48
2 Vector and Tensor Analysis
Problems
2.1 Simplify the following expressions as much as possible:
(a) δ ij δ ij
(b) δ mn δ ik δ jk δ ji δ nm
(c) ij k δ ik
(d) ij k ij m δ km
(e)
∂
∂x i
x i x j
2.2 Let a = a i e i , A = A ij e i e j , and B = B ij k e i e j e k , where the e i are orthogonal
unit vectors. Express a · A · B in dyadic notation.
2.3 Show that T · a = a · T
T , where a is a vector and T is a second-order tensor.
2.4 Let
A =
1 −4
2 3
,
b =
−1
2
,
in which the components are defined in the xy-plane. Compute A · b and
b · A using (a) matrix algebra and (b) dyadic analysis. Hint: Both ways
should give the same results.
2.5 Using dyadic analysis, show that A : (B · C) = B : (A · C
T ), where A, B,
and C are second-order tensors.
2.6 The fourth-order identity tensor is defined by
˜
I = e i e j e i e j .
Show that T = T : ˜
I, where T is a second-order tensor.
2.7 A tensor is defined by
T = 2e 1 e 1 + e 2 e 2 − 4e 3 e 3 + 3e 2 e 3 + 3e 3 e 2 .
Compute the eigenvalues and eigenvectors of T and show that the eigenvectors are mutually orthogonal.
2.8 In two dimensions, show that Eq. (2.50) is consistent with (2.48) 2 .
2.9 Show that the dot product of two orthogonal tensors is an orthogonal tensor.
2.10 If A is a second-order tensor, show that
∂
∂A
tr(A · A) = 2A
T .
Hint: Use dyadic analysis in Cartesian coordinates.
2 Vector and Tensor Analysis
Problems
2.1 Simplify the following expressions as much as possible:
(a) δ ij δ ij
(b) δ mn δ ik δ jk δ ji δ nm
(c) ij k δ ik
(d) ij k ij m δ km
(e)
∂
∂x i
x i x j
2.2 Let a = a i e i , A = A ij e i e j , and B = B ij k e i e j e k , where the e i are orthogonal
unit vectors. Express a · A · B in dyadic notation.
2.3 Show that T · a = a · T
T , where a is a vector and T is a second-order tensor.
2.4 Let
A =
1 −4
2 3
,
b =
−1
2
,
in which the components are defined in the xy-plane. Compute A · b and
b · A using (a) matrix algebra and (b) dyadic analysis. Hint: Both ways
should give the same results.
2.5 Using dyadic analysis, show that A : (B · C) = B : (A · C
T ), where A, B,
and C are second-order tensors.
2.6 The fourth-order identity tensor is defined by
˜
I = e i e j e i e j .
Show that T = T : ˜
I, where T is a second-order tensor.
2.7 A tensor is defined by
T = 2e 1 e 1 + e 2 e 2 − 4e 3 e 3 + 3e 2 e 3 + 3e 3 e 2 .
Compute the eigenvalues and eigenvectors of T and show that the eigenvectors are mutually orthogonal.
2.8 In two dimensions, show that Eq. (2.50) is consistent with (2.48) 2 .
2.9 Show that the dot product of two orthogonal tensors is an orthogonal tensor.
2.10 If A is a second-order tensor, show that
∂
∂A
tr(A · A) = 2A
T .
Hint: Use dyadic analysis in Cartesian coordinates.