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Axes Transformation Rules and Tensor Notation Rules for Cartesian Tensors
Author: John M. Cimbala, Penn State University
Latest revision: 20 September 2007
Axes Transformation Rules
Let
unprimed
coordinates
x
1
,
x
2
, and
x
3
denote the
original
axes.
x
1
x
2
x
1
′
x
2
′
α
11
22
21
12
Let
primed
coordinates
x
1
′
,
x
2
′
, and
x
3
′
denote the
rotated
axes.
Let
C
ij
be the
cosine matrix
, defined as
cos
ij
ij
C
≡
, where
ij
is defined as the angle
between the
old
(
i
) and
new
(
j
) axes.
(
Note
: The figure to the right is in two dimensions
only, but can be extended easily to three
dimensions.)
Tensor
order
Number of
free indices
Number of
elements
Mathematical
name
Common
name
Transformation rule
0
0
1
zeroorder tensor
scalar
A
′
=
A
1
1
3
firstorder tensor
vector
A
′
m
=
C
im
A
i
2
2
9
secondorder tensor
“tensor”
A
′
mn
=
C
im
C
jn
A
ij
3
3
27
thirdorder tensor

A
′
mnp
=
C
im
C
jn
C
kp
A
ijk
4
4
81
fourthorder tensor

A
′
mnpq
=
C
im
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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 CIMBALA

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