two-dimensional flux

If is a vector field and a directed curve, then the two dimensional flux of across is the integral

where is a unit normal vector to the curve , pointing 90 degree clockwise from the tangent vector ()

Info

Note the orientation convention is arbitrary, and this clockwise convention is also opposite to the one customarily used in Physics, where and form a right-handed coordinate system for motion along . The choice of here is the most natural for applying Green’s theorem to flow problems.

Example: Use geometric method to compute the 2D flux across the unit circle

The vector field is parallel to normal vector and on unit circle. Thus, the flux is

Relation with line Integral

With line integral, if is a vector field and a directed curve, then

We can interpret this as work done by along .

Notice that we are integrating the tangential component of along the curve . In component notation, if , then the above reads

Analogously, we may integrate , the normal component of along the curve .