Geology Reference
In-Depth Information
its transpose. This allows to write immediately
the inverse transformation:
2
4
Two tensors are often encountered in compu-
tation based on index notation. The first one is
the
Kronecker delta
, •
ij
, which simply represents
a component of the identity matrix:
3
5
D
2
4
3
5
2
4
3
5
r
™
¥
i
j
k
sin ™ cos¥ cos™ cos ¥
sin ¥
sin ™ sin
¥
cos™ sin
¥
cos
¥
cos™
0 if i
¤
j
1 otherwise
sin ™
0
•
ij
D
(A1.21)
(A1.20)
The second tensor is the
Levi
-
Civita tensor
,
"
ijk
, which is defined to be zero if any two of the
indices
ijk
are equal, and otherwise either
C
1or
1 according as
ijk
is an even or odd permutation
of 1,2,3. Thus, in terms of the Levi-Civita tensor,
the components of the cross product
C
D
A
B
can be written as follows:
A1.5 Index Notation
Index notation is a standard way to represent
the Cartesian components of vectors and tensors
in continuum mechanics. In this context,
x
1
x
,
x
2
y
,
x
3
z
, the base versors (
i
,
j
,
k
) are substi-
tuted by (
e
1
,
e
2
,
e
3
), and the generic component of
a vector
A
is indicated simply by
A
i
. This notation
combines with the
Einstein summation conven-
tion
, which requires that duplicated indices in
expressions are always summed. For example,
C
i
D
©
ijk
A
j
B
k
(A1.22)
Regarding the differential operators, in index
notation the components of the gradient of a
scalar field ¥ are simply:
@¥
@x
i
I
i
D
1;2;3
A
i
B
i
A
1
B
1
C
A
2
B
2
C
A
3
B
3
A
x
B
x
C
A
y
B
y
C
A
z
B
z
.
r
¥/
i
The divergence of a vector field
A
D
A
(
r
) will
be written as follows:
Some ambiguity is possible where powers of
an indexed quantity occur. Therefore, an expres-
sion such as:
@A
i
@x
i
r
A
X
A
i
Finally, the components of the curl will be
expressed as:
i
with the summation convection is always written
in the form:
.
r
A/
i
©
ijk
@A
j
@x
k
I
i
D
1;2;3
A
i
A
i
