Intro to Continuum Mechanics: Explaining Rotation

[stanley.st]

Hello!

I read somewhere about intro to continuum mechanics. There was a vector $$\vec{\mu}$$ and displacement vector $$\delta\vec{\mu}$$. As vector $$\vec{\mu}$$ move, it will get new position

$$\vec{\mu}'=\vec{\mu}+\delta\vec{\mu}$$

$$\vec{\mu}'=\vec{\mu}+\frac{\partial\vec{\mu}}{\partial x_i}\delta x_i=\vec{\mu}+\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}-\frac{1}{2}\frac{\partial\vec{\mu}}{\partial x_j}\right)\delta x_i=\vec{\mu}+\left[\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}+\frac{\partial\vec{\mu}}{\partial x_j}\right)+\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)\right]\delta x_i$$

Last component

$$\frac{1}{2}\left(\frac{\partial\vec{\mu}}{\partial x_i}-\frac{\partial\vec{\mu}}{\partial x_j}\right)$$

represent rotation. Can you explain me that? I don't understand this rotation.

 

[Studiot]

Look up the Curl of a vector in 2 dimensions

http://en.wikipedia.org/wiki/Curl_(mathematics )