View Full Version : Cross product in higher dimensions
Mikkel Bo Hansen mihan99
Oct29-04, 02:07 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nHi,\nI have read in your archive, that it is impossible to form a cross =\nproduct in higher dimensions than 3 or 7. That is a bit devastating. I =\nhave a differentiation of an inner product (or dot product), where both =\nvectors are of dimension n, n>>3. And I use:\n\nDEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n\nBut if DELxG does not exist, then I have a problem. Is this the right =\nway to do it? I know that this is the right way in three dimensions, but =\nhow about n dimensions?\n\nBest regards\n\n-- Mikkel\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hi,
I have read in your archive, that it is impossible to form a cross =
product in higher dimensions than 3 or 7. That is a bit devastating. I =
have a differentiation of an inner product (or dot product), where both =
vectors are of dimension n, n>>3. And I use:
DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
But if DELxG does not exist, then I have a problem. Is this the right =
way to do it? I know that this is the right way in three dimensions, but =
how about n dimensions?
Best regards
-- Mikkel
Van Jacques
Oct30-04, 08:10 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nMikkel Bo Hansen mihan99 wrote:\n> Hi,\n> I have read in your archive, that it is impossible to form a cross =\n> product in higher dimensions than 3 or 7. That is a bit devastating.\nI =\n> have a differentiation of an inner product (or dot product), where\nboth =\n> vectors are of dimension n, n>>3. And I use:\n>\n> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>\n> But if DELxG does not exist, then I have a problem. Is this the right\n=\n> way to do it? I know that this is the right way in three dimensions,\nbut =\n> how about n dimensions?\n>\n> Best regards\n>\n> -- Mikkel\n\nThe cross product is antisymmetric, and the way to do such products\nis to use the exterior product from multilinear algebra, which can\nbe found in any text on differential geometry. In n dimensions\na 2-vector is\n\na /\\ b = a@b - b@a , where @ = tensor product, and will be\n\nommited hereafter. Thus a/\\a = 0. a,b, are vectors in n dimensions.\n\nOne can form a 3-vector a /\\ b /\\ c and in general\n\na_1 /\\ a_2 /\\ ... /\\ a_k = a k-vector or k-form in A_k = antisymmetric\n\nk-tensors. One also has the Hodge * operator;\n\n* : A_k --> A_(n-k) : a --> *a = e(a)\n\nwhere e in A_n is the antisymmetric Levi-Civita tensor.\n\nThis leads to differential k-forms and integrals over k-dimensional\nsubspaces of n-space.\nIn 3D, the cross product is a x b = *(a/\\b) = e(a,b) = - e(b,a).\n\nVan\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Mikkel Bo Hansen mihan99 wrote:
> Hi,
> I have read in your archive, that it is impossible to form a cross =
> product in higher dimensions than 3 or 7. That is a bit devastating.
I =
> have a differentiation of an inner product (or dot product), where
both =
> vectors are of dimension n, n>>3. And I use:
>
> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>
> But if DELxG does not exist, then I have a problem. Is this the right
=
> way to do it? I know that this is the right way in three dimensions,
but =
> how about n dimensions?
>
> Best regards
>
> -- Mikkel
The cross product is antisymmetric, and the way to do such products
is to use the exterior product from multilinear algebra, which can
be found in any text on differential geometry. In n dimensions
a 2-vector is
a /\ b =[/itex] a@b - b@a , where @ = tensor product, and will be
ommited hereafter. Thus a/\a = . a,b, are vectors in n dimensions.
One can form a 3-vector a /\ b /\ c and in general
a_1 /\ a_2 /\ ... /\ a_k = a k-vector or k-form in A_k = antisymmetric
k-tensors. One also has the Hodge * operator;
* : A_k --> A_(n-k) : a --> [itex]*a = e(a)
where e in A_n is the antisymmetric Levi-Civita tensor.
This leads to differential k-forms and integrals over k-dimensional
subspaces of n-space.
