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Re: Outer product in mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg51282] Re: [mg51261] Outer product in mathematica
• From: Matteo Delfino <delfino at studenti.ph.unito.it>
• Date: Tue, 12 Oct 2004 01:57:45 -0400 (EDT)
• References: <200410110525.BAA05003@smc.vnet.net>
• Reply-to: delfino at studenti.ph.unito.it
• Sender: owner-wri-mathgroup at wolfram.com

Jonas Sourlier ha scritto:
> Hi there
> 
> In the lections at my University I have learnt that the outer product
> of two Vectors is defined as follows:
> 
> (a)       x (b)       = 0
> (a, b)    x (c, d)    = ad - bc
> (a, b, c) x (d, e, f) = (bf - ce, cd - af, ae - bd)
> 
> For two four-dimensional Vectors the outer product produces a
> six-dimensional Vector (handled as a skew-symmetric 4x4-matrix).
> The general, axiomatic definition of the outer product says that it is
> 
> graduately anti-commutative: u x v = (-1)^(k+l) (v x u)
> bilinear: (u + v) x w = u x w + v x w
> 
> for two vectors u, v and w with dimensions k, l, m.
> 
> 
> Now, my question: The outer product seems to be implemented in
> Mathematica with the function Outer. But whatever I've tried so far
> with Outer I didn't manage to get the outer product of two vectors as
> described above.
> 
> Outer[Times, {a,b,c},{d,e,f}]    produces the 3x3-Matrix
> 
> {{a d, a e, a f}, {b d, b e, b f}, {c d, c e, c f}}
> 
> How can I calculate the outer product of two vectors with Mathematica?
> 
> Thank a lot for helping me!
> Jonas
Outer product (or tensor product) and cross product are not the same 
thing.Outer product is defined between two tensors and produces a tesor
  of higher rank,cross product takes two vector (read:tensor of the same 
rank) and produces another vector (read:tensor of the same rank as 
before).In mathematica cross product is defined as Cross[] and Outer 
product as Outer[].

YOU MUST USE CROSS[].

DEFINITION OF CROSS PRODUCT BY OUTER PRODUCT

It is still possible to extend the definition of Cross[] product to 
tensors of any rank,try searching google for informations such as:

http://www.math.wisc.edu/~milewski/321f04/fwnotestensors.pdf

http://groups.google.it/groups?q=%22tensor+product%22+%22cross+product%22&hl=it&lr=&selm=gAnda.419%24DS2.99500%40newsfep1-win.server.ntli.net&rnum=1


Hope It helps.
Delfino Matteo

• References:
  • Outer product in mathematica
    • From: Jonas Sourlier <aeroswiss@gmx.ch>