Lecture VI
Calculus of Vector Functions
R
d
2
�
R
Recall that
d
�
denotes
the
first-order derivative
of
�
R
(
t
),
and that
dt
2
de-
dt
notes
the second-order
derivative
of
�
R
(
t
).
We
introduce
new notations for these
R
�
˙
R
¨
�
R
(
t
) =
a
1
(
t
)
ˆ
i
+
a
2
(
t
)
ˆ
+
a
3
(
t
)
ˆ
functions:
d
�
=
R
(
t
)
and
d
2
�
=
R
(
t
).
Let
�
j
k
.
dt
dt
2
Then
the following differentiation rules stand:
�
˙
j
+
˙
a
3
(
t
)
ˆ
1.
R
(
t
) =
a
˙
1
(
t
)
ˆ
i
+ ˙
a
2
(
t
)
ˆ
k
.
¨
�
a
1
(
t
)
ˆ
i
+ ¨
j
a
3
(
t
)
ˆ
2.
R
(
t
)
= ¨
a
2
(
t
)
ˆ
+ ¨
k
.
A
(
t
)
and
�
Let
�
B
(
t
)
be differentiable
vector functions,
and let
a
(
t
)
be
a
differentiable scalar
function.
The
following
differentiation rules stand:
dt
(
�
�
�
˙
�
˙
3.
d
A
(
t
) +
B
(
t
))
=
A
+
B
.
4.
d
A
(
t
))
=
a
˙(
t
)
�
�
˙
A
(
t
) +
a
(
t
)
A
(
t
).
dt
(
a
(
t
)
�
dt
(
�
�
�
˙
�
A
·
B
.
5.
d
A
(
t
)
·
B
(
t
))
=
A
·
B
+
�
�
˙
6.
d
A
(
t
)
×
�
�
˙
B
+
�
B
(
t
))
=
A
×
�
A
×
B
.
dt
(
�
�
˙
R
(
t
)
be
a
unit vector function,
�
Theorem
1 (Unit-Vector
Theorem)
Let
�
R
(
t
) =
ˆ
u
(
t
)
for all
t
.
Then:
u
is
perpendicular to
that
of
ˆ
1.
The
direction
of
ˆ
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- Two '04
- Duorg
- Math, Derivative, following differentiation rules
-
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