How can I "picture" tensors?
Frank Astier
I was introduced to tensors during my student years, with a lot of
abstraction -- not much room for geometrical intuition.
Too bad, because a vector, for example, can be easily represented as an
"arrow", and it becomes easier to work with this "picture" in mind.
Is it possible to have a analogous "picture" for tensors?
Frank
jddescr...@my-deja.com
See the recent thread on Quaternions. Quaternions
are a very general picture of a two index tensor
somewhat similar to but much more easily pictured
than how a complex number represents a 2D transformation.
Another approach, that is less complete, is the Vector
Analysis of Gibbs. Gibbs specializes quaternions
to diadics but has some of the same pictures. The base
picture is in a rotation space , what Gibbs calls an
axial vector space. We have there a quaternion axis of
rotation and of stretch. Hamilton envisioned the picture
as representing the division of two vectors and presented
all the [simple] pictures you might want. One book that
has been referenced here and has some of the pictures
for the complex variable specialization is "Visual Complex
Analysis". It even has a tiny section on quaternions.
The visualizations are quite well done. Good seeing. JD
------------------------------------------------------------------------
Sent via Deja.com http://www.deja.com/
Before you buy.
Paul Hunt
fas...@us.oracle.com wrote:
> I'm looking for a "picture" of tensors.
>
> I was introduced to tensors during my student years, with a lot of
> abstraction -- not much room for geometrical intuition.
>
> Too bad, because a vector, for example, can be easily represented as
an
> "arrow", and it becomes easier to work with this "picture" in mind.
>
> Is it possible to have a analogous "picture" for tensors?
>
> Frank
>
To conceptualize tensors I started simple then
worked up like this:
dX_i = T_i,j * dx_j represents transformation
T operating on vector dx to produce vector dX.
Simplest is a linear transformation such as a
a rotation about the origin. When dimension
is 2, T is representable as a rotation matrix:
| cosA -sinA |
T = | sinA cosA | where A is the angle.
Then the x and X are two-component vectors,
which locate a point before and after it's
rotation through A, while T is an operation
reflecting certain properties of the space
(cartesian plane).
If PX/Px denotes partial of X respecting x,
then the tensor component T_1,1 = PX_1/Px_1.
Therefore the tensor is showing the space is
"treating" x_1 and x_2 differently under a
rotation about the origin.
For the strain tensor e_i,j, the components
of its matrix are given by:
e_i,j = (1/2)[ (Pu_i/Px_j) + (Pu_j/Px_i) ],
where u is a vector locating the initial x,
following deformation of an elastic media.
But e reflects properties of the media, not
properties of the vectors X or x for which
u is the difference (u = X - x). In other
words something's happened between X and x,
which yields u, but what happened is e_i,j.
You might say that vectors possess specific
point properties, while tensors possess the
local properties of that space in which the
vectors are precisely located (embedded).
Finally there may be more than one action
so that products of tensors are needed to
fully describe the total of effects upon
a vector.
Even furthermore you start fooling around
with the tensors in themselves and discover
things like the strain cannot be arbitarily
specified because the e_i,j consists of 6
equations in 3 unknowns u_i, for 3-space.
Thereafter you have inherent properties of
tensors to deal with, just like in vectors
you have a norm or magnitude inherent.
/ph
- - - - - - -
Judy & Irv Dick
In each of 3 directions the tendency for the band to stretch can be a
different value.The tensor describes the tendency
Hope this helps
Paul Hunt
fas...@us.oracle.com wrote:
> I'm looking for a "picture" of tensors.
>
> I was introduced to tensors during my student years, with a lot of
> abstraction -- not much room for geometrical intuition.
>
> Too bad, because a vector, for example, can be easily represented as
an
> "arrow", and it becomes easier to work with this "picture" in mind.
>
> Is it possible to have a analogous "picture" for tensors?
>
> Frank
>
Just one more thing to say on visualization. Some
folks do regard tensors as vector-bundles, and I'm
inclined to accept that view since the applications
clearly show that we are doing an inner product of
a row and column vector to yeild a row vector.
