Talk:Musical isomorphism

Pseudo-Riemannian case

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 is any expert who could do this, many thanks in advance! --Ernsts (talk) 12:59, 2 May 2009 (UTC)

Isn't pseudo-Riemannian case exactly the same? Kallikanzarid (talk) 07:51, 17 October 2010 (UTC)

Original source?

The isomorphisms seem to appear already in H. Whitney's "Geometric integration theory" from the late fifties. From whose work do they originate? --130.230.19.134 (talk) 12:31, 3 August 2009 (UTC)

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 manifolds.68.192.250.151 (talk) 04:17, 12 November 2012 (UTC)

Dot Product

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:

$\mathbf {v} \cdot \mathbf {w} =\mathbf {v} ^{\flat }\mathbf {w} =\mathbf {w} ^{\flat }\mathbf {v} .$

Danielkwalsh (talk) 10:56, 5 August 2009 (UTC)

This is false: you need the inner product $g(\cdot ,\cdot )$ (aka (pseudo)-Riemannian metric) to define musical isomorphism. Kallikanzarid (talk) 07:37, 17 October 2010 (UTC)

The formulas are actually correct, but miss the point somewhat: to define the musical isomorphism, we need the dot product a.k.a. metric. Tkuvho (talk) 11:01, 17 October 2010 (UTC)

That's exactly what I said :) Kallikanzarid (talk) 11:57, 7 November 2010 (UTC)

Gradient, divergence and curl:

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.

Regarless, the cross proudct formuls does not need both to be raised and lowered, as the Hodge dual works in either the vector or covector domain. — Arthur Rubin (talk) 20:06, 30 September 2009 (UTC)

-- 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. --IkamusumeFan (talk) 08:38, 19 October 2014 (UTC)

Canonical isomorphism

The first paragraph states that a musical isomorphism can also be called canonical isomorphism. This is in disagreement with http://en.wikipedia.org/wiki/Canonical 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 (UTC) —Preceding unsigned comment added by MBarao (talk • contribs)

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 providing there's only one inner product defined; I'm not sure about the infinite case. Dependent Variable (talk) 02:45, 18 October 2010 (UTC)

Perhaps "canonical" comes from symplectic geometry, e.g. canonical coordinates. The cotangent manifold is the (cough cough, ahem) "canonical" example of a symplectic manifold. Slap an inner product on there, and le viola. 67.198.37.16 (talk) 16:20, 29 May 2024 (UTC)

(2,0) or (0,2) ?

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. Tkuvho (talk) 05:49, 18 October 2010 (UTC)

Note that sections of the tangent bundle are described in terms of the notation (1,0) at Tensor_field#Notation. Tkuvho (talk) 14:14, 18 October 2010 (UTC)

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 Riemannian Manifolds, introduced on p. 12, while Carroll uses the latter custom in his Lecture Notes on General Relativity. But yes, if one usage is more common on Wikipedia, it should be used as standard, with a word of explanation every time it appears if there's any chance of ambiguity. Dependent Variable (talk) 22:55, 18 October 2010 (UTC)

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. Tkuvho (talk) 02:37, 19 October 2010 (UTC)

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). Dependent Variable (talk) 20:28, 19 October 2010 (UTC)

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 standard textbooks do anymore about this problem. Tkuvho (talk) 20:54, 19 October 2010 (UTC)

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 as much as it is by authors who call it (0,1). It's just an arbitrary difference in notation. Dependent Variable (talk) 02:05, 20 October 2010 (UTC)

where does this terminology come from and who uses it?

It would be good if someone could include the following in the article:

who introduced the terminology mulical isomorphism?

where is this terminology used?

Also, why does this need a separate article from raising and lowering indices? --345Kai (talk) 15:57, 13 August 2011 (UTC)

^^Ditto what Kai asked. Rschwieb (talk) 15:00, 17 November 2011 (UTC)

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. NeilOnWiki (talk) 12:09, 31 August 2019 (UTC)

You get not only basis independence, but the freedom to talk about tangent and cotangent bundles abstractly. It's easy to forget that these are distinct, when just raising and lowering indexes. 67.198.37.16 (talk) 16:28, 29 May 2024 (UTC)

Gradient, Curl and Divergence by the view of Musical isomorphism

By musical isomorphism and Hodge dual, one may define some conceptions in vector calculus elegantly. For example, in a 3 dimensional Euclidean space, let f be a function, and let F be a vector field, and let v, w be vectors. Then gradient, curl, divergence and cross product can be defined as follows:

$\nabla f=\mathrm {grad} f=(\mathbf {d} f)^{\sharp }$

$\nabla \times F=\mathrm {curl} F=[*(\mathbf {d} F^{\flat })]^{\sharp }$

$\nabla \cdot F=\mathrm {div} F=*\mathbf {d} (*F^{\flat })$

$v\times w=[*(v^{\flat }\wedge w^{\flat })]^{\sharp }$

All above are well known results. One may check for details.

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. --IkamusumeFan (talk) 08:20, 19 October 2014 (UTC)

References

Confusing section title

In § Extension to k-vectors and k-forms, we see a discussion of p-vectors and p-covectors, but no mention of k-vectors or k-forms. This will surely be confusing to non-specialists! Are we to equate p and k? And where and what are the forms? 2001:8003:4002:3700:F5BB:B979:4046:834C (talk) 12:28, 30 December 2019 (UTC)

Pedagogy

I do not approve of the sequence of the discussion of the musical isomorphisms.

In the article, first the expression of the musical isomorphisms is described in local coordinates.

Then the mathematical description is given for what is really happening.

The sequence ought to be exactly the opposite.

Currently the article gives no reason for the first description of the flat and sharp isomorphisms. That is pedagogically backwards.

First, the article ought to discuss the natural isomorphism between a finite-dimensional vector space and its dual space in the presence of an inner product.

Only then should the article define the musical isomorphisms in terms of this isomorphism at each point of a manifold endowed with a (pseudo-)Riemannian metric. Because that is what is happening.

And only then should the article explain how to compute this isomorphism in local coordinates. 2601:200:C000:1A0:ED91:8933:953C:2AB0 (talk) 20:52, 16 May 2021 (UTC)

Thanks

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

hey while you can sign that comment manually, it is also possible to do so by adding three tildas to the end. Gaismagorm (talk)

Definition is mathematically invalid

The article says:

Given a vector field X = Xⁱ e_i and denoting g_ij Xⁱ = X_j, we define its flat by:

${\begin{aligned}\flat :\mathrm {T} M&\longrightarrow \mathrm {T} ^{*}M\\X&\longmapsto g_{ij}X^{i}\mathbf {e} ^{j}\\&\longmapsto X_{j}\mathbf {e} ^{j}\end{aligned}}$

And something similar for sharp. However, the tangent bundle does not contain vector fields as elements, it contains individual vectors as elements. Mathwriter2718 (talk) 02:35, 11 July 2024 (UTC)

I fixed this. Mathwriter2718 (talk) 20:44, 17 July 2024 (UTC)

Merge proposal

I propose merging Musical isomorphism and Raising and lowering indices, though I am not sure which of the two pages should be moved into the other. They are about essentially the same process: the musical isomorphisms are what you take to raise or lower an index. Mathwriter2718 (talk) 20:46, 17 July 2024 (UTC)

I think it is clear that they should be merged. I think a merge in either direction would be equally good. Gumshoe2 (talk) 21:16, 17 July 2024 (UTC)