Talk:Vector calculus

Note to author

To the author of the page: Don't you think that the description of vector calculus as a "collection of formulas and techniques" is somewhat oversimplified? Indeed, its origin is _not_ a compillation of formulas for use in engineering, but rather the search for the composition structure of "higher order numbers", started by Gauss (and Argand and others) applied to the complex numbers and then Hamilton, Grasmann, Gibbs (and others) to "3rd and 4th order" numbers (what we call today 3-vectors and quaternions, respectively). And this is a "trascendental" motivation, compared to the development of formulas for engineering! But this is just a comment. Cheers!

Jose A. Vallejo Faculty of Sciences UASLP (Mexico)

'Most of the results are more easily understood using the concepts of differential geometry' - well, not using the ideas as currently expressed on WP they're not.

A statement that curl is just dF is NOT going to solve any fluid mechanics! There is a real problem here (yes, it may be ME) but the differential geometry concepts are not sufficiently handlable by a non-graduate mathematician, given present WP presentations.

Interestingly, from errors made in some classical formulae (eg confusion of total & partial time derivatives in Liouville's theorem) by those using diff geom techniques, the mathematicians would be wise to swot up the old techniques too! Linuxlad 17:11, 11 Apr 2005 (UTC)

What about Green's Theorem? Shouldn't that be included as an important theorem of vector calculus? Begreen 01:48, 6 March 2006 (UTC)

That's It?

This seems to be a pitifully inadequate description of vector calculus to me. It looks like a good lead in, but the article about the techniques is missing! (Arundhati Bakshi (talk • contribs)) 12:34, 20 March 2006 (UTC)

weird link

Did anyone read that article about the improper use of nabla, written by Chen-to Tai? What a weird, pretentious article. Tai comes across as a bastard. His point is that you can't take a dot-product of nabla with a vector. But his quotes from literature are bizarre. Kreyszig, a well-respected author, comments himself on this (correctly), but Tai doesn't think Kreyszig has commented well enough, so Tai's response (my paraphrase): "the further Kreyszig explains, the more the student becomes confused". Almost no author escapes from Tai's wrath. Well, good for you Tai.

So, I suggest that link be removed, or at least demoted to the talk page (with a warning next to it).Lavaka 23:14, 16 February 2007 (UTC)

I agree that the link is not very useful, so I removed it. I guess the article is meant to be read in conjunction with Tai's other article, which appears to be interesting reading. By the way, Chen-To Tai also seems to be well respected; the IEEE has an award named after him. -- Jitse Niesen (talk) 01:13, 17 February 2007 (UTC)

I would ad that it is improper to refer to fellow users as bastards. Please be respectful, even of those you disagree with. Thank you. Sunshine Warrior04 (talk) 08:46, 26 October 2011 (UTC)

Conventional vector algebra

I moved this section to Talk:Cross product#Conventional vector algebra. Paolo.dL 09:21, 25 July 2007 (UTC) --Physis 04:26, 10 November 2007 (UTC)

Hypercorrect verbalization of divergence theorem

The article summarizes the divergence theorem like this:

Divergence theorem

$\iiint \limits _{V}\left(\nabla \cdot \mathbf {F} \right)dV=\iint \limits _{\partial V}\mathbf {F} \cdot d\mathbf {S} ,$

The integral of the divergence of a vector field over some solid equals the integral of the flux through the surface bounding the solid.

My emphasis added, typeset with red.

I conjecture that the formula is correct, but the verbalization contains a "hypercorrect", overlapping invocation of notion integral. Maybe the correct verbalization would be,

“ The integral of the divergence of a vector field over some solid equals the integral of flux through the surface bounding the solid. ”

In summary; the notion of flux already contains the notion of integral.

I am new in this field, maybe I am not right.

Physis 04:26, 10 November 2007 (UTC)....pp

You're correct, and I've updated the wording. —Toby Bartels (talk) 16:11, 22 May 2016 (UTC)

VECTOR

I asm very much week in vvvvectior basic subject —Preceding unsigned comment added by 220.225.86.85 (talk) 12:32, 11 April 2009 (UTC)

Free Textbook, Lectures

I have found this free textbook helpful as a student: Calculus, by Michael Corral (Schoolcraft College) (updated recently, 2009-03-29). It is available at his website [1] and is licensed under GNU Free Documentation License, Version 1.2, as stated on the site. Perhaps this link could be added to the bottom of the main page for other students.

Similarly, I have found the free Berkeley recordings of lectures in Multivariable Calculus (Michael Hutchings) to be worthwhile for study. They can be viewed at Earth or at the original source, Berkeley Webcasts - Video and Podcasts: Mathematics 53, although Berkeley's site is much less descriptive. The videos are licensed under a Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License as stated at Berkeley's site. (Note: this is my first Wikipedia edit, so if I have broken any guidelines, feel free to correct me). Vienneau (talk) 21:11, 28 June 2009 (UTC)

Vector Algebra = Linear Algebra?

