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 > > ============================================================================ == >