The components of the linear strain tensor in spherical coordinates are calculated using three methods. The first is a change of basis method and does not require elaborate concepts. The second requires the concept of covariant derivative, and the third requires the concept of a Lie derivative. The calculation provides an opportunity to become familiar with both tensorial methods and associated computational tools in the context of an elementary physical situation. The components of the linear strain tensor in an arbitrary curvilinear orthonormal basis are interpreted geometrically as half the sum of two scalar products.

1.
B. F.
Schutz
,
Geometrical Methods of Mathematical Physics
(
Cambridge U. P.
, Cambridge,
1980
).
2.
Y.
Choquet-Bruhat
and
C.
DeWitt-Morette
with M. Dillard-Bleick,
Analysis, Manifolds and Physics
, revised ed. (
North-Holland
, New York,
1982
).
3.
T.
Frankel
,
The Geometry of Physics
(
Cambridge U. P.
, Cambridge,
1997
).
4.
M.
Nakahara
,
Geometry Topology and Physics
(
Institute of Physics
, Bristol,
1990
).
5.
C. W.
Misner
,
K. S.
Thorne
, and
J. A.
Wheeler
,
Gravitation
(
Freeman
, San Francisco,
1973
).
6.
N.
Straumann
,
General Relativity
(
Springer
, Berlin,
2004
).
7.
R. M.
Wald
, “
Resource Letter TMGR-1: Teaching the mathematics of general relativity
,”
Am. J. Phys.
74
(
6
),
471
477
(
2006
).
8.
L. D.
Landau
and
E. M.
Lifshitz
,
Theory of Elasticity
(
Pergamon
, New York,
1959
).
9.
L. D.
Landau
and
E. M.
Lifshitz
,
The Classical Theory of Fields
(
Pergamon
, Oxford,
1962
), 2nd ed.
10.
A. E. H.
Love
,
The Mathematical Theory of Elasticity
, 2nd ed. (
Cambridge U. P.
, Cambridge,
1906
), Secs. 20 and 22.
11.
Our conventions for horizontal indices positions are those of Ref. 5, pp.
204
210
.
12.
See also Ref. 5. p. 207, part (c) of Exercise 8.2.
13.
See also Ref. 5, p. 213, part (a) of Exercise 8.6.
14.
See, for example, Ref. 3, pp.
125
154
.
15.
Reference 2, pp.
177
178
, Ref. 3, p.
625
, and Ref. 6, p.
644
.
16.
A.
Harvey
,
E.
Schucking
, and
E. J.
Surowitz
, “
Redshifts and Killing vectors
,”
Am. J. Phys.
74
(
11
),
1017
1024
(
2006
).
17.
C.
Cattaneo
, “
Attempt at a relativistic elasticity theory
,”
C.R. Acad. Sci., Ser. A
272
,
1421
1424
(
1971
).
18.
Y.
Choquet-Bruhat
and
L.
Lamoureux-Brousse
, “
On the equations of relativistic elasticity
,”
C.R. Acad. Sci., Ser. A
276
,
1317
1320
(
1973
).
19.
V.
Vitelli
,
J. B.
Lucks
, and
D. R.
Nelson
, “
Crystallography on curved surfaces
,”
Proc. Natl. Acad. Sci. U.S.A.
103
(
33
),
12323
12328
(
2006
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.