Question

Let Ω ⊂ R

d be a bounded mooth domain and u, w ∈ C
1
c
(Ω) (ee Problem 1. 10
for the definition of compactly upported function). Conider a vector field v = (v1,. . . , vd) : Ω → R
d

uch 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