In that view you think of the tensor as a collection
of row vectors, defining some action or property of
the space, which may be applied to arbitary nearby
coordinate column vectors.
Please do post updates as you seek for satisfactory
"pictures" of tensors.
/ph
- - - - - -
Lewis Mammel
>
> Lewis Mammel <lhma...@ameritech.net> wrote:
>
> >john baez wrote:
> >
> >> However, many physics books use these words in the opposite way when
> >> discussing tensors, so you always have to be on your guard. My book
> >> with Muniain includes a discussion of why different people use these
> >> words in different ways when talking about tensors. There's a simple
> >> reason.
> >
> >So we're supposed to buy your book to find out? Is that it?
> >
> >Lew Mammel, Jr.
>
> Well, you don't expect him to give away what he is simultaneously
> billing for, do you?
Nobody likes to be strung along, least of all me. I don't suppose
I should complain, though, as his original remark was very stimulating,
even if I do go to my grave without knowing what the damn reason is.
"betcha didn't know the South won the Civil War!"
Lew Mammel, Jr.
Jackie & Barry
Lewis Mammel wrote:
> Nobody likes to be strung along, least of all me. I don't suppose
> I should complain, though, as his original remark was very stimulating,
> even if I do go to my grave without knowing what the damn reason is.
Wow!
John Baez puts out lots of useful stuff on these news groups for free.
He also has a huge amount free stuff on his web site. If he occasionally
suggests one of his own books as being useful I don't think that counts
as "stringing someone along". His motivation his clearly to share the
love he has for the subject, not the little money he might make from
selling some ng readers a few copies of his book.
And you don't have to go to your grave not knowing, you just have to put
in fraction of the effort that he did writing his book, posting to the
ng's and creating his web site.
Barry
Edward Green
>Lewis Mammel wrote:
>
>> Nobody likes to be strung along, least of all me. I don't suppose
>> I should complain, though, as his original remark was very stimulating,
>> even if I do go to my grave without knowing what the damn reason is.
>
>
>Wow!
>
>John Baez puts out lots of useful stuff on these news groups for free.
>He also has a huge amount free stuff on his web site. If he occasionally
>suggests one of his own books as being useful I don't think that counts
>as "stringing someone along". His motivation his clearly to share the
>love he has for the subject,
And just occasionally to raz somebody, he being, by all accounts,
human.
>not the little money he might make from
>selling some ng readers a few copies of his book.
I can't speak for John Baez's motivation, but I would not
underestimate the value to an academic in moving a few more copies of
his book. My absolute nadir experience in this regard was being sold
a small abstruse mathematical text, about the size of "my first
reader", as a required text for a graduate seminar (~$70), which we
never delved into, the chief motivation of the instructor being,
explicitly, to help out his friend and Bell Labs in moving a few more
copies of a work which, there being no accounting for human tastes
and economic motivation, was not exactly selling like John Le Carre.
>And you don't have to go to your grave not knowing, you just have to put
>in fraction of the effort that he did writing his book, posting to the
>ng's and creating his web site.
So consider yourself justly chastised, oh ungrateful one. ;)
Tom Roberts
> > In article <38074FE9...@lucent.com>
> > Tom Roberts <tjro...@lucent.com> wrote:
> >> : but is rather a notational quirk involving
> >> : sub- and super-scripts. "Contravariant" always relates to a "push
> >> : forward" and "covariant" is always related to a "pull back".
> It's worth knowing, if you ever talk to mathematicians, that in mathematics
> "covariant" refers to quantities that "push forward", while
> "contravariant" refers to quantities that "pull back".
Ouch! I had thought I was using the same nomenclature as Baez and Muniain.
Looks like I have more homework....
Tom Roberts tjro...@lucent.com
Robert Low
>
> Just for the rest of us, what does it mean to "push forward" and "pull
> back"?
You have two spaces, say M and N and a function f:M->N.
If a structure on M naturally induces (through the
action of f) a structure on N, then this pushes forward.
If a structure on N naturally induces (through the
action of f) one on M, this pulls back.
