Vector Forms of Green’s Theorem

A group of scientists is studying water movement in a pond to track pollutants discharged from a nearby factory.

Because the pond’s shoreline is irregular and difficult to access, measuring water movement along the entire boundary is impractical.

This is where the vector form of Green’s Theorem becomes useful. It relates the circulation around the shoreline curve, C, to the curl of the water-flow field inside pond D.

First, the water velocity is measured at different points inside the pond. This data defines a continuous vector field F, modeled by an equation, that represents the pond's overall water flow.

Next, the curl of this vector field, F, is calculated. This measures the local rotation of the flow and results in a vector pointing perpendicular to the water's surface.

For this field, the curl vector is then combined with the unit vector k by a dot product. This gives a constant value of 2.

As the double integral simply gives the area of the region D, the total circulation along C is simply 2 times the pond’s area.

This method reveals how pollutants are rotating and spreading within the pond.