CONTINUUM MECHANICS for ENGINEERS

2.29 Transcribe the left-hand side of the following equations into indicial
notation and verify that the indicated operations result in the expressions
on the right-hand side of the equations for the scalar φ, and
vectors u and v.
(a) div(φv) = φ div v + v ⋅ gradφ
(b) u × curl v + v × curl u = – (u ⋅ grad) v – (v ⋅ grad) u + grad (u ⋅ v)
(c) div (u × v) = v ⋅ curl u – u ⋅ curl v
(d) curl (u × v) = (v ⋅ grad)u – (u ⋅ grad) v + u div v – v div u
(e) curl (curl u) = grad (div u) – 2u 2.30 Let the volume V have a bounding surface S with an outward unit
normal ni. Let xi be the position vector to any point in the volume or
on its surface. Show that
(a)
(b)
(c) λw⋅ nˆdS = w ⋅grad
λdV
, where w = curl v and λ = λ(x).
∫S ∫
(d) eˆ × x, eˆ , nˆdS
2Vδ where and are coordinate base
vectors.
Hint: Write the box product
2.32 For the position vector x i having a magnitude x, show that x ,j = x j/x
and therefore,
(a)
(b) (x –1 ) ,ij =
(c)
∫S
x n dS = δ V
i j ij
( x⋅x)⋅ nˆdS
= 6V
∫S V
[ ] = ∫S
i j ij
[ eˆ ˆ , ˆ =
i × x,ej n]
( eˆ ˆ ˆ
i × x)⋅ ( ej × n)
and transcribe into indicial notation.
2.31 Use Stokes’ theorem to show that upon integrating around the space
curve C having a differential tangential vector dxi that for φ(x).
x
x
,ij
,ii
δ xx
= − 3
x x
2
=
x
ij i j
3xixj ij
− 5
x x
δ
3
φ ,idxi = 0 ∫ C
ê i
ê j