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1 Preliminaries
V
(a · b − b · a)dV =
S
(a∇ · b + a × ∇ × b − b∇ · a − b × ∇ × a) · ndS
=
S
[n · a∇ · b − n · b∇ · a + (n × a) · ∇ × b − (n × b) · ∇ × a]dS
1.19 Given a second order tensor T i j is
T i j =
⎡
⎣
T 11 T 12 T 13
T 21 T 22 T 23
T 31 T 32 T 33
⎤
⎦ =
⎡
⎣
1 1 0
1 2 2
0 2 3
⎤
⎦
Find (1) T ii ; (2) T i j T i j ; (3) T i j T ji .
1.20 Suppose that the second order tensor is
T i j =
⎡
⎣
2 4 −2
0 3 2
4 2 5
⎤
⎦
Decompose T i j into the sum of a symmetric tensor and an antisymmetric tensor.
1 Preliminaries
V
(a · b − b · a)dV =
S
(a∇ · b + a × ∇ × b − b∇ · a − b × ∇ × a) · ndS
=
S
[n · a∇ · b − n · b∇ · a + (n × a) · ∇ × b − (n × b) · ∇ × a]dS
1.19 Given a second order tensor T i j is
T i j =
⎡
⎣
T 11 T 12 T 13
T 21 T 22 T 23
T 31 T 32 T 33
⎤
⎦ =
⎡
⎣
1 1 0
1 2 2
0 2 3
⎤
⎦
Find (1) T ii ; (2) T i j T i j ; (3) T i j T ji .
1.20 Suppose that the second order tensor is
T i j =
⎡
⎣
2 4 −2
0 3 2
4 2 5
⎤
⎦
Decompose T i j into the sum of a symmetric tensor and an antisymmetric tensor.