensional vector spaces.

    The next step is to look at the "d" operator applied to 1-forms. This will yield a 2-form. You have written down the expression for applying d to a a 1-form in local coordinates. In 3-dimensions one can think of a 1-form as a vector field. At any point the When you apply d to that vector field, following your equation, you get a 2-form. 2-forms at a point, in dimension 3, also form a 3-dimensional vector space. So, we are applying "d" to a 1-form (a vector) to obtain a 2-form (also a vector or perhaps a convector), which suggests that we may be obtaining something like the "curl" of ordinary vector analysis. Upon noting that the wedge product is anti-commutative and comparing with the classical expression for the curl, confirms our suspicions. If you apply "d" to a 2-form (also viewed as a vector field) you get the divergence, a 3-form (and 3-forms in 3-dimensional are 1-dimensional hence identifiable as scalars).

    This is all tied to the theory of integration of differential forms over cochains, which yields the generalized Stokes's Theorem which is just a generalization of the fundamental theorem of calculus.

    The best accessible source for this is Spivak's little book Calculus on Manifolds, which I highly recommend.
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    Thank you DrRocket, much appreciated, and it does make things a little bit clearer.

    I was thinking, from the point of view of ( generalized ) Stokes Theorem - 1-forms in 3-space are really just vector fields; if I visualise a bunch of vectors "sticking out", then their end points ( all taken together ) form an oriented surface, i.e. a 2-form. So would it be permissible - and does it make sense - to visualise the exterior derivative as an oriented "boundary" of sorts ? This seems to work in higher dimensions as well - for example, if I "foliate" a bunch of oriented surfaces, I get an oriented volume of sorts. Or if I bunch together a set of scalars, I get a line segment. Generally, the boundary of a field of objects will be a construct that has a dimension increased by 1.
    Also, a boundary itself has of course no boundary, which is just precisely Poincare's Lemma dd = 0.
    To me that gives intuitive meaning to Stokes Theorem - that the integral of a form along a boundary is simply the same as the integral of its exterior derivative over the enclosed region. In fact, doesn't that theorem precisely mean that the exterior derivative is in some sense a "boundary" ?

    Do you think that makes sense, or is that reasoning flawed in some way ?

    It is actually fascinating just how much meaning is contained in something as simple as Stokes Theorem. Mathematics never ceases to amaze me !
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    Quote Originally Posted by Markus Hanke View Post
    To me that gives intuitive meaning to Stokes Theorem - that the integral of a form along a boundary is simply the same as the integral of its exterior derivative over the enclosed region. In fact, doesn't that theorem precisely mean that the exterior derivative is in some sense a "boundary" ?
    The boundary of an interval is its end points. The fundamental theorem of calculus says that the integral of the derivative of a function f (df) is just the difference of f evaluated at the end points (the boundary). Think about it a minute and you see that this is Stokes Theorem for an interval and that Stokes Theorem is just a generalization of the fundamental theorem of calculus.

    Stokes theorem in general says that the integral of df over a manifold is equal to the integral of f over the boundary of the manifold, where f is a differential form of the appropriate degree. The boundary of a manifold is a topological notion and it is not defined in terms of the exterior derivative -- though one must be a little careful her as we are really dealing with an oriented boundary.

    It is important to note that these ideas are formulated in terms of calculus on manifolds, not necessarily presented as subsets of any Euclidean space, so the idea is more general than integrating over a boundary that defines some "enclosed region" and in fact if you think about it, a curve, for instance, might be the boundary of more than one manifold -- for instance the equator bounds both the northern and southern hemispheres.

    I strongly recommend that you take a look at Spivak's book Calculus on Manifolds. It is clear, accessible (about a junior/senior level), short, and usualy available at a reasonable price on the used book market.
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    Quote Originally Posted by DrRocket View Post
    The boundary of an interval is its end points. The fundamental theorem of calculus says that the integral of the derivative of a function f (df) is just the difference of f evaluated at the end points (the boundary). Think about it a minute and you see that this is Stokes Theorem for an interval and that Stokes Theorem is just a generalization of the fundamental theorem of calculus.

    Stokes theorem in general says that the integral of df over a manifold is equal to the integral of f over the boundary of the manifold, where f is a differential form of the appropriate degree. The boundary of a manifold is a topological notion and it is not defined in terms of the exterior derivative -- though one must be a little careful her as we are really dealing with an oriented boundary.