I noticed that in the discussion of the non-differential operators of vector calculus that the "vector algebra" bold psuedo-link goes nowhere. I left wikipedia and searched google for "vector algebra" and was taken to a page that defined "vector algebra" as "linear algebra". Can I change the page so that "vector algebra" is replaced by a link the Wikipedia article on "linear algebra"? —Preceding unsigned comment added by 65.50.39.118 (talk) 05:46, 7 September 2010 (UTC)

No, the phrase "vector algebra" as used here pertains to three-dimensional space and includes the concept of the cross product which is not defined in other dimensions. Here there is a traditional interpretation of "vector algebra", as explained in the introduction, harking back to the textbook Vector Analysis. A century ago the faculties of physics and engineering needed some of the methods of quaternion algebra without entering the extra fourth dimension. Somewhat later the idea of an n-dimensional linear algebra in vector spaces was developed where inner product generalized the dot product. This article vector calculus also pertains to the three-dimensional context, and the generalization is multivariable calculus as stated.Rgdboer (talk) 20:13, 4 December 2011 (UTC)

slight extension

I made a slight extension to illustrate what the triple products look like:

• Triple scalar product:

$\mathbf {v} _{1}\cdot \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)$;

• Triple vector product:

$\mathbf {v} _{1}\times \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)$ or $\left(\mathbf {v} _{1}\times \mathbf {v} _{2}\right)\times \mathbf {v} _{3}$.

• Also a note on the perp dot product $\mathbf {v} _{1}\bot \cdot \mathbf {v} _{2}$ was included, for completeness.

Hopefully this will not be reverted - the edits are not incorrect or pointless. Typical readers will aready have an idea of what triple products are before cliking the links.--Maschen (talk) 11:54, 3 December 2011 (UTC)

\oiint template

Well, Gauss'/Divergance and Stokes'/Curl theorems would render as:

$\scriptstyle S$ $\mathbf {A} \cdot {\rm {d}}\mathbf {S} =\iiint _{V}\nabla \cdot \mathbf {A} {\rm {d}}V$

$\scriptstyle S$ $\nabla \times \mathbf {A} \cdot {\rm {d}}\mathbf {S} =\oint _{C}\mathbf {A} \cdot {\rm {d}}{\boldsymbol {\ell }}$

generated by

:{{oiint
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{A} \cdot {\rm d}\mathbf{S}=\iiint_V \nabla\cdot\mathbf{A}{\rm d}V</math>
}}

:{{oiint
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\nabla\times\mathbf{A} \cdot {\rm d}\mathbf{S}=\oint_C \mathbf{A}\cdot{\rm d}\boldsymbol{\ell}</math>
}}

I'm not forcefully saying to add this template to replace the current double integrals (over closed surfaces) symbols, as the initial author of those templates for sake of vanity, rather because other editors have worked extremely hard to render them the quality they are now. Opinions?-- F = q(E + v × B) 20:35, 31 January 2012 (UTC)

No objections? I'll just do it and see what happens... -- F = q(E + v × B) 14:23, 16 February 2012 (UTC)

Links and relationships

This page should clarify it's relationship to multivariable calculus, and to differential geometry of curves. Brent Perreault (talk) 13:43, 21 December 2012 (UTC)

Vector vs Vector Field

article says cross product is an operation on vector fields -- is this correct? aren't vectors used in that operation? same question for other operations mentioned.--Jrm2007 (talk) 18:50, 7 July 2016 (UTC)

Oliver Heaviside

I do not get it. So you know Oliver Heaviside invented vector calculus, and according to you it is not open for discussion, you deleted my post. It was not even in the article, it was on the article talk page. I just want to say if that is so ( that you know I am right), you are worse than religion. Was it Galileo, looking into the night sky using a telescope and the church knew he was correct, but alas it was not open for discussion. I was informed that Heaviside invented vector calculus on the Tom Bearden website. Debate me. K00la1dx (talk) 12:35, 20 July 2022 (UTC)

The second paragraph of the article begins with Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside. Apparently, you are unable to read more than the first paragraph of an article.

This is not a blog, and civility is required for discussions in talk pages of Wikipedia. So there is nothing to debate with you. D.Lazard (talk) 13:11, 20 July 2022 (UTC)

The way your civility bands together without putting forward what is true and honest, is a complete insult to the anglo-saxon. I said it and I will say it again Heaviside invented vector calculus. Not Heaviside and Gibbs, not Heaviside and Hamilton. Heaviside. This article has a complete bias. Maybe we should put more info about Heaviside in the article. Do you want to ask permission from your Pope? K00la1dx (talk) 16:23, 20 July 2022 (UTC)