# Calculate the Curl of a Velocity vector field

## Homework Statement

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.

## Homework Equations

3. The Attempt at a Solution [/B]

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

Delta2

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Delta2
Homework Helper
Gold Member
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:
themagiciant95
Thanks so much =)

Delta2