Curl of a 3D vector field on a surface
- Author:
- Art Busch
- We typically are interested in integrating the curl of F over a surface as a surface integral.
- The curl of F is a vector. Like any vector, you can write this vector as a sum of orthogonal vectors.
- So we start by writing (shown above in black) as a sum of a vector parallel to the unit normal vector n, and a vector perpendicular to .
- The component of the curl that is parallel to n describes a kind of microscopic circulation on the surface.
- The rate of this rotation is the magnitude of the blue vector shown above, which is equal to
- The component of the curl that is perpendicular to is parallel to the surface.
- This component represents a rotational motion across surface itself. We typically ignore this, although it could be interpreted as a force that warps, moves, or causes the surface to break.