Section 1.6
Solid Mechanics Part III
Kelly
30
1.6 Vector Calculus 1 - Differentiation
Calculus involving vectors is discussed in this section, rather intuitively at first and more
formally toward the end of this section.
1.6.1
The Ordinary Calculus
Consider a
scalar-valued function of a scalar
, for example the time-dependent density
of a material
)
(
t
ρ
=
.
The calculus of scalar valued functions of scalars is just the
ordinary calculus.
Some of the important concepts of the ordinary calculus are reviewed
in Appendix B to this Chapter, §1.B.2.
1.6.2
Vector-valued Functions of a scalar
Consider a
vector-valued function of a scalar
, for example the time-dependent
displacement of a particle
)
(
t
u
u
=
.
In this case, the derivative is defined in the usual
way,
t
t
t
t
dt
d
t
Δ
−
Δ
+
=
→
Δ
)
(
)
(
lim
0
u
u
u
,
which turns out to be simply the derivative of the coefficients
1
,
i
i
dt
du
dt
du
dt
du
dt
du
dt
d
e
e
e
e
u
≡
+
+
=
3
3
2
2
1
1
Partial derivatives can also be defined in the usual way.
For example, if
u
is a function of
the coordinates,
)
,
,
(
3
2
1
x
x
x
u
, then
1
3
2
1
3
2
1
1
0
1
)
,
,
(
)
,
,
(
lim
1
x
x
x
x
x
x
x
x
x
x
Δ
−
Δ
+
=
∂
∂
→
Δ
u
u
u
Differentials of vectors are also defined in the usual way, so that when
3
2
1
,
,
u
u
u
undergo
increments
3
3
2
2
1
1
,
,
u
du
u
du
u
du
Δ
=
Δ
=
Δ
=
, the differential of
u
is
3
3
2
2
1
1
e
e
e
u
du
du
du
d
+
+
=
and the differential and actual increment
u
Δ
approach one another as
0
,
,
3
2
1
→
Δ
Δ
Δ
u
u
u
.
1
assuming that the base vectors do not depend on
t