Graphics Reference
In-Depth Information
which expresses how fast the function y is changing relative to the variable x for any value of x.
But what happens if the function is defined in terms of a vector equation such as
f
t
=
x t
i
+
y t
j
+
z t
k
?
(8.1)
For example, Eq. (8.1) could describe a curve in R
3
such as a helix, and
d
f
dt
would define the
tangential slope of the curve at a point determined by t.
To define the first differential of a vector-based equation, consider the curve shown in
Fig. 8.1, where two position vectors
f
t and
f
t
+
t identify two points P t and Qt
+
t,
respectively.
Y
P
(
t
)
δ
t
)
Q
(
t
+
f
(
t
)
δ
f
(
t
+
t
)
Z
X
f
=
f
(
t
)
Figure 8.1.
From Fig. 8.1 we can state
+
−
PQ
f
t
=
f
t
+
t
Therefore,
−
PQ
=
f
t
+
t
−
f
t
If t is small relative to the magnitude of
−
PQ,
−
PQ can be replaced by
f
:
f
=
f
t
+
t
−
f
t
(8.2)
1
t
, we obtain
Multiplying Eq. (8.2) by
f
t
=
f
t
+
t
−
f
t
(8.3)
t
Expanding Eq. (8.3) into its components gives
f
t
=
x t
+
t
−
x t
y t
+
t
−
y t
z t
+
t
−
z t
i
+
j
+
k
t
t
t
