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MathGroup Archive 2001

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Re: Curl

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30481] Re: Curl
  • From: "Alan Mason" <amason2 at austin.rr.com>
  • Date: Thu, 23 Aug 2001 02:15:46 -0400 (EDT)
  • References: <9lvh0j$4o5$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Carlo,
The curl is special to three dimensions and is trivially implemented there.
It is related to the adjoint operation * on differential forms in 3
dimensions by h1 x h2 = *(h1 ^ h2) where ^ denotes the ordinary wedge
product of differential forms.  Here h1 and h2 are one-forms (identified
with vectors using the standard inner product) in R^3.  The adjoint operator
* is defined in arbitrary dimension n and takes r-forms to (n-r)-forms, with
orientation (sign) taken into account.

The divergence, unlike the curl, does extend to n dimensions.  In fact, if
omega is an n-1 form, then d omega is an n form and thus given by d omega =
F dx1^...^dxn, F a scalar function.  A one-form theta can be written as *
omega, and F is just the divergence of theta.

A reference is "Functions of Several Variables", by W. H. Fleming.

Alan
"Carlo Gabrieli" <gabrielic at popmail.inwind.it> wrote in message
news:9lvh0j$4o5$1 at smc.vnet.net...
> Hi,
> I found on Roman Maeder "Computer Science with Mathematica" elegant
> implementations of Grad, Div, Laplacian, Jacobian but not of Curl. I
> know that the Mathematica package VectorAnalysis has a Curl function
> but I'd like to learn how to implement a Curl function for an
> arbitrary number of Cartesian coordinates
>
> Thanks in advance to all
> Best Regards
> Carlo Gabrieli
> snail mail: Carlo Gabrieli
> Via San Giovanni d'Acri 15
> 30126 Lido di Venezia (VE)
> Tel.& FAX: 011-39-41-5264157
>
> e-mail: gabrieli at iuav.it
> gabrieli at flux.isdgm.ve.cnr.it
> gabrielic at libero.it
> gabrielic at inwind.it
>
> web pages: http://www.omitech.it/MERLIN/conn.htm
>
>
>
============================================================================
==
>
> "If you can't explain your research to your grandmother, then you
> don't understand it yourself"
>
> Richard Feynman
>
>
> "Update your bumper stickers, kids: Mac OS 8 = Windows 2010"
>
> David Pogue
>
>
============================================================================
==
>



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