Newsgroups: sci.math,sci.physics
From: roberta@tucson.Princeton.EDU (Robert Wade Anderson)
Subject: Co/Contravariance terminology
Date: Thu, 19 Jan 1995 15:58:27 GMT
I have a quick question that I've been wodering about: Why are certain vectors
called "contravariant" and some called "covariant"? It was my understanding
that co/contra variance referred to different ways of "taking" the coordinates
of the vector relative to a basis. So I would think that this means "vector
expressed in terms of its co/contravariant components" but I think this is
not wholly the case. Is there more to the story? Can a vector be *inherently*
co/contravariant?
Thanks, e-mail appreciated.
-Bob
==============================================================================
From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,sci.physics
Subject: Re: Co/Contravariance terminology
Date: 19 Jan 1995 18:08:23 GMT
In article <1995Jan19.155827.4631@princeton.edu>,
Robert Wade Anderson wrote:
>I have a quick question that I've been wodering about: Why are certain vectors
>called "contravariant" and some called "covariant"?
[snip]
> Is there more to the story? Can a vector be *inherently*
>co/contravariant?
Hmm, then Yes, then No. A vector is an element of a vector space, period. It
only become co/contra variant when you expect it to be transformed. You see, the
vectors in question aren't just abstract copies of Euclidean space; rather, they
are associated to (for example) manifolds. As such, whenever you have a
function from one manifold to another, you expect to have some correspondence
between the vector spaces attached to the one and the vector spaces
attached to the other. Just how the correspondences go is what separates
the "cos" from the "contras".
The classic example of a co-variant vector space is the tangent space to
a manifold M at a point p. If N is another manifold and
f: M --> N is a differentiable function, let q=f(p) (so q is a point
in N). Then the derivative f'(p) is a linear map carrying the
tangent space T_p(M) of M at p to the tangent space T_q(N) of N
at q. You know it's a linear map going in this direction if you think
about the matrix of partial derivatives used to represent this linear map;
it has as many columns as the dimension of M and as many rows as the
dimension of N . The fact that f induced a map "going the same way"
is why we call the tangent vectors "co-variant". (More precisely, the
tangent-space functor is a covariant functor from pointed manifolds to
vector spaces. But you probably don't like to hear it that way.)
The classic example of a contravariant vector space is the dual to a
vector space. If V is a vector space, V* is defined to be the collection
of all linear maps V--> R to the underlying field. It forms a vector
space in its own right and in fact is isomorphic to (has the same dimension as)
V, if V is finite dimensional. But notice that if f: V--> W is a linear
map, then we get an induced map f* _not_ from V* to W* but rather the
other way around: f*: W* --> V* is defined by taking an element phi in W*
and defining f*(phi) to be the following element of V* :
f*(phi) is the linear map V--> R which carries each v in V to
the real number phi(f(v)). (It takes longer to write it out in English
than it ought to; it's really quite natural once you get the hang of it.)
The fact that f induced a map "going the other way" is why we
call the dual vectors "contra-variant". (More precisely, the dual-space
functor is a contravariant functor from vector spaces to vector spaces).
And yes, I guess you can say the difference is in how you express things
in terms of a basis but frankly I've noticed that focusing on bases tends
to obscure an understanding of vector spaces rather than enlighten, so
I won't get into that.
By the way, it has been suggested that homology is out of synch and ought
to be renamed. "homology" ought to be called "cohomology" and what we
now call "cohomology" ought to be "contrahomology". But I wouldn't
hold out for a change any time soon.
dave
==============================================================================
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math,sci.physics
Subject: Re: Co/Contravariance terminology
Date: 19 Jan 1995 22:24:41 GMT
In article <3fm9qn$t6j@mp.cs.niu.edu>, rusin@washington (Dave Rusin) writes:
>By the way, it has been suggested that homology is out of synch and ought
>to be renamed. "homology" ought to be called "cohomology" and what we
>now call "cohomology" ought to be "contrahomology". But I wouldn't
>hold out for a change any time soon.
Hilton and Wylie did everything but that in their book. Worth a laugh.
--
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
==============================================================================
From: baez@math.ucr.edu (john baez)
Newsgroups: sci.math,sci.physics
Subject: Re: Co/Contravariance terminology
Date: 24 Jan 1995 16:20:29 -0800
In article , Van wrote:
>In physics, the tangent space is usually called contra-variant, and
>the dual cotangent space covariant, though the opposite makes more
>sense. See Spivak "A Comprehensive Intro to Differential Geometry" Vol. 1
>for a discusion of this, as well as a discussion of the basic ideas
>of Co/Contravariant vector fields.
