Formal definition of curl in two dimensions
What we're building to
Formalizing fluid rotation
Letting the size of the region change
Average rotation per unit area
Key idea: Maybe if you take , which measures the fluid flow around a region, and divide it by the area of that region, it can give you some notion of the average rotation per unit area.
Defining two-dimensional curl
- is a two-dimensional vector field.
- is some specific point in the plane.
- represents some region around the point . For instance, a circle centered at .
- indicates the area of .
- indicates we are considering the limit as the area of goes to , meaning this region is shrinking around .
- is the boundary of , oriented counterclockwise.
- is the line integral around , written as instead of to emphasize that is a closed curve.
One more feature of conservative vector fields
- If a vector field represents fluid flow, you can quantify "fluid rotation in a region" by taking the line integral of that vector field along the border of that region.
- To go from the idea of fluid rotation in a region to fluid flow around a point (which is what curl measures), we introduce the idea of "average rotation per unit area" in a region. Then consider what this value approaches as your region shrinks around a point.
- In formulas, this gives us the definition of two-dimensional curl as follows:
- This relationship between curl and closed-loop line integrals implies that irrotational fields and conservative fields are one and the same.