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[Phys-l] two types of vector



On 04/09/2009 02:00 PM, chuck britton wrote:

The Poynting vector (don't ALL vectors point ;-) ...

I know it's bad luck to answer a rhetorical question, and I enjoy
a parenthetical pun as much as the next guy, but ... the answer is
no, not all vectors are pointy vectors.

The other type of vector is familiar from the contour lines on a
topographical map. Such a vector is called a one-form. It is
properly called a vector because it upholds all the axioms that
define a vector space.

We can mention some ideas, and show the correspondence between them:

pointy vector one-form
column vector row vector
ket bra
contravariant covariant
tip and tail contour lines
rank-1 tensor rank-1 differential form

Quite commonly you have a _metric_ i.e. a machine that will tell you
the length of a vector and the angle between two vectors. In such a
case, there is a one-to-one correspondence between each pointy vector
and its corresponding one-form, and people tend to get careless about
making the distinction.

HOWEVER there are some important cases where you don't have a usable
metric. This includes systems that have a topology, but do not have
a geometry (because they do not have a metric).

Thermodynamics is a spectacular example. You don't know the angle
between the "pressure axis" and the "temperature axis" (or any other
angle) and there is no way to take a dot product or compare lengths
in different directions. Therefore you cannot convert one-forms to
pointy vectors or vice versa. That makes it super-useful to have a
bag of tricks for dealing with both types of vectors, one-forms as
well as pointy vectors.

Many of the key equations in thermodynamics are in fact vector
equations. Most of the things you were told were "infinitesimals"
are more properly formalized as vectors. Recognizing them as vectors
strips off many layers of misconceptions.

A partially similar situation arises in general relativity. There
usually is a metric, but it might be distinctly not worth the trouble
to figure out what the metric is doing under the given conditions.
This puts a premium on techniques for dealing with vectors that do
not require using the metric.

Even without a metric, you can compute a gradient vector (which is
a one-form). You can use the gradient to construct and evaluate a
multi-dimensional Taylor series. You can compute the scalar product
(i.e. the contraction) between a row vector and a column vector
(e.g. a Dirac bra_ket) (or e.g. the number of contour lines pierced
by a pointy vector).

http://www.av8n.com/physics/intro-vector.htm