Let Ω ⊂ R
d be a bounded smooth domain and u, w ∈ C
1
c
(Ω) (see Problem 1.10
for the definition of compactly supported functions). Consider a vector field v = (v1, . . . , vd) : Ω → R
d
such that v ∈ C
1
(Ω) and ∇ · v = 0. Prove that
Z
Ω
(v(x) · ∇u(x)) w(x) dx = −
Z
Ω
(v(x) · ∇w(x)) u(x) dx.
