97
A P P E N D I X A
Vector Calculus
For a scalar function C.x/ that takes a vector argument x D
x
x
; x
y
; x
z
, the gradient is com-
puted as
@C.x/
@x
D
2
6
6
6
6
6
6
6
6
6
6
4
@C.x/
@x
x
@C.x/
@x
y
@C.x/
@x
z
3
7
7
7
7
7
7
7
7
7
7
5
: (A.1)
e derivative of a vector function f.x/ D
f
x
.x/; f
y
.x/; f
z
.x/
with respect to the vector
x is given by
@f.x/
@x
D
2
6
6
6
6
6
6
6
6
6
6
4
@f
x
.x/
@x
x
@f
x
.x/
@x
y
@f
x
.x/
@x
z
@f
y
.x/
@x
x
@f
y
.x/
@x
y
@f
y
.x/
@x
z
@f
z
.x/
@x
x
@f
z
.x/
@x
y
@f
z
.x/
@x
z
3
7
7
7
7
7
7
7
7
7
7
5
: (A.2)
98 A. VECTOR CALCULUS
DERIVATIVE CHAIN RULES
@
@x
.
f .g.x//
/
D
@f .u/
@u
@g .x/
@x
@
@x
.
f .x/g.x/
/
D
@f .x/
@x
g.x/ C
@g . x/
@x
f .x/
@
@x
f .x/
g.x/
D
@f .x/
@x
g.x/
@g.x/
@x
f .x/
g.x/
2
:
(A.3)
VECTOR EQUALITIES
Ox D
x
jjxjj
jjxjj D
p
x x D
q
x
2
x
C x
2
y
C x
2
z
@jjxjj
@x
D
x
T
jjxjj
D Ox
T
@Ox
@x
D
@
x
jjxjj
@x
D
Ijjxjj xOx
T
jjxjj
2
D
I OxOx
T
jjxjj
:
(A.4)
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