Biomedical Engineering Reference
In-Depth Information
In ordinary vector differentiation, we meet the gradient of a scalar field
f
, defined in
Cartesian coordinates as
∂
f
∂
z
e
z
where
e
x
,
e
y
and
e
z
areCartesian unit vectors.When dealingwith functions ofmany variables
it proves useful to make a generalization and write the gradient of (for example)
U
as
∂
f
∂
x
e
x
+
∂
f
∂
y
e
y
+
grad
f
=
⎛
⎞
∂
U
∂
q
1
∂
U
∂
q
2
.
∂
U
∂
q
p
⎝
⎠
grad
U
=
(4.6)
so grad
U
is a column matrix that stores all the partial derivatives. This 'vector' will occur
many times through the text, and I am going to give it the symbol
g
(for gradient).
The second derivatives can be collected into a symmetric
p
p
matrix that is called the
Hessian
of
U
and I will give this the symbol
H
. In the case where
p
×
=
3, we have
⎛
⎝
⎞
⎠
∂
2
U
∂
q
1
∂
2
U
∂
q
1
∂
q
2
∂
2
U
∂
q
1
∂
q
3
∂
2
U
∂
q
2
∂
q
1
∂
2
U
∂
q
2
∂
2
U
∂
q
2
∂
q
3
H
=
(4.7)
∂
2
U
∂
q
3
∂
q
1
∂
2
U
∂
q
3
∂
q
2
∂
2
U
∂
q
3
The Taylor expansion then becomes
1
2
ξ
T
H
ξ
ξ
T
g
U
(
q
)
−
U
(
q
e
)
=
+
+···
(4.8)
Both the gradient and the Hessian have to be evaluated at the point
q
e
, and so you will
sometimes see the equation written with an 'e' subscript:
1
2
ξ
T
H
e
ξ
ξ
T
g
e
+
U
(
q
)
−
U
(
q
e
)
=
+···
The superscript T as in ξ
T
indicates the transpose of a matrix; the transpose of a column
matrix is a row matrix. The Hessian is often referred to as the force constant matrix.
Finally, if I denote the 3
N
Cartesian coordinates
X
1
,
X
2
, ...,
X
3
N
, we usually write the
transformation from Cartesian coordinates to internal coordinates as
=
q
BX
(4.9)
where the rectangular matrix
B
is called the Wilson B-matrix. The
B
matrix has
p
rows and
3
N
columns.