For example, vectors push forward, differential
forms pull back.
Bruce Scott TOK
john baez <ba...@charity.ucr.edu> wrote:
>It's worth knowing, if you ever talk to mathematicians, that in mathematics
>"covariant" refers to quantities that "push forward", while
>"contravariant" refers to quantities that "pull back". This convention
>applies to lots of things besides tensors, and it makes sense insofar as
>"pushing fowards" is "going with the flow" hence "covariant", while
>"pulling back" is "going against the flow" hence "contravariant".
I am interested in the basics of this "push forward" and "pull back"
language. It is starting to enter gyrokinetic theory in plasma physics
(this is a Lie transform formalism which is very similar to "GR for
phase space").
John, do you have anything online for this?
--
cu,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
Nathan Urban
> Edward Green wrote:
> > Well, you don't expect him to give away what he is simultaneously
> > billing for, do you?
> Nobody likes to be strung along, least of all me. I don't suppose
> I should complain, though, as his original remark was very stimulating,
> even if I do go to my grave without knowing what the damn reason is.
Well, there are these things called "libraries"...
I don't think he's stringing you along. I have the same attitude to
people who ask FAQ questions. I'll give a terse answer and refer them
to the FAQ for further details if they're interested. I won't give
a detailed explanation if one can be found in the FAQ. That's why
there _is_ a FAQ: so people don't have to keep repeating explanations.
I expect that writing a book is similar -- you do it so you don't have
to keep telling people the same old thing.
Nathan Urban
> I am interested in the basics of this "push forward" and "pull back"
> language.
Read the latter half of this:
http://www.deja.com/=dnc/msgid.xp?MID=%3C7gts48$ipa$1...@crib.corepower.com%3E
It briefly explains the concept.
Lewis Mammel
>
> In article <380A840D...@ameritech.net>, Lewis Mammel <lhma...@ameritech.net> wrote:
>
> > Edward Green wrote:
>
> > > Well, you don't expect him to give away what he is simultaneously
> > > billing for, do you?
>
> > Nobody likes to be strung along, least of all me. I don't suppose
> > I should complain, though, as his original remark was very stimulating,
> > even if I do go to my grave without knowing what the damn reason is.
>
> Well, there are these things called "libraries"...
I have access to a public library, a Community College Library, and several
Borders Bookstores, which function sometimes as libraries. None of these has
Baez's book.
> I don't think he's stringing you along. I have the same attitude to
> people who ask FAQ questions.
Do you even know what the question is? There seems to be enough information
from several contributors to this thread to make a reasoanble answer to it,
although John dropped his "simple reason" teaser after most of those were
already present. So is it just what's obvious from the general information?
Or is there some cute twist or something? Do you know? ... if you know what
the question was, that is.
> I'll give a terse answer and refer them
> to the FAQ for further details if they're interested. I won't give
> a detailed explanation if one can be found in the FAQ. That's why
> there _is_ a FAQ: so people don't have to keep repeating explanations.
> I expect that writing a book is similar -- you do it so you don't have
> to keep telling people the same old thing.
He started it by throwing out a teaser, then another one after that.
"You call it wonderful, I call it crass" - go look that one up.
Lew Mammel, Jr.
Nathan Urban
> Nathan Urban wrote:
> > > Nobody likes to be strung along, least of all me. I don't suppose
> > > I should complain, though, as his original remark was very stimulating,
> > > even if I do go to my grave without knowing what the damn reason is.
> > Well, there are these things called "libraries"...
> I have access to a public library, a Community College Library, and several
> Borders Bookstores, which function sometimes as libraries. None of these has
> Baez's book.
Try a university library, or if you can't get to one, get one of the
other libraries to obtain it from a university through interlibrary loan.
> > I don't think he's stringing you along. I have the same attitude to
> > people who ask FAQ questions.
> Do you even know what the question is?
Yes, I know what the question is. I posted a URL to an old DejaNews
article of mine on the subject. But if you're going to be a jerk about
it, then I regret having done so.
> There seems to be enough information
> from several contributors to this thread to make a reasoanble answer to it,
> although John dropped his "simple reason" teaser after most of those were
> already present. So is it just what's obvious from the general information?