    It is important to note that these ideas are formulated in terms of calculus on manifolds, not necessarily presented as subsets of any Euclidean space, so the idea is more general than integrating over a boundary that defines some "enclosed region" and in fact if you think about it, a curve, for instance, might be the boundary of more than one manifold -- for instance the equator bounds both the northern and southern hemispheres.

    I strongly recommend that you take a look at Spivak's book Calculus on Manifolds. It is clear, accessible (about a junior/senior level), short, and usualy available at a reasonable price on the used book market.
    Thanks DrRocket, I see what you are saying. Looking at it that way, at least the first Maxwell equation dF=0 makes sense - it simply means that field lines are either closed loops, or extend into infinity; in either case, they don't have "boundaries".

    I stopped at the local library yesterday on my way to work, and lo and behold they have a copy of Spivak's book in the maths section ( we are not far away from the local university here ). I will definitely go through it in detail. First though I am planning to finish Darling's book, since I endeavour to always finish what I started even if I don't get all the finer details.
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    Quote Originally Posted by Markus Hanke View Post
    Thanks DrRocket, I see what you are saying. Looking at it that way, at least the first Maxwell equation dF=0 makes sense - it simply means that field lines are either closed loops, or extend into infinity; in either case, they don't have "boundaries".

    I stopped at the local library yesterday on my way to work, and lo and behold they have a copy of Spivak's book in the maths section ( we are not far away from the local university here ). I will definitely go through it in detail. First though I am planning to finish Darling's book, since I endeavour to always finish what I started even if I don't get all the finer details.
    I have had a chance to look at Darling's book. My advice is to set it aside and read Spivak first, then perhaps go back to Darling. I find Darling's book rather strange from a pedagogical perspective in that he jumps to rather advanced material without establishing the foundations, while at the same time presuming only limited mathematical background and skipping the proofs of the implicit function theorem and the inverse function theorem which are critical to the foundations of the subject. Altogether, a rather uneven approach.

    The major inplication of dF=0 is that F=dG for some G. This is true because you are workin on a manifold (ordinary Euclidean space) that is topologically simply connected. What this says is that F arises as the gradient of some potential. In the language of differential forms this means that all closed differentials are exact differentials, and that fact depends on the topology of the manifold. You will see this discussed in Spivak in terms of "star shaped regions".

    I'm not sure what you mean by saying that the field lines might be closed loops, since one implication is that the line integral around any closed path is 0, which is equivalent to the line integral being a function of the end points, and not of the details of the path.

    In general, one has to be a bit careful when speaking of "field lines". You have to think a bit to even make sense of the term, despite the fact that it is commonly used. A field is just a curve the tangent to which is the direction of the vectors in the vector field with which one is concerned. It fails to provide information as to the magnitude. The important underlying concept is the vector field itself, and any notion of field lines needs to be tied to it.

    The notion of field lines is helpful but can also lead to confusion in the hands of those who are less than expert. There are all sorts of useful insights and misconceptions tied to field lines and "magnetic reconnection" in the context of plasma physics.
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    Quote Originally Posted by DrRocket View Post
    I have had a chance to look at Darling's book. My advice is to set it aside and read Spivak first, then perhaps go back to Darling.
    Ok, I think I will heed your advice on this, simply because you are the expert in this particular area, and thus understand the bigger picture much better than I do. Also, I am very much struggling with Darling's book, since in a lot of places the level of abstraction is just too high for me to make sense of the concepts to be introduced, never mind actually applying them to computational problems; thus it is useful to go back to the basics first.

    I'm not sure what you mean by saying that the field lines might be closed loops, since one implication is that the line integral around any closed path is 0, which is equivalent to the line integral being a function of the end points, and not of the details of the path.
    I meant that in a purely physical sense; the field lines of magnetic fields form closed loops, i.e. there are no magnetic "charges" or "monopoles". On the other hand, the field lines of the electric field of a point charge extends into infinity. In both cases, the lines don't just "end" anywhere, i.e. there are no boundaries to such fields in vacuum.
    But yes, you are of course right in that this is not exactly rigorous.
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    By the way, when browsing in the library I chanced across a booked called The Geometry of Physics, by Theodore Frankel. It looks very interesting, and seems written at a level which I could follow without major problems. Are you by any chance familiar with that book ? Would you recommend it ?
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    Quote Originally Posted by Markus Hanke View Post
    By the way, when browsing in the library I chanced across a booked called The Geometry of Physics, by Theodore Frankel. It looks very interesting, and seems written at a level which I could follow without major problems. Are you by any chance familiar with that book ? Would you recommend it ?
    I know the book and own a copy. It is a reasonable attempt to cover the material suggested by the title. I would recommend it for some purposes, but not for the purpose of learning differential geometry.

