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Re: contravariant and covariant objects



Joe Heafner wrote:

http://www.mathpages.com/rr/s5-02/5-02.htm>:

"Figure 1 shows an arbitrary coordinate system, with
the axes X1 and X2, as well as the contravariant and
covariant components of the position vector P with
respect to these coordinates. As can be seen, the jth
component of the "contravariant path" from O to P
consists of a segment parallel to jth coordinate
axis, whereas the jth component of the "covariant
path" consists of a segment perpendicular to all the
axes other than the jth. This is the essential
distinction between the contravariant and covariant
ways of expressing a vector (or, more generally, a
tensor)."

1) That's interesting. That makes clearer the way
contravariant and covariant objects are typically
used in General Relativity, in situations where you
do have a metric and could define a dot product if
you wanted to, but you have to be careful because
as likely as not you've got a non-orthonormal basis.
You could in principle orthonormalize it (via Gram-
Schmidt) if you wanted to, but that would introduce
other problems. There is a fundamental dilemma: in
polar coordinates, do you want your basis vector to
represent a unit change in the theta coordinate
(d(theta)) or do you want it to represent a step of
unit length in the theta direction (r d(theta))???

2) On the other hand, I'm leery of adopting the
mathpages picture as the "essential distinction"
because it uses the idea of "perpendicular" to define
what is meant by covariant. That sacrifices the
most-crucial application of the idea, namely, cases
where you have no metric at all, i.e. not the vaguest
idea of what "perpendicular" means.

To me the essential idea is that a one-form is
something that you can multiply by a vector
!!without!! using any idea of metric (and the
product is a scalar).

The canonical example of this is a gradient, which
is a one-form. It is represented by the contour
lines on a topo map. You can calculate the scalar
product graphically and geometrically, by counting
the number of contour lines pierced by a displacement
vector.


However, Weinreich (p. 7) emphasizes that there is
also a contravariant/covariant distinction in a change of scale of the
units along the axes.

Oh yeah. Vectors and one-forms behave completely
differently under a change of scale.

An often-encountered but less-often-understood example
of this arises in computer programming, in optimization
problems. Suppose you are searching for a minimum of
some objective function (V) by doing "gradient descent".
You take a step such that
X_i := X_i - (partial V)/(partial X_i)

For starters, that's dimensionally unsound. It's
trying to add a one-form to a vector. You can see
how broken it is if you change from measuring X in
meters to measuring X in inches. The numerical value
of each X_i gets bigger, but the numerical value of
each (partial V)/(partial X_i) gets smaller.

The algorithm as written above is widely used but has
all sorts of horrible performance problems. It can
be improved by multiplying the gradient by something
with the right units (such as the inverse of a second
derivative).