I suspect that the physicists you are referring to wouldn't recognize a
tangent space if it bit them on the leg. Instead, they talk about
a vector as "being" a bunch of components. In modern notation
we have
A = A^i partial_i
The physicist sees the A^i with its index upstairs, calls that the
vector, and calls it contravariant 'cause it transforms in an
certain sort of way under coordinate transformations. The mathematician
sees the partial_i, calls that a vector, and calls it covariant 'cause
it transforms the opposite way! (Well, a certain sort of
mathematician... a modern category-theoretic sort of mathematician has a
better reason for calling tangent vectors covariant.) Meanwhile, of
course, the vector A, knowing nothing of coordinates, sits there
happily remaining the same no matter what coordinates happen to be used
to describe it.
For a slightly less flippant explanation, you could try my book Gauge
Fields Knots and Gravity, where I tried my best to explain this stuff,
and also of course the mystery of "active" vs "passive" coordinate
transformations.
==============================================================================-
Newsgroups: sci.math,sci.physics
From: vanjac@netcom.com (Van)
Subject: Re: Co/Contravariance terminology
Date: Tue, 24 Jan 1995 18:31:40 GMT
In article <3fm9qn$t6j@mp.cs.niu.edu>,
Dave Rusin wrote:
>In article <1995Jan19.155827.4631@princeton.edu>,
>Robert Wade Anderson wrote:
>>I have a quick question that I've been wodering about:Why are certain vectors
>>called "contravariant" and some called "covariant"?
>> Is there more to the story? Can a vector be *inherently*
>>co/contravariant?
>
>The classic example of a co-variant vector space is the tangent space to
>a manifold M at a point p.
In physics, the tangent space is usually called contra-variant, and
the dual cotangent space covariant, though the opposite makes more
sense. See Spivak "A Comprehensive Intro to Differential Geometry" Vol. 1
for a discusion of this, as well as a discussion of the basic ideas
of Co/Contravariant vector fields.
--
Van - Internet address - vanjac@netcom.com
==============================================================================
From: Bruce Bowen
Subject: Re: Tensor Subscripts & Superscripts
Date: Thu, 16 Sep 1999 21:45:34 GMT
Newsgroups: sci.math
To: atp@shell9.ba.best.com
Check out the following two websites for a simple geometric
interpretation of covariance and controvariance that's useful for most
of what you need it for.
http://www.seanet.com/~ksbrown/kmath416.htm
http://home.pacbell.net/bbowen/covariant.htm
-Bruce bbowen@pppppppacbell.net (remove the stutter)
In article <37d6cd68$0$224@nntp1.ba.best.com>,
Alexander Poquet wrote:
> Hi. I am currently in the process of teaching myself
> tensor calculus and I find I am not fully grasping the
> difference between their superscripts and subscripts.
> The difference is downplayed in the book I am using,
> possibly because it focuses on continuum mechanics;
> I personally am interested in tensors from a more
> mathematical standpoint.
Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.
==============================================================================
From: johncp@magicnet.net (John C. Polasek)
Subject: Re: Tensor Subscripts & Superscripts
Date: Mon, 20 Sep 1999 04:16:26 GMT
Newsgroups: sci.math
Alex: perhaps this will give you some insight. Covariant and
contravariant tensors appear often in scalar products. Consider the
product W = xi Kij fj, where
xi is the displacement you apply to an object characterised by
Kij a strain/stress tensor (mixed)
fj as the resulting forces generated.
Physically, you push an amount xi covariant into the tensor Kij which
can redirect your effort, with the resulting forces fj contravariant.
This product is W a scalar, the energy content. The x is covariant
because naturally it varies as the coordinate axes. But for constant
energy, then the fj will vary oppositely, or contravariantly.
At least that's how I look at it and should help you out some.
John C. Polasek
>
>In article <37d6cd68$0$224@nntp1.ba.best.com>,
> Alexander Poquet wrote:
>> Hi. I am currently in the process of teaching myself
>> tensor calculus and I find I am not fully grasping the
>> difference between their superscripts and subscripts.
>> The difference is downplayed in the book I am using,
>> possibly because it focuses on continuum mechanics;
>> I personally am interested in tensors from a more
>> mathematical standpoint.
>
>
>Sent via Deja.com http://www.deja.com/
>Share what you know. Learn what you don't.