Calculate the Curl of a Velocity vector field

1. Dec 3, 2017

themagiciant95

1. The problem statement, all variables and given/known data

The velocity of a solid object rotating about an axis is a field $$\bar{v} (x,y,z)$$
Show that $$\bar{\bigtriangledown }\times \bar{v} = 2\,\bar{\omega }$$, where $$\bar{\omega }$$ is the angular velocity.

2. Relevant equations

3. The attempt at a solution

I tried to use the Stoke's theorem using an infinitesimal element with trapezoidal shape, but i was stuck with calculations. Which is the best way to resolve the equation ? It would be fantastic if you could explain me the geometric intuition behind the problem

2. Dec 3, 2017

Delta²

Just use the relationship $\vec{v}(x,y,z)=\vec{\omega}\times(x\vec{i}+y\vec{j}+z\vec{k})$ and some identities of vector calculus about the curl operator.

The main identity you ll use is the first one found here : https://en.wikipedia.org/wiki/Curl_(mathematics)#Identities. Notice that you ll treat $\vec{\omega}$ as a constant vector in this identity so it will be $\nabla\cdot\vec{\omega}=0$ , $\vec{F}\cdot\nabla \vec{\omega}=0$ e.t.c

Last edited: Dec 3, 2017
3. Dec 3, 2017

themagiciant95

Thanks so much =)