In 3D, the cross product is a x b = *(a/\b) = e(a,b) = - e(b,a).
Van
Aage Andersen
Oct30-04, 08:10 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n> I have read in your archive, that it is impossible to form a cross =\n> product in higher dimensions than 3 or 7. That is a bit devastating. I =\n> have a differentiation of an inner product (or dot product), where both =\n> vectors are of dimension n, n>>3. And I use:\n>\n> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>\n> But if DELxG does not exist, then I have a problem. Is this the right =\n> way to do it? I know that this is the right way in three dimensions, but =\n> how about n dimensions?\n\nThe wedge product is a generalisation to n dimensions.\n\nhttp://www.mrao.cam.ac.uk/~clifford/publications/\n\nAage\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I have read in your archive, that it is impossible to form a cross =
> product in higher dimensions than 3 or 7. That is a bit devastating. I =
> have a differentiation of an inner product (or dot product), where both =
> vectors are of dimension n, n>>3. And I use:
>
> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>
> But if DELxG does not exist, then I have a problem. Is this the right =
> way to do it? I know that this is the right way in three dimensions, but =
> how about n dimensions?
The wedge product is a generalisation to n dimensions.
http://www.mrao.cam.ac.uk/~clifford/publications/
Aage
Paul Victor Birke
Nov2-04, 12:38 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nHello Aage\n\nCan you please tell me which paper hiding in there it is?\n\ntuck\n\nPaul\n\nAage Andersen wrote:\n\n>>I have read in your archive, that it is impossible to form a cross =\n>>product in higher dimensions than 3 or 7. That is a bit devastating. I =\n>>have a differentiation of an inner product (or dot product), where both =\n>>vectors are of dimension n, n>>3. And I use:\n>>\n>>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>>\n>>But if DELxG does not exist, then I have a problem. Is this the right =\n>>way to do it? I know that this is the right way in three dimensions, but =\n>>how about n dimensions?\n>\n>\n> The wedge product is a generalisation to n dimensions.\n>\n> http://www.mrao.cam.ac.uk/~clifford/publications/\n>\n> Aage\n>\n>\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello Aage
Can you please tell me which paper hiding in there it is?
tuck
Paul
Aage Andersen wrote:
>>I have read in your archive, that it is impossible to form a cross =
>>product in higher dimensions than 3 or 7. That is a bit devastating. I =
>>have a differentiation of an inner product (or dot product), where both =
>>vectors are of dimension n, n>>3. And I use:
>>
>>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>>
>>But if DELxG does not exist, then I have a problem. Is this the right =
>>way to do it? I know that this is the right way in three dimensions, but =
>>how about n dimensions?
>
>
> The wedge product is a generalisation to n dimensions.
>
> http://www.mrao.cam.ac.uk/~clifford/publications/
>
> Aage
>
>
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Mikkel Bo Hansen mihan99" <mihan99@student.sdu.dk> wrote in message news:<3BC166246158F845A8ABBACA9EEE1E2D18D374@ADM-EXCH0C.adm.c.sdu.dk>...\n> Hi,\n> I have read in your archive, that it is impossible to form a cross =\n> product in higher dimensions than 3 or 7. That is a bit devastating. I =\n> have a differentiation of an inner product (or dot product), where both =\n> vectors are of dimension n, n>>3. And I use:\n>\n> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>\n> But if DELxG does not exist, then I have a problem. Is this the right =\n> way to do it? I know that this is the right way in three dimensions, but =\n> how about n dimensions?\n>\n> Best regards\n>\n> -- Mikkel\n\nThe cross product exists, in a sense, in all dimensions as the\nClifford outer product (sometimes called the wedge product). It\'s\njust that in most multidimensional geometries this structure tends to\nbe an object unto itself and cannot be related to other objects in the\nsame Clifford algebra. As an example, in n = 3 the outer product of\ntwo vectors forms a bivector which can be related to a vector by\nClifford duality. Hence, we have the classical notion of an axial\nvector.\n\nIn n = 4, however, a different situation arises, since bivectors are\ndual to bivectors, so the the concept of an axial vector makes no real\nsense in four dimensions. A very good example of this type of object\nis the Maxwell tensor from relativity. It is essentially a four\ndimensional curl, having a dual where the E and B fields are switched.\n\nBivector fields generated by the outer product of two vectors will\ncarry over to all numbers of dimensions, so that should be the best\nway to approach it.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Mikkel Bo Hansen mihan99" <mihan99@student.sdu.dk> wrote in message news:<3BC166246158F845A8ABBACA9EEE1E2D18D374@ADM-EXCH0C.adm.c.sdu.dk>...
