Page 68 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 49
EXAMPLE 2.16: The Euclidean metric in a rectangular coordinate system—see
Equation 2.12—is obtained from Equation 2.17 by setting gy = Sij, etc.

COMMENT 2.5 : Assume that a transformation ca n b e established from a

given coordinate system {s^} t o a rectangular coordinate system {s^}, an d

let J — (dUi/dsi) be the Jacobian matrix of this transformation. Then,

the matrix g = (gij) o f the Euclidean metric in th e {sj} coordinate system

T
can be expressed as g — J J.

EXAMPLE 2.17: Consider the transformation of systems from cylindrical {sj}
coordinates to rectangular {~s j} coordinates in Equation 2.7. The Jacobian is

and, hence, the Euclidean metric for cylindrical coordinates is given by

i-e., gn = 933 = 1, 922 = s\, and g^ = 0 (i =£ j), which is the same result
as that obtained in Example 2.13. Of course, other (non-Euclidean) metrics
could be considered for the cylindrical coordinate system, as well.

COMMENT 2.6: Suppose that a transformation between a system o f gener-

alized curvilinear coordinates {s^} and an underlying system of rectangular

coordinates {¥,} exists. Then, th e metric coefficients ca n be expressed a s

In Equation 2.16 above we used the metric coefficients g^ to represent
the coefficients in the distance \ds\ for a general coordinate system. The
metric coefficients g^ play an important role in several other operations of
spatiotemporal geometry related to the modeling of natural phenomena, such
as the determination of a suitable metric (the issue is discussed also in the
section entitled "Restrictions on spatiotemporal geometry imposed by physical
laws" on p. 56). The gradient, Laplacian, and divergence operators for a
scalar field X(p) and a vector field X(p) can be expressed in terms of general
curvilinear coordinates as follows