contravariant and covariant objects

["John S. Denker" <jsd@MONMOUTH.COM>]

Joe Heafner wrote:
> ... beginning to make sense now, especially the difference
> between covariant and contravariant vectors.

I've been thinking about that lately, too.  Here are my
notes:

For vectors in ordinary non-curved Euclidean space, there
is no distinction between covariant and contravariant
vectors, so the issue isn't usually mentioned when
vectors are first introduced.

Most people first encounter the words covariant and
contravariant in General Relativity class ... but the
ideas crop up in many other situations.  In general
the distinction is important whenever you have
 ++ no dot product, which implies
   no notion of "length" of vectors, and
   no notion of "angle" between vectors, especially
   no notion of "perpendicular" vectors;  also
 ++ no way of transposing a row vector into a column vector

In more detail:

 *) Equivalent terminology:
   -- contravariant versus covariant
   -- column vector versus row vector
   -- vector        versus one-form
   -- bra <...|     versus ket |...>

 *) A gradient is the perfect example of a one-form.
Similarly, in wave mechanics, "k" shouldn't be called
the wave-vector but rather the wave-one-form.  See
Reference 1 for pictures and additional examples.

(Note the reference cites the E-field as an example of
a 1-form, but I'm not 100% happy with that;  I suppose
it's OK for electrostatics but it's not relativistically
correct.)

 *) In thermodynamics, in general you have no idea
whether a change in volume is "perpendicular" to a
change in pressure or not.  Consequently, although
you can specify a point in state-space by a vector
of variables, you cannot take dot products between
such vectors.

 *) Electromagnetism in anisotropic media is analogous
to ordinary dynamics in curved space.  See Reference 2.

 *) Quantum Mechanics distinguishes a bra <...| from
a ket |...>.  Here the problem is not a lack of a dot
product, but rather the presence of complex numbers.
The bra is not the transpose but rather the _conjugate_
transpose of the ket.  And it's nice to be able to
write the outer product as |ket><bra|.

 *) There are many other situations where you might
want to distinguish row vectors from column vectors.
One example is non-square matrices, which show up when
you have an overdetermined or underdetermined system of
linear equations ... which is fairly common in pattern-
recognition and data-analysis tasks.

Note that Matlab and Scilab systematically distinguish
row vectors from column vectors;  they won't take the
transpose except when you explicitly command it.

Not having a dot product isn't the end of the world.
Even if you aren't allowed to take transposes, you can
always form the scalar product between a row vector and a
column vector, i.e. <bra| times |ket>.  What you can't
do is form the scalar product between two row vectors,
or between two column vectors.  Such products exist
but they're not scalars;  rather, they are higher-grade
objects.

 *) I believe the connection to Clifford Algebra works
like this:

  a) Starting from Clifford Algebra, delete all forms
of multiplication of vectors (and multivectors) except
for the wedge product.  Keep the wedge product (which
includes multiplication by scalars).

  b) Extend the grade-structure to include negative grades.
A one-form is a grade=-1 object.  The wedge product of two
one forms is a two-form, i.e. a grade=-2 object.  The
(wedge) product of a vector with a one-form is a grade=0
object, i.e. a scalar.

  c) This naturally associates an p-vector (grade=p) with a
p-form (grade=-p).  This is not to be confused with the Hodge
dual, which associates a vector with a pseudovector, and
more generally associates an object of grade=p with an object
of grade=(D-p), where D is the dimensionality of the space
you're working in.  This use of the Hodge dual requires a
metric (a dot product) and also requires knowing D (which is
suspicious all by itself, since most of the laws of physics
don't care what D is).

==============
References:

1) Bernard Jancewicz (Wroclaw U.)
"ANSWER TO QUESTION 55: ARE THERE PICTORIAL EXAMPLES THAT
DISTINGUISH COVARIANT AND CONTRAVARIANT VECTORS?"
Jul 1998. 4pp. e-Print Archive: gr-qc/9807044
  http://xxx.lanl.gov/abs/gr-qc/9807044

2) B. Jancewicz: "A variable metric electrodynamics.
The Coulomb and Biot- Savart laws in anisotropic media,"
Ann. of Phys. 245, 227-274 (1996)