I don't think you can piece it together easily from what other people
have posted so far. Contrary to some people's expectations, one can't
get _everything_ on a silver platter over the Internet.
> > I'll give a terse answer and refer them
> > to the FAQ for further details if they're interested. I won't give
> > a detailed explanation if one can be found in the FAQ. That's why
> > there _is_ a FAQ: so people don't have to keep repeating explanations.
> > I expect that writing a book is similar -- you do it so you don't have
> > to keep telling people the same old thing.
> He started it by throwing out a teaser, then another one after that.
Would you prefer a "Go read the FAQ" type of response, or "The answer
to your question is X in brief, go read the FAQ for details" type
of response, or no response at all? Personally I would prefer some
information to none. I certainly do not expect someone who answers my
questions on Usenet to spend as much time as _I_ want in answering them.
> "You call it wonderful, I call it crass" - go look that one up.
So what would you call someone who's rude to the people who try to
help him?
Neither Baez nor I nor anyone else here is under any obligation to spend
time explaining things to your own satisfaction. On Usenet, you get
what you pay for. You get _more_ than what you pay for, because it's not
costing you anything and people are freely giving their time to respond.
What right do you have to complain about the depth of the response?
I'm sure Baez isn't being a jerk to try to force you to buy his book;
he's probably just sick of explaining the same thing to people when he's
written an entire book on the subject. That, as I pointed out, is _why_
people write textbooks -- to put all the information in one place so they
don't have to explain it to everyone one-by-one. It sure ain't for the
lucrative royalties (at least in the vast majority of scientific texts).
Penny314
Any book by john baez is well worth its' cost.
Covariant tensors have contravariant
components because when you transform the full object ( tensor and its'
components)
you should get the original object.
pennysmith
Penny314
Also look at any basic book on category theory.
best
pennysmith
Bruce Scott TOK
I got this:
Deja.com - Internal Server Error (404)
We're sorry, but we couldn't find the page you asked for! Feel free to
browse through the channels at the bottom of the page, or search for a
topic that interests you.
Thanks for using Deja.com!
I find generally that Deja News has become rather less than entirely
useful since it was bought out and "improved".
Nathan Urban
> In article <7ugbqr$qa2$1...@crib.corepower.com>,
> Nathan Urban <nur...@vt.edu> wrote:
> >In article <7ufit5...@s5tok.rzg.mpg.de>, b...@rzg.mpg.de (Bruce Scott TOK) wrote:
> >> I am interested in the basics of this "push forward" and "pull back"
> >> language.
> >Read the latter half of this:
> > http://www.deja.com/=dnc/msgid.xp?MID=%3C7gts48$ipa$1...@crib.corepower.com%3E
> >It briefly explains the concept.
> I got this:
> Deja.com - Internal Server Error (404)
Hmm. It works fine for me. You might try it again. But if not, I'll
paste in the relevant piece:
Let's say that we have a function phi that maps a manifold M into another
manifold N, phi:M->N. If we have some scalar function f defined on N
(e.g., f:N->R), we can use the function to "pull it back" from being a
function defined on N to a new function, its "pullback under phi" phi^* f,
that is defined on M. To do this, you simply say that the value of phi^*
f at a given point x in M is equal to the value of f at the point phi(x)
in N that the function maps x to: (phi^* f)(x) = f(phi(x)).
Interestingly, there is in general no way to take a function defined on
M and use phi to "push it forward" to become a function defined on N.
(If you try to take a point in N and figure out what the value of the
"pushed-forward function" should be there, you can't, because there might
be zero or many points in M that phi maps to that point.) Consequently,
a function is called "contravariant" by mathematicians since you can
make it go "backwards" from N to M, but not vice versa.
