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Sounds as though we have a clash of traditions in physics and in differential geometry. Also, I am a bit worried about your use of the term "argument". If we speak of arguments, then we pass to the dual. Thus, the metric, which is is a 2-covector, takes a pair of arguments which are both vectors.
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which says "Various functions in mathematics are also canonical, (...) the canonical isomorphism between a finite-dimensional vector space and its double dual. Although a finite-dimensional vector space and its dual space are isomorphic, there is no canonical isomorphism.". MBarão 22:38, 20 July 2010
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The convention is to treat the tangent spaces as primary and use the unqualified word "vector" as a synonym for tangent vectors, and the word "covector" or "dual vector" etc. for vectors of the cotangent spaces. I think that viewpoint is held by authors who call a tangent vector's valence (0,1) just
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There is need to extend this article on pseudo-riemann manifolds, as tge
Minkowski manifold, too. That is sharp and flat should also be defined for indefinite (i. e. mixed) but nondegenerate signatures of the symmetric bilinearform g. Raising and lowering are used for minkowski space, too. If there
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Let's take the tangent bundle and its sections, the proverbial vector fields. Are we to view these as of the "vector" variety, or of the "covector" variety? It seems a little odd to think of a vector field as of valence "covector" because you "evaluate" it on a 1-form. I don't remember what the
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There isn't in general a canonical isomorphism between a finite vector space and its dual space. But if an inner product is defined on a vector space, then this inner product does provide a canonical isomorphism, as described in this article. At least, this is true for finite vector spaces, and
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It seems to me that if the gradient, curl and divergence operators, along with the cross product are defined in terms of musical isomorphisms, so should the dot product. It seems to me that this could be written in two ways:
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Unfortunately both customs are current; some authors give the number of vector arguments first (or uppermost), others the number of covector arguments first (or uppermost). For example, Lee uses the former custom in
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I don't think anyone calls it the "musical isomorphism". Also, unlike an important theorem or definition, it will not be found to originate in any given work, since it is implicit anywhere there are
Riemannian
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This article helps to make some differential geometry accessible. The subject (usually written for graduate mathematicians) here is physics-friendly - many thanks for that. 17:35, 11 June 2021 92.7.63.73
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1488:{\displaystyle {\begin{aligned}\flat :\mathrm {T} M&\longrightarrow \mathrm {T} ^{*}M\\X&\longmapsto g_{ij}X^{i}\mathbf {e} ^{j}\\&\longmapsto X_{j}\mathbf {e} ^{j}\end{aligned}}}
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442:-- Usually we define gradient, divergence and curl over vector fields, and thus there was no symbol of covector in the equations. In fact, the equations are all well known results. --
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One may supplement these results to the article. Since the results have been given in the related articles, I think it is also fine to just leave them in this talk page. Thanks. --
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581:. But yes, if one usage is more common on Knowledge, it should be used as standard, with a word of explanation every time it appears if there's any chance of ambiguity.
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That's just how the valence of a tensor is defined, isn't it? The number of vector arguments, the number of covector arguments (in whichever order they're listed).
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I've removed the entire section, as some of the "vectors" are clearly covectors, and so the raising and lowering operations are unnecessary and incorrect.
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By musical isomorphism and Hodge dual, one may define some conceptions in vector calculus elegantly. For example, in a 3 dimensional
Euclidean space, let
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And something similar for sharp. However, the tangent bundle does not contain vector fields as elements, it contains individual vectors as elements.
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Having a separate article worked well for me as a reader. Eg. I found it very helpful to have a notation independent of indices and specific bases.
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The isomorphisms seem to appear already in H. Whitney's "Geometric integration theory" from the late fifties. From whose work do they originate? --
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Currently the article gives no reason for the first description of the flat and sharp isomorphisms. That is pedagogically backwards.
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The page refers to dx^i dx^j as a (2,0) tensor. Shouldn't this be (0,2) ? The custom is to list vectors first, then covectors.
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The first paragraph states that a musical isomorphism can also be called canonical isomorphism. This is in disagreement with
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304:{\displaystyle \mathbf {v} \cdot \mathbf {w} =\mathbf {v} ^{\flat }\mathbf {w} =\mathbf {w} ^{\flat }\mathbf {v} .}
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hey while you can sign that comment manually, it is also possible to do so by adding three tildas to the end.
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I think it is clear that they should be merged. I think a merge in either direction would be equally good.
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at each point of a manifold endowed with a (pseudo-)Riemannian metric. Because that is what is happening.
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The formulas are actually correct, but miss the point somewhat: to define the musical isomorphism, we
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Knowledge. If you would like to participate, please visit the project page, where you can join
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abstractly. It's easy to forget that these are distinct, when just raising and lowering indexes.
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as much as it is by authors who call it (0,1). It's just an arbitrary difference in notation.
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Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems
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be vectors. Then gradient, curl, divergence and cross product can be defined as follows:
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Note that sections of the tangent bundle are described in terms of the notation (1,0) at
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providing there's only one inner product defined; I'm not sure about the infinite case.
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the expression of the musical isomorphisms is described in local coordinates.
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It would be good if someone could include the following in the article:
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the mathematical description is given for what is really happening.
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831:{\displaystyle \nabla f=\mathrm {grad} f=(\mathbf {d} f)^{\sharp }}
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Gradient, Curl and
Divergence by the view of Musical isomorphism
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All above are well known results. One may check for details.
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918:{\displaystyle \nabla \times F=\mathrm {curl} F=^{\sharp }}
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is any expert who could do this, many thanks in advance! --
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where does this terminology come from and who uses it?
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1134:: CS1 maint: multiple names: authors list (
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697:^^Ditto what Kai asked.
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