UNIVERSITY OF CALIFORNIA BERKELEY
Structural Engineering,
Department of Civil Engineering
Mechanics and Materials
Fall 2002
Professor: S. Govindjee
A Quick Overview of Curvilinear Coordinates
1
Introduction
Curvilinear coordinate systems are general ways of locating points in Eu
clidean space using coordinate functions that are invertible functions of the
usual
x
i
Cartesian coordinates. Their utility arises in problems with obvious
geometric symmetries such as cylindrical or spherical symmetry. Thus our
main interest in these notes is to detail the important relations for strain and
stress in these two coordinate systems. Shown in Fig. 1 are the definitions of
the coordinate functions. Note that while the definition of the cylindrical co
ordinate system is rather standard, the definition of the spherical coordinate
system varies from book to book. Both systems to be studied are orthogonal.
The precise definitions used here are:
Cylindrical
x
1
=
r
cos(
θ
)
x
2
=
r
sin(
θ
)
x
3
=
z
(1)
r
=
x
2
1
+
x
2
2
θ
=
tan
−
1
(
x
2
/x
1
)
z
=
x
3
(2)
Spherical
x
1
=
r
sin(
ϕ
) cos(
θ
)
x
2
=
r
sin(
ϕ
) sin(
θ
)
x
3
=
r
cos(
ϕ
)
(3)
r
=
x
2
1
+
x
2
2
+
x
2
3
ϕ
=
cos
−
1
(
x
3
√
x
2
1
+
x
2
2
+
x
2
3
)
θ
=
tan
−
1
(
x
2
/x
1
)
(4)
2
Basis Vectors
For convenience in some of the equations to be given later we will denote our
curvilinear coordinates as
z
k
where (
z
1
, z
2
, z
3
) = (
r,
θ
, z
) in the cylindrical
case and (
z
1
, z
2
, z
3
) = (
r,
ϕ
,
θ
) in the spherical case. For the basis vectors we
will introduce for two types of basis vectors. The natural basis vectors and
1
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r
ϕ
z
θ
r
θ
x3
x2
x1
x1
x2
x3
Figure 1: Definition of the cylindrical and spherical coordinate systems.
the physical basis vectors. Both bases are orthogonal but the physical basis
has the additional property of orthonormality. The basic definitions are
g
k
=
∂
x
i
∂
z
k
e
i
(5)
e
k
=
g
k
g
k
.
(6)
To di
ff
erentiate between the physical basis vectors and the usual Cartesian
ones we typically write
e
r
,
e
θ
,
· · ·
etc. For the cylindrical coordinate system
one has:
g
1
→
cos(
θ
)
sin(
θ
)
0
g
2
→
−
r
sin(
θ
)
r
cos(
θ
)
0
g
3
→
0
0
1
(7)
and
e
r
=
g
1
e
θ
=
1
r
g
2
e
z
=
g
3
.
(8)
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 Spring '10
 EdgarKnobloch
 mechanics, Sin, Coordinate system, Polar coordinate system, Coordinate systems, basis vectors, Curvilinear coordinates

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