Question

In what way is the measurement of the net flow of a vector field
Field[x, y] = {m[x, y], n[x, y]}
along C related to the measurement of the net flow of its spin field
spinField[x, y] = {-n[x, y], m[x, y]}
across C?

Answer 1

Answer :

Field[x, y] = {m[x, y], n[x, y]}

The integral of field is used to measure the area of enclosed region by this vector field.

So in this case , integral

If this net integral is positive then field is in counterclockwise direction.

and if negative then field is in clockwise direction.

spin Field[x, y] = {-n[x, y], m[x, y]}

Integral of spin field :

  

If this integral is positive then the field is inside to outside and

if this integral is negative than the field is outside to inside as

Spin field is always perpendicular to the vector field.

So,

Trajectories of field vector and spin vector are at right angle.