However, you _can_ make _vectors_ go forward. If you have a vector
v_x at some point x in M, then draw any curve through x whose tangent
at x is equal to the vector v_x. (What the curve looks like away from
x doesn't matter.) Map all the points in this curve from M to N using
the phi; you'll get a new curve in N. Find the mapped point phi(x) and
construct the tangent to the new curve there; the resulting vector is the
pushforward phi_* v_x of v_x from M to N. (Note the subscript asterisk
instead of the superscript for the pullback notation.) Consequently,
vectors are called "covariant" by mathematicians since they go
"with" the function from M to N, instead of backwards, against it.
(Physicists use the opposite convention and call them "contravariant"
because they originally focused on how the _components_ of the vector
transformed rather than the vector itself, namely oppositely; I don't
like the physicist convention but you need to be aware of it.)
One-forms or covectors go backward, they're contravariant like functions.
A covector field can be defined by describing what number it produces
when you give it any vector. You can take a covector field w on N to
a covector field phi^* w on M by first taking an arbitrary vector at a
point x in M, pushing it forward to N, finding out what number results
when you insert that vector into w defined on N, and defining phi^* w at
x in M to be that number. (Exercise: think about why there's no way to
push forward covectors.)
In general, I'll call things with all upstairs indices (like vectors)
"covariant", things with all downstairs or no indices (like covectors
or functions) "contravariant", and mixtures as "mixed".
> I find generally that Deja News has become rather less than entirely
> useful since it was bought out and "improved".
Indeed. It's a shame, because there's a real need for a _permanent_
and _reliable_ archive of Usenet.
Erik Max Francis
> Deja.com - Internal Server Error (404)
...
> I find generally that Deja News has become rather less than entirely
> useful since it was bought out and "improved".
Try again. Internal server errors are never permanent.
--
Erik Max Francis | icq 16063900 | whois mf303 | email m...@alcyone.com
Alcyone Systems | irc maxxon (efnet) | web http://www.alcyone.com/max/
San Jose, CA | languages en, eo | icbm 37 20 07 N 121 53 38 W
USA | Tue 1999 Oct 19 (32%/950) | &tSftDotIotE
__
/ \ The hour which gives us life begins to take it away.
\__/ Seneca
ph...@interpac.net
>In article <7ugbqr$qa2$1...@crib.corepower.com>,
>Nathan Urban <nur...@vt.edu> wrote:
>>In article <7ufit5...@s5tok.rzg.mpg.de>, b...@rzg.mpg.de (Bruce Scott TOK) wrote:
>>
>>> I am interested in the basics of this "push forward" and "pull back"
>>> language.
>>
>>Read the latter half of this:
>>
>> http://www.deja.com/=dnc/msgid.xp?MID=%3C7gts48$ipa$1...@crib.corepower.com%3E
>>
>>It briefly explains the concept.
>
>I got this:
>
>Deja.com - Internal Server Error (404)
>
> We're sorry, but we couldn't find the page you asked for! Feel free to
> browse through the channels at the bottom of the page, or search for a
> topic that interests you.
>
> Thanks for using Deja.com!
>
>I find generally that Deja News has become rather less than entirely
>useful since it was bought out and "improved".
>
>--
>cu,
>Bruce
>
Well my story begins about a year ago when I
picked up on deja-news and began using it as
my primary newsreader. Deja-News provided a
full-page view of messages, but now you have
to click on "original usenet format," to see
80-character screen, otherwise deja.com uses
1/3 of the screen for splashy ad space.
Deja.com is not the same organization as the
original deja-news.com. Deja is commercial,
that's the bottom line. They are competing
with ebay, yahoo, and others. Someday they
may no longer offer free archive access.
During this transition from a news-oriented
service to a product-orientated seller-site,
they've rewritten their system's html's many
times ... and apparently they are still not
satisfied. With each rewrite their system
goes down (or wacky) for up to three days.
My experience since the big buy-out has been
really frustrating access periodically, that
amounts to about a dozen resolves on my part
to unsubrscribe. They've never once posted
a home-page stating that their system would
be down for X hours due to improvements and
testing ... They have always left the caller
hanging onto a mystifying discrepency. Last
week it was for two times that all messages
began on 9/9/99 ... Now that kind of thing
can easily freak an average user aware that
a 9/9/99 bug is in the wild. First time it
happened to me I jumped out of deja and ran
a virus check on my machine.
