Section 1.15
Solid Mechanics Part III
Kelly
124
1.15 Tensor Calculus 2: Tensor Functions
1.15.1
Vector-valued functions of a vector
Consider a vector-valued function of a vector
)
(
),
(
j
i
i
b
a
a
=
=
b
a
a
This is a function of three independent variables
3
2
1
,
,
b
b
b
, and there are nine partial
derivatives
j
i
b
a
∂
∂
/
.
The partial derivative of the vector
a
with respect to
b
is defined to
be a second-order tensor with these partial derivatives as its components:
j
i
j
i
b
a
e
e
b
b
a
⊗
∂
∂
≡
∂
∂
)
(
(1.15.1)
It follows from this that
1
−
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
∂
∂
a
b
b
a
or
ij
j
m
m
i
a
b
b
a
δ
=
∂
∂
∂
∂
=
∂
∂
∂
∂
,
I
a
b
b
a
(1.15.2)
To show this, with
)
(
),
(
j
i
i
j
i
i
a
b
b
b
a
a
=
=
, note that the differential can be written as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
=
∂
∂
∂
∂
=
∂
∂
=
3
1
3
2
1
2
1
1
1
1
1
1
a
b
b
a
da
a
b
b
a
da
a
b
b
a
da
da
a
b
b
a
db
b
a
da
j
j
j
j
j
j
i
i
j
j
j
j
Since
3
2
1
,
,
da
da
da
are independent, one may set
0
3
2
=
=
da
da
, so that
1
1
1
=
∂
∂
∂
∂
a
b
b
a
j
j
Similarly, the terms inside the other brackets are zero and, in this way, one finds Eqn.
1.15.2.
1.15.2
Scalar-valued functions of a tensor
Consider a scalar valued function of a (second-order) tensor
j
i
ij
T
e
e
T
T
⊗
=
=
),
(
φ
φ
.
This is a function of nine independent variables,
)
(
ij
T
φ
φ
=
, so there are nine different
partial derivatives:

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