DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

Cartesian coordinates
5.4
5.2
5.0
4.8
0.010 0.000 0.010 0.020
Energy per unit length U (upper curves) and tension
T (lower curves) as functions of the sign-preserving
square root of the state parameter: the full lines rep-
resent the actual values derived from the Witten mi-
croscopic model, the dashed line being the values ob-
tained with the Carter–Peter macroscopic model. The
fit is almost perfect up to the point where U and T
both increase for positive values of the state param-
eter. This is satisfying since the macroscopic model
there ceases to be valid because of instabilities.
the shape of the energy per unit length and tension
as function of the state parameter √ w together
with a comparison with the same functions
in the Witten conducting string model. It
is clear from the figure that the fit is valid in most
of the parameter space except where the string
itself is unstable. The string equation of state is
different according to the timelike or spacelike
character of the current. For the former it is
U = T + m 2
⋆ exp 2 m 2
− T /m 2
⋆ − 1 ,
while for positive w we have
T = U −m 2 ⋆
1 − exp
−2
U − m 2
/m 2 ⋆
.
See conducting string, cosmic string, current
carrier (cosmic string), current generation (cosmic
string), current instability (cosmic string),
phase frequency threshold, summation convention,
Witten conducting string, worldsheet geometry.
Cartesian coordinates A coordinate system
in any number of spatial dimensions, where the
coordinates {x i } define orthogonal coordinate
lines. In such a system, Pythagoris’ theorem
holds in its simplest form:
© 2001 by CRC Press LLC
ds 2 = δij dx i dx j ,
where the summation convention is assumed for
i and j over their range.
Or, in nonflat spaces or spacetimes, a system
in which the coordinates have many of
the properties of rectangular coordinates, but
Pythagoris’ theorem must be written as:
ds 2 = gij (x k )dx i dx j ,
where gij (x k ) is the coordinate dependent metric
tensor. In this case the description is reserved
for coordinates that all have an infinite range,
and/or where the metric coefficients gij (x k ) are
“near” δij everywhere.
Cartesian coordinates [in a plane] A relationship
between the points of the plane and
pairs of ordered numbers called coordinates.
The pair of numbers corresponding to each point
of the plane is determined by the projection of
the point on each of two straight lines or axes
which are perpendicular to each other. Cartesian
coordinates thus establish a nonsingular relationship
between pairs of numbers and points
in a plane. The point in which the two axes intersect
is called the origin. The horizontal axis
is called the x-axis, and the vertical axis is called
the y-axis. Named after Rene Descartes (1596–
1650).
Cartesian coordinates [in space] A relationship
that is established between the points of
space and trios of ordered numbers called coordinates.
Cartesian coordinates are a coordinate
system in which the trio of numbers corresponding
to each point of space is determined by the
projection of the point on each of three straight
lines or axes which are perpendicular to each
other and intersect in a single point. The positive
direction of the z-axis is generally set, such that
the vectorial product of a non-null vector along
the positive x-axis times a non-null vector along
the positive y-axis generates a vector along the
positive z-axis; this is called a right-handed coordinate
system. Named after Rene Descartes
(1596–1650).
Casagrande size classification A classification
of sediment by particle size (diameter). The
basis for the Unified Soils Classification commonly
used by engineers.