The good part of deja.com is that they still
own all of deja-news archives, but the bad
thing is that they're patently unreliable.
End of story is that I went on to subscribe
http://www.remarq.com, and install the old
standy: FreeAgent (http://www.forteinc.com).
Free Agent along with a few free nntp-server
sites obtained from newzbot, gives reliable
access to the usenet forums, but still does
not answer the need for access to archives.
Hope this helps resolve your question about
deja.com access from another viewpoint.
/ph
- - - - - - -
Lewis Mammel
> ... Consequently,
> vectors are called "covariant" by mathematicians since they go
> "with" the function from M to N, instead of backwards, against it.
> (Physicists use the opposite convention and call them "contravariant"
> because they originally focused on how the _components_ of the vector
> transformed rather than the vector itself, namely oppositely; I don't
> like the physicist convention but you need to be aware of it.)
This is a gloss. In the first place, tensors do not depend on manifolds
for their definition. They are simple constructs of linear algebra, which
I don't believe is the preserve of physicists. Finite Dimensional Vector
Algebra by Marcus (1973) gives the tradtional definition, where the tensor
product of p copies of V and q copies of V* ( dual to V ) is called
p-contravariant and q-covariant. This is a very abstract hard-core math
book, which gives L(V1,...,Vm:R)* as an example of a tensor space, calling it
the "dual product space" and commenting that it is "sometimes given as a
defintion [ of a tensor space ]," as I indeed had always accepted, having
learned it from Vector Spaces of Finite Dimension by Shephard.
So how is it that this nomenclature is no longer operative? This whole business
gets curiouser and curiouser the more I look into it. It really seems that the
"contravariance" that you mention just has nothing whatsoever to do with the
traditional usage. Manifolds, Tensor Analysis, and Applications, by Abraham, Marsden
and Ratiou ( 1983 ) ( BTW from my library at work ) also adheres to the traditional
definition of contravariant, although these authors, for some reason, give an idiosyncratic
definition of tensors, equating the tensor space to the space of multilinear forms instead
of its dual. They compensate by calling L(V1*,...,Vp*,V1,...,Vq:R) p-contravariant and
q-covariant. It works out the same since L(V*) = V** == V is contravariant.
So anyway, do the "old" times go all they up through the 80's, or what? It's funny
that Lichnerowicz, who gives a really old-time component based tensor definition,
comments equably ( re co- and contravariant ) "Naturally this terminology varies
according to which of the spaces En and En* is considered to be given first."
Lew Mammel, Jr.
jmfb...@aol.com
nur...@crib.corepower.com (Nathan Urban) wrote:
>In article <7ui2bq...@s4bds.rzg.mpg.de>, b...@rzg.mpg.de (Bruce Scott
>
>> In article <7ugbqr$qa2$1...@crib.corepower.com>,
>> Nathan Urban <nur...@vt.edu> wrote:
>
>> >In article <7ufit5...@s5tok.rzg.mpg.de>, b...@rzg.mpg.de (Bruce Scott
TOK) wrote:
>
>> >> I am interested in the basics of this "push forward" and "pull back"
>> >> language.
>
>> >Read the latter half of this:
>> >
http://www.deja.com/=dnc/msgid.xp?MID=%3C7gts48$ipa$1...@crib.corepower.com%3E
>> >It briefly explains the concept.
>
>> I got this:
>
>> Deja.com - Internal Server Error (404)
>
>paste in the relevant piece:
>> I find generally that Deja News has become rather less than entirely
>> useful since it was bought out and "improved".
>
>Indeed. It's a shame, because there's a real need for a _permanent_
>and _reliable_ archive of Usenet.
There was a discussion about this in alt.folklore.computers.
It is not a good idea to depend on a business for this.
A volunteer is starting (or trying to start) to save a few
newsgroups that deal with our folklore. Somebody may
want to to try do a similar thing with these newsgroups.
/BAH
Subtract a hundred and four for e-mail.
