Global Positioning System Reference
In-Depth Information
x
0
dx
x
0
dx
2
1
2
3
4
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∂
2
y
∂x
2
∂y
∂x
1
2!
=
+
+
+···
y
f (x
o
)
(A.113)
The linear portion is given by the first two terms
x
0
∂y
∂x
y
¯
=
f (x
o
)
+
dx
(A.114)
Th
e derivative is evaluated at the point of expansion
x
0
. At the point of expansion,
th
e linearized and the nonlinear functions are tangent. They separate by
=
−¯
ε
y
y
(A.115)
as
x
departs from the expansion point
x
0
. The linear form (A.114) is a sufficiently
ac
curate approximation of the nonlinear relation (A.112) only in the vicinity of the
po
int of expansion.
The expansion of a two-variable function
[35
Lin
—
0.7
——
No
*PgE
z
=
f(x,y)
(A.116)
is
x
0
,y
0
x
0
,y
0
∂z
∂x
∂z
∂y
z
=
f (x
0
,y
0
)
+
dx
+
dy
+···
(A.117)
[35
Th
e point of expansion is
P(x
=
x
0
,y
=
y
0
)
. The linearized form
x
0
,y
0
x
0
,y
0
∂z
∂x
∂z
∂y
¯
=
+
+
z
f(x
0
,y
0
)
dx
dy
(A.118)
re
presents the tangent plane on the surface (A.116) at the expansion point. A gener-
ali
zation for the expansion of multivariable functions is readily seen. If
n
functions
ar
e related to
u
variables as in
f
1
(
x
)
f
2
(
x
)
.
f
n
(
x
)
f
1
(x
1
,x
2
,
···
,x
u
)
f
2
(x
1
,x
2
,
···
,x
u
)
y
=
f
(
x
)
=
=
(A.119)
.
f
n
(x
1
,x
2
,
···
,x
u
)
the linearized form is
x
0
∂
f
∂
x
y
=
f
(
x
0
)
+
d
x
(A.120)
where
