Appendix B: Curvilinear coordinate systems
r
(
q
1
, q
2
)
e
1
e
2
q
1
increasing
q
2
increasing
Many problems can be approached more simply by choosing a coordinate system that
fits a given geometry. Instead of writing the position vector
r
as a function of Cartesian
coordinates (
x, y, z
),
r
is now written as function of three new coordinates:
r
(
q
1
, q
2
, q
3
).
The coordinate lines are swept out by varying one of the coordinates, keeping the other two
constant. We will deal only with the by far most important case of
orthogonal
coordinate
systems, in which the coordinate lines always intersect one another at right angles.
Evidently,
∂
r
∂q
i
is a vector which points in the direction of the ith coordinate line,
see the figure. If each of these vectors are normalized to unity, we obtain the local basis
system:
ˆ
q
i
=
1
h
i
∂
r
∂q
i
,
h
i
≡
vextendsingle
vextendsingle
vextendsingle
vextendsingle
∂
r
∂q
i
vextendsingle
vextendsingle
vextendsingle
vextendsingle
.
(42)
The quantities
h
i
(
q
1
, q
2
, q
3
) are called
scale factors
or
metric coefficients
. The fact that
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 Fall '11
 Eggers
 Geometry, Coordinate system, Polar coordinate system, Coordinate systems, ∂r, cylindrical polars

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