MATH 603 Homework Set V
Due Monday, 17 November 2014
1. Let v denote a smooth vector field defined on a bounded, open subset R of R3 . Prove
that
Z
Z
∇v dV =
v ⊗ n dA
R
∂R
where n denotes the unit normal vector field to the boundary ∂R of R.
2. Let u denote a smooth vector field on R3 . Derive the following identities.
(a) Div{(∇u)u} = ∇u · ∇uT + u · (∇Div u)
(b) ∇u · ∇uT = Div{(∇u)u − (Div u)u} + (Div u)2 .
3. Let E[·] denote a constant, fourth-order elasticity tensor. Consider the Navier system
of partial differential equations
ü(t, x) = Div E[∇u(t, x)].
(1)
A progressive wave is a vector field of the form:
u(t, x) := φ(x · m − ct)a
(2)
where c is a constant scalar wave speed, and a and m are the constant vector wave
amplitude and wave direction, respectively.
(a) Show that (2) satisfies (1) if and only if
A(m)a = c2 a
(3)
where A(m) denotes the Acoustic Tensor defined by the relation:
A(m)a := E[a ⊗ m]m.
(b) Show that the Acoustic Tensor satisfies:
i.
A(m)T = A(m)
ii.
a · A(m)a = a ⊗ m · E[a ⊗ m].
1
(4)