What is the cross product integral?

Question

I understand the integral $\int \vec{u} \cdot \mathrm{d} \vec{v}$ is a line integral but what is the integral $\int \vec{u} \times \mathrm{d} \vec{v}$ and how does it work? For example, how would I evaluate an integral such as $\int \vec{v} \times \mathrm{d} \vec{v}$?

Answer 1

In 3D $\vec u \times d\vec v$ is a vector, each component of which is proportional to $\left| {d\vec v} \right|$: so you integrate each component.
In 2D it is just a vector in the $z$ direction, and you integrate its magnitude and assign the result to the$z$ component.
As for the meaning $\vec u \times \vec v = \left| {\vec u} \right|\vec \omega $ is the angular momentum (of a unitary mass particle, with position v. $\vec u$), $\vec u \times d\vec v$ is its change, and the integral therefore is the change from one point to another.