> Hi,
> I have read in your archive, that it is impossible to form a cross =
> product in higher dimensions than 3 or 7. That is a bit devastating. I =
> have a differentiation of an inner product (or dot product), where both =
> vectors are of dimension n, n>>3. And I use:
>
> DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>
> But if DELxG does not exist, then I have a problem. Is this the right =
> way to do it? I know that this is the right way in three dimensions, but =
> how about n dimensions?
>
> Best regards
>
> -- Mikkel
The cross product exists, in a sense, in all dimensions as the
Clifford outer product (sometimes called the wedge product). It's
just that in most multidimensional geometries this structure tends to
be an object unto itself and cannot be related to other objects in the
same Clifford algebra. As an example, in n = 3 the outer product of
two vectors forms a bivector which can be related to a vector by
Clifford duality. Hence, we have the classical notion of an axial
vector.
In n = 4, however, a different situation arises, since bivectors are
dual to bivectors, so the the concept of an axial vector makes no real
sense in four dimensions. A very good example of this type of object
is the Maxwell tensor from relativity. It is essentially a four
dimensional curl, having a dual where the E and B fields are switched.
Bivector fields generated by the outer product of two vectors will
carry over to all numbers of dimensions, so that should be the best
way to approach it.
Paul Victor Birke
Nov3-04, 10:05 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hello Aage\n\nCould you tell me exactly what paper in there I should have a peek at?\n\ntuck\n\nPaul\n\nAage Andersen wrote:\n>>I have read in your archive, that it is impossible to form a cross =\n>>product in higher dimensions than 3 or 7. That is a bit devastating. I =\n>>have a differentiation of an inner product (or dot product), where both =\n>>vectors are of dimension n, n>>3. And I use:\n>>\n>>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>>\n>>But if DELxG does not exist, then I have a problem. Is this the right =\n>>way to do it? I know that this is the right way in three dimensions, but =\n>>how about n dimensions?\n>\n>\n> The wedge product is a generalisation to n dimensions.\n>\n> http://www.mrao.cam.ac.uk/~clifford/publications/\n>\n> Aage\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello Aage
Could you tell me exactly what paper in there I should have a peek at?
tuck
Paul
Aage Andersen wrote:
>>I have read in your archive, that it is impossible to form a cross =
>>product in higher dimensions than 3 or 7. That is a bit devastating. I =
>>have a differentiation of an inner product (or dot product), where both =
>>vectors are of dimension n, n>>3. And I use:
>>
>>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>>
>>But if DELxG does not exist, then I have a problem. Is this the right =
>>way to do it? I know that this is the right way in three dimensions, but =
>>how about n dimensions?
>
>
> The wedge product is a generalisation to n dimensions.
>
> http://www.mrao.cam.ac.uk/~clifford/publications/
>
> Aage
Aage Andersen
Nov5-04, 08:20 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n> Could you tell me exactly what paper in there I should have a peek at?\n\n>> The wedge product is a generalisation to n dimensions.\n\n\nYou may perhaps start here:\nhttp://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html\n\nAlso the papers of David Hestenes are interesting.\n\nhttp://modelingnts.la.asu.edu/\n\nhttp://modelingnts.la.asu.edu/html/Oersted-ReformingTheLanguage.html\n\nhttp://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf\n\nhttp://modelingnts.la.asu.edu/pdf/SpaceTimeCalc.pdf\n\nhttp://modelingnts.la.asu.edu/pdf/NFMPchapt1.pdf\n\nhttp://modelingnts.la.asu.edu/pdf/NFMPchapt2.pdf\n\nThe wedge product or outer product of two vectors a, b is denoted a ^ b.\n\nAage\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Could you tell me exactly what paper in there I should have a peek at?