Nathan Urban
> Nathan Urban wrote:
> > ... Consequently,
> > vectors are called "covariant" by mathematicians since they go
> > "with" the function from M to N, instead of backwards, against it.
> > (Physicists use the opposite convention and call them "contravariant"
> > because they originally focused on how the _components_ of the vector
> > transformed rather than the vector itself, namely oppositely; I don't
> > like the physicist convention but you need to be aware of it.)
> This is a gloss. In the first place, tensors do not depend on manifolds
> for their definition.
True, but tensor fields do (and tensor fields are what appear in
relativity), and you wanted to know what Baez and others were talking
about when they mentioned pullbacks and pushforwards.
> They are simple constructs of linear algebra, which
> I don't believe is the preserve of physicists.
What do physicists have to do with it? I'm saying that the convention
on _what these objects are called_ depends on who you're talking to.
Of course _how the objects themselves behave_ can't depend on who you're
talking to.
> Finite Dimensional Vector
> Algebra by Marcus (1973) gives the tradtional definition, where the tensor
> product of p copies of V and q copies of V* ( dual to V ) is called
> p-contravariant and q-covariant. This is a very abstract hard-core math
> book, which gives L(V1,...,Vm:R)* as an example of a tensor space, calling it
> the "dual product space" and commenting that it is "sometimes given as a
> defintion [ of a tensor space ]," as I indeed had always accepted, having
> learned it from Vector Spaces of Finite Dimension by Shephard.
> So how is it that this nomenclature is no longer operative?
What nomenclature? The abstract definition of a tensor space?
That's certainly in use. "Contravariant" and "covariant"? Those are
also in use, though which term means what is something that depends on
who you're talking to -- people have opposite conventions on this matter.
> This whole
> business gets curiouser and curiouser the more I look into it. It really
> seems that the "contravariance" that you mention just has nothing whatsoever
> to do with the traditional usage.
"Covariance" and "contravariance" have pretty consistent meanings, namely
in category theory terms of functors that "preserve order of composition"
or "reverse order of composition". What that means in a concrete case
depends on the category in which you are working.
Think about what "covariant" and "contravariant" mean in terms of tensors.
In a loose sense, they "transform oppositely under changes in coordinate
system". But what exactly does that mean? Can that be made precise in
a rigorous way? It turns out that it can:
How does a vector transform? Via a linear transformation of vector
spaces. In all generality, given two linear transformations L:U->V and
M:V->W, we may compose them to give a product LM:U->W (defined by
LM(u) = M(L(u)).
How does a covector transform? Via a linear transformation of dual
spaces. Let's examine L more closely. It is a map from U to V.
From it, we may construct a new map, the adjoint L* of L, which is a
linear transformation of dual spaces L*:V*->U*. How do we define it?
If we take some covector f in V*, L* transforms it to the unique covector
L*(f) in U* satisfying L*(f)(u) = f(L(u)) for all vectors u in U.
Already we see some backwardness, which is made more obvious when you
consider compositions: if you take the adjoint (LM)* of the product LM,
you get the product M*L*. So if you transform a vector by composition of
two linear transformations, you can transform a covector by composition of
two adjoint transformations _in the opposite order_. This is the rigorous
meaning of "covectors transform in a manner opposite to that of vectors".
Consequently, if we arbitrarily take vectors as "fundamental", we can
say that they transform in the "normal way" and hence covariantly.
Covectors transform in the "opposite way" (as compared to how vectors
transform) and hence contravariantly.
All of this generalizes to higher-order tensors, of course.
As to the physicist vs. mathematician thing: it boils down to active
vs. passive transformations. You may view a transformation as either
active (mapping vectors to vectors) or passive (leaving vectors the
same but changing the basis to get new components). If you consider
the matrices that effect these transformations, the matrix for the
active transformation is the inverse of the matrix for the passive
transformation. And since (AB)^-1 = B^-1 A^-1, whether the transformation
goes "normally" or "oppositely" thus depends on whether you're looking
at the _vectors_ (the "mathematician way") or their _components_ (the
"physicist way"). However, I think mathematicians didn't always do it
the same way either (and probably still don't).
