CONTINUUM MECHANICS for ENGINEERS

2.33 Show that for arbitrary tensors A and B, and arbitrary vectors a and b,
(a) (A ⋅ a) ⋅ (B ⋅ b) = a ⋅ (AT ⋅ B) ⋅ b
(b) b × a = (B – BT ) ⋅ a, if 2bi = εijkBkj (c) a ⋅ A ⋅ b = b ⋅ AT ⋅ a
2.34 Use Eqs 2.4-11 and 2.4-12 as necessary to prove the identities
(a) [Aa, Ab, Ac] = (det A) [a, b, c]
(b) AT ⋅ (Aa × Ab) = (det A) (a × b)
for arbitrary vectors a, b, c, and tensor A.
2.35 Let φ = φ (xi) and ψ = ψ(xi) be scalar functions of the coordinates.
1
2
Recall that in the indicial notation φ ,i represents φ and φ ,ii
represents 2 φ. Now apply the divergence theorem, Eq 2.8-1, to the
field φψ, i to obtain
( )
∫ ,i i
i ,i ,ii
S ∫ , +
V
φψ ndS= φ ψ φψ dV
Transcribe this result into symbolic notation as
∫ ∫ ∫
φ ψ φ ∂ψ
2
⋅ ˆndS dS = φ⋅ ψ + φ ψ dV
S ∂n
S V
which is known as Green’s first identity. Show also by the divergence
theorem that
and transcribe into symbolic notation as
which is known as Green’s second identity.
( )
( ,i ,i ) i
,ii − ,ii
S
V
∫ ∫
( )
φψ −ψφ
ndS= φψ ψφ dV
( )
φ ∂ψ
ψ
∂
∂φ
⎛
⎞
2 2
⎜ − ⎟ dS = φ ψ −ψ
φ dV
∫S⎝n ∂n⎠∫V