Biomedical Engineering Reference
In-Depth Information
=
(
M
xx
a
x
+
M
xy
a
y
+
M
xz
a
z
)
e
x
+
(
M
yx
a
x
+
M
yy
a
y
+
M
yz
a
z
)
e
y
+
(
M
zx
a
x
+
M
zy
a
y
+
M
zz
a
z
)
e
z
=
b
x
e
x
+
b
y
e
y
+
b
z
e
z
.
(7.33)
Using matrix notation we can write:
∼
=
M
∼
, in full:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
b
x
M
xx
M
xy
M
xz
a
x
b
y
=
M
yx
M
yy
M
yz
a
y
b
z
M
zx
M
zy
M
zz
a
z
⎡
⎤
M
xx
a
x
+
M
xy
a
y
+
M
xz
a
z
⎣
⎦
=
M
yx
a
x
+
M
yy
a
y
+
M
yz
a
z
.
(7.34)
M
zx
a
x
+
M
zy
a
y
+
M
zz
a
z
Along with the earlier specified matrix
M
the transposed matrix
M
T
is defined
according to (taking a mirror image along the principal diagonal):
⎡
⎣
⎤
⎦
M
xx
M
yx
M
zx
M
T
=
.
(7.35)
M
xy
M
yy
M
zy
M
xz
M
yz
M
zz
The tensor
M
T
is associated with the matrix
M
T
. Notice that
T
T
M
T
,
∼
=
M
∼
is equivalent to
∼
=
∼
b
=
M
·
a
b
=
a
·
M
T
.
is equivalent to
The inverse of the tensor
M
is denoted by
M
−
1
. By definition:
M
−
1
M
·
=
I
,
(7.36)
with
I
the unit tensor,
I
=
e
x
e
x
+
e
y
e
y
+
e
z
e
z
. The inverse of matrix
M
is denoted
with
M
−
1
. By definition:
M M
−
1
=
I
,
(7.37)
with
I
the unit matrix.