> So anyway, do the "old" times go all they up through the 80's, or what?
I don't know the history of this terminology.
> It's funny
> that Lichnerowicz, who gives a really old-time component based tensor
> definition, comments equably ( re co- and contravariant ) "Naturally this
> terminology varies according to which of the spaces En and En* is considered
> to be given first."
Yes, that's what I was getting at above. If you consider En as
"fundamental" and En* as "dual to it", then you say that the latter
transforms oppositely to the former and hence "contra" variantly.
But since a dual space is a vector space in its own right and its
transformations are hence linear transformations on a vector space, you
could take it as "fundamental" and call its transformations "covariant".
Lewis Mammel
>
> In article <380D55C0...@ameritech.net>, Lewis Mammel <lhma...@ameritech.net> wrote:
>
> > Finite Dimensional Vector
> > Algebra by Marcus (1973) gives the tradtional definition, where the tensor
> > product of p copies of V and q copies of V* ( dual to V ) is called
> > p-contravariant and q-covariant. This is a very abstract hard-core math
> > book, which gives L(V1,...,Vm:R)* as an example of a tensor space, calling it
> > the "dual product space" and commenting that it is "sometimes given as a
> > defintion [ of a tensor space ]," as I indeed had always accepted, having
> > learned it from Vector Spaces of Finite Dimension by Shephard.
>
> > So how is it that this nomenclature is no longer operative?
>
> What nomenclature? The abstract definition of a tensor space?
> That's certainly in use. "Contravariant" and "covariant"? Those are
> also in use, though which term means what is something that depends on
> who you're talking to -- people have opposite conventions on this matter.
Of course I mean co- and contravariance. I was citing a recent hard-core
math book which adheres to the so-called "physicists convention" for this
nomenclature. I was illustrating just how abstract the approach was to show
that this book is in no way an applied math book. So here's a recent pure math
book on linear algebra that adheres to the old, applied math convention. Well,
it is almost 30 years old actually, so maybe the math community is shifting
away from it in more recent years. It's hard to know what "modern" implies.
Are there more recent linear algebra textbooks which promulgate the
modernized convention?
> Think about what "covariant" and "contravariant" mean in terms of tensors.
> <snip>
>
> Consequently, if we arbitrarily take vectors as "fundamental", we can
> say that they transform in the "normal way" and hence covariantly.
> Covectors transform in the "opposite way" (as compared to how vectors
> transform) and hence contravariantly.
This certainly makes sense, and I agree it seems clear that the original
convention was due to the identification of a vector with its contravariant
( under change of basis ) components. Most of this usage is in the context
of component calculation, so I suppose the two conventions will continue
to coexist.
Lew Mammel, Jr.
Lewis Mammel
>
> Nathan Urban wrote:
>
> >
> > Consequently, if we arbitrarily take vectors as "fundamental", we can
> > say that they transform in the "normal way" and hence covariantly.
> > Covectors transform in the "opposite way" (as compared to how vectors
> > transform) and hence contravariantly.
>
> convention was due to the identification of a vector with its contravariant
> ( under change of basis ) components. Most of this usage is in the context
> of component calculation, so I suppose the two conventions will continue
> to coexist.
Just to expand a litle further, and in favor of the "modern" viewpoint, I
would say that the identification of a vector with its contravariant components
is somewhat self contradictory. These components naturally associate with the
dual basis, since the forms which map a vector to each of its contravariant
components ARE the dual basis. Lichnerowicz ( in the Methuen Monograph, Tensor
Calculus ) names the basis dual to ( e_1, ...,e_n ) as ( x^1, ..., x^n ), which
emphasizes the point. The translators, J.W. Leech and D.J. Newman, seem not to
have understood the meaning here, as they inexplicably add a footnote "explaining"
that "The dual basis arrived at in this manner obviously depends on the choice
of the vector x." There is no "vector x" ! These are the forms which map ANY
vector x to each of its contravariant components. BTW, this error is something that
I identified many years ago when studying this book, but I think it serves to
illustrate how confusing these basic ideas are.
Lew Mammel, Jr.