Section 1.16
Solid Mechanics Part III
Kelly
135
1.16 Curvilinear Coordinates
Up until now, a rectangular Cartesian coordinate system has been used, and a set of
orthogonal unit base vectors
i
e
has been employed as the basis for representation of
vectors and tensors.
This basis is
independent of position
and provides a simple
formulation.
Two exceptions were in §1.6.10 and §1.14.4, where cylindrical and
spherical coordinate systems were used.
These differ from the Cartesian system in that
the cylindrical and spherical base vectors do depend on position.
However, although the
directions of these base vectors may change with position, they are always orthogonal to
each other.
In this section, arbitrary bases, with base vectors not necessarily orthogonal
nor of unit length, are considered.
It will be seen how these systems reduce to the special
cases of orthogonal (e.g. cylindrical and spherical systems) and Cartesian systems.
1.16.1 Curvilinear Coordinates
A Cartesian coordinate system is defined by the fixed base vectors
3
2
1
,
,
e
e
e
and the
coordinates
)
,
,
(
3
2
1
x
x
x
, and any point
p
in space is then determined by the position
vector
i
i
x
e
x
=
(see Fig. 1.16.1
1
).
This can be expressed in terms of
curvilinear
coordinates
)
,
,
(
3
2
1
Θ
Θ
Θ
by the transformation (and inverse transformation)
( )
)
,
,
(
,
,
3
2
1
3
2
1
Θ
Θ
Θ
=
Θ
=
Θ
i
i
i
i
x
x
x
x
x
(1.16.1)
In order to be able to solve for the
i
Θ
given the
i
x
, and to solve for the
i
x
given the
i
Θ
,
it is necessary and sufficient that the following determinants are non-zero – see Appendix
1.A.2 (the first here is termed the
Jacobian
J
of the transformation):
J
x
x
x
x
J
j
i
j
i
j
i
j
i
1
det
,
det
=
∂
Θ
∂
=
⎥
⎦
⎤
⎢
⎣
⎡
∂
Θ
∂
Θ
∂
∂
=
⎥
⎦
⎤
⎢
⎣
⎡
Θ
∂
∂
≡
,
(1.16.2)
the last equality following from (1.15.2, 1.10.18d).
If
1
Θ
is varied while holding
2
Θ
and
3
Θ
constant, a space curve is generated called a
1
Θ
coordinate curve
.
Similarly,
2
Θ
and
3
Θ
coordinate curves may be generated.
Three
coordinate surfaces
intersect in
pairs along the coordinate curves.
On each surface, one of the curvilinear coordinates is
constant.
Note
:
•
This Jacobian is the same as that used in changing the variable of integration in a volume
integral, §1.7; from Cartesian coordinates to curvilinear coordinates, one has
∫
∫
Θ
Θ
Θ
→
V
V
d
d
Jd
dx
dx
dx
3
2
1
3
2
1
1
superscripts are used here and in much of what follows for notational consistency (see later)