What is the geometric significance of curl in vector fields?

[Char. Limit]

Homework Statement

OK, so I understand how to calculate this stuff. But I want to know the geometric significance of a line integral over a vector field, a double integral over a vector field, and of course curl.

Homework Equations

$$\int_C \vec{F} \cdot d\vec{r}$$

$$\int \int_C \vec{F} \cdot d\vec{r}$$

$$\nabla \times \vec{F}$$

 

[Rick88]

Taking the curl of a vector field allows you to calculate how much rotation that field possesses.

Example taken by Wikipedia:
Suppose the vector field describes the velocity field of a fluid flow (maybe a large tank of water or gas) and a small ball is located within the fluid or gas (the centre of the ball being fixed at a certain point). If the ball has a rough surface it will be made to rotate by the fluid flowing past it. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the value of the curl at this point

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