>> The wedge product is a generalisation to n dimensions.
You may perhaps start here:
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html
Also the papers of David Hestenes are interesting.
http://modelingnts.la.asu.edu/
http://modelingnts.la.asu.edu/html/Oersted-ReformingTheLanguage.html
http://modelingnts.la.asu.edu/pdf/PrimerGeometricAlgebra.pdf
http://modelingnts.la.asu.edu/pdf/SpaceTimeCalc.pdf
http://modelingnts.la.asu.edu/pdf/NFMPchapt1.pdf
http://modelingnts.la.asu.edu/pdf/NFMPchapt2.pdf
The wedge product or outer product of two vectors a, b is denoted a ^ b.
Aage
Gregory L. Hansen
Nov5-04, 08:20 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nIn article <3BC166246158F845A8ABBACA9EEE1E2D18D374@ADM-EXCH0C.adm.c.sdu.dk>,\nMikkel Bo Hansen mihan99 <mihan99@student.sdu.dk> wrote:\n>\n>Hi,\n>I have read in your archive, that it is impossible to form a cross =\n>product in higher dimensions than 3 or 7. That is a bit devastating. I =\n>have a differentiation of an inner product (or dot product), where both =\n>vectors are of dimension n, n>>3. And I use:\n>\n>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F\n>\n>But if DELxG does not exist, then I have a problem. Is this the right =\n>way to do it? I know that this is the right way in three dimensions, but =\n>how about n dimensions?\n>\n>Best regards\n>\n>-- Mikkel\n>\n\n\nThe cross product gives an axis of rotation, a line that\'s perpendicular\nto both of the input vectors. But in higher dimensions there\'s no single\nline that\'s mutually perpendicular and that can serve as an axis of\nrotation. It\'s a lot like finding a line perpendicular to the x axis in\nthree dimensions-- the y axis and z axis are perpendicular to it, as well\nas any line in between. The wedge product returns a plane of rotation,\nwhich remains valid in higher dimensions.\n\n--\n"Not that there\'s anything wrong with just lying around on your back. In\nits way, rotting is interesing too... It\'s just that there are other ways\nto spend your time as a cadaver." -- Mary Roach, "Stiff", 2003.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>In article <3BC166246158F845A8ABBACA9EEE1E2D18D374@ADM-EXCH0C.adm.c.sdu.dk>,
Mikkel Bo Hansen mihan99 <mihan99@student.sdu.dk> wrote:
>
>Hi,
>I have read in your archive, that it is impossible to form a cross =
>product in higher dimensions than 3 or 7. That is a bit devastating. I =
>have a differentiation of an inner product (or dot product), where both =
>vectors are of dimension n, n>>3. And I use:
>
>DEL(F.G) =3D Fx(DELxG) + Gx(DELxF) + (F.DEL)G + (G.DEL)F
>
>But if DELxG does not exist, then I have a problem. Is this the right =
>way to do it? I know that this is the right way in three dimensions, but =
>how about n dimensions?
>
>Best regards
>
>-- Mikkel
>
The cross product gives an axis of rotation, a line that's perpendicular
to both of the input vectors. But in higher dimensions there's no single
line that's mutually perpendicular and that can serve as an axis of
rotation. It's a lot like finding a line perpendicular to the x axis in
three dimensions-- the y axis and z axis are perpendicular to it, as well
as any line in between. The wedge product returns a plane of rotation,
which remains valid in higher dimensions.
--
"Not that there's anything wrong with just lying around on your back. In
its way, rotting is interesing too... It's just that there are other ways
to spend your time as a cadaver." -- Mary Roach, "Stiff", 2003.
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