Proof for mean value of derivatives

[Himathfriends]

I want to prove that mean value of del(u) equals del(ū)
u stands for velocity
ū stands for mean value of velocity

 

[skeeter]

what does del(u) mean?

 

[Himathfriends]

what does del(u) mean?

Average (∇u) = ∇ū
ū= average u