Hi, if the velocity of a partical is u = (u,v,w) in cartesian coordinates (x,y,z), and (ur, uθ, uz) in cylindrical polar (r,θ,z), could someone please explain why:

dv/dx - du/dy = (1/r) d/dr(r uθ) ?

I tried using the chain rule to get:
dv/dx - du/dy = (cosθ)dv/dr - (sinθ)du/dr ...........(1)

Then use
r = cosθ x + sinθ y
θ = -sinθ x + cosθ y

To write u =
urr + uθθ in terms of x and y,
then equate components with u =ux + vy to get:

u = urcos
θ - uθsinθ
v = ursin
θ + uθcosθ

Then subbing those values into (1) and simplifying, I end up with it equal to

I don't know where the extra r terms come from (or if they just cancel out somehow??) Could someone please point out if I'm doing something wrong?