    It is not a good substitute for Spivak's book, and it is a lot more expensive. I would generally caution against trying to learn mathematics from books written by physicists, particularly mathematics that is inherently a bit abstract. They tend to oversimplify and in doing so say sometimes say things that are simply untrue, or to somehow stumble to the correct answer by means of completely invalid logic, which can be even more confusing and damaging to someone who is trying to learn. Once you understand the mathematics then such books can help to understand the application to physics, but only when you can read between the lines to see unstated connections to the real mathematics and when you are prepared to identify and correct mistakes in the mathematics. This statement does not apply to texts designed solely to help one do specific computations; i.e. to mathematical methods as opposed to actual mathematics. So, for instance if one is interested in seeing material on special coordinate systems of use in very specific problems Arfken's book is a good one. But beware of any theoretical statements.

    Let me give you one well-known example. Dirac was a genius and a first-rate intellect, particularly when it comes to quantum mechanics. His book is a classic of physics, and immensely valuable. I recommend it highly. On the other hand, von Neumann's book Mathematical Foundations of Quantum Mechanics is also a classic, and in no small part was written to put on firm footing much of the work in Dirac's book which is based on the "convenient fiction" that any Hermitian operator is diagonalizable (it is not necessarily). Similarly the recent book by Weinberg, Quantum Mechanics, is an excellent physics text, but his treatment of Hilbert space is at best a bit misleading. Learn physics from physicists. Learn mathematics from mathematicians. Moreover, in either case I am of the opinion that one ought to first go to texts written by known masters of the subject or at least to texts that have been around for a long time and are highly regarded in the community. Spivak's books have that regard. So do the texts by Helgason, Warner and others in the list that can be found in thread on recommended texts.

    For a relatively simple introduction to differential geometry you might look at do Carmo's Differential Geometry of Curves and Survaces, and then go on to his more advanced books. But I would still recommend that you read Spivak's Calculus on Manifolds first. Spivak also has a more advanced set of 5 volumes on differential geometry which is very good and is highly regarded in the mathematics community. It has the disadvantage of being 5 volumes each about the size of the New York phone book but is has a very readable style.

    For my money of the best single volume treatments of differential geometry is the book by Helgason. It is extremely good, but it does require that one have some background and mathematical maturity. One aspect of differential geometry is that it makes use of abstract algebra, topology and calculus of several variables simultaneously and you really need a decent background in each of these subjects before you are ready for differential geometry in the general setting of a differential manifold. do Carmo's book avoids some of this by focusing on the case of curves and surfaces only, as opposed to a general abstract manifold. For applications in physics, say to general relativity, you need the more general setting. There is really no way around this preparatory material. Differential geometry is a huge subject.
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    Quote Originally Posted by DrRocket View Post
    I know the book and own a copy. It is a reasonable attempt to cover the material suggested by the title. I would recommend it for some purposes, but not for the purpose of learning differential geometry.
    That sounds reasonable.

    But I would still recommend that you read Spivak's Calculus on Manifolds first.
    Right so, that's what I am going to do. I'm up the walls in both of my jobs at the moment, but expect to be able to make a start on this perhaps the week after next or so. If I get stuck somewhere in Spivak's book, I hope it will be ok if I open a thread here and pick your brains a little

    Thanks again for your help and the recommendations. Much appreciated !
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    Quote Originally Posted by Markus Hanke View Post
    By the way, when browsing in the library I chanced across a booked called The Geometry of Physics, by Theodore Frankel.
    What's it about? Since geometry is a branch of physics a good title for a book would be The Physics of Geometry. I'm reminded of a website that covers this point at http://users.wfu.edu/brehme/space.htm
    There is a fiction abroad in the world that geometry is a branch of mathematics. In fact, it is one of the three foundations of physics (the other two being time and matter-energy).
    The author is one of those names normally associated with relativity. He's the creator of the famous Brehme diagram.
    The most important thing to keep in mind is that you don't know everything and nobody else does either.
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