Vector Forms of Green’s Theorem

The study of fluid motion often involves understanding how local rotational behavior relates to global circulation. In the context of a pond with pollutants, direct measurement of water movement along an irregular shoreline can be impractical. Green’s Theorem in vector form provides an alternative by relating the circulation around a closed boundary to properties of the flow within the enclosed region.

Measurements of water velocity at different points define a continuous vector field that represents the overall motion of the fluid. This field assigns a vector to each point in the region, capturing both magnitude and direction. The circulation along the boundary curve can be expressed as a line integral of this vector field:

\begin{equation*}\oint_{C} \textbf{F} \cdot d\textbf{r}\end{equation*}

The internal behavior of the fluid is described by the curl of the vector field, which measures local rotation. This quantity is defined as:

\begin{equation*}\text{curl } \textbf{F} = \nabla \times \textbf{F}\end{equation*}

The curl produces a vector perpendicular to the surface, indicating the axis and strength of rotation at each point. When combined with a unit vector normal to the surface, the relevant scalar component of rotation is obtained.

Green’s Theorem connects the boundary circulation to the accumulation of curl over the region:

\begin{equation*}\oint_{C} \textbf{F} \cdot d\textbf{r} = \iint_{D} (\text{curl } \textbf{F}) \cdot \textbf{k} \, dA\end{equation*}

If the curl has a constant component in the normal direction, the expression simplifies. For example, if this component equals a constant value, the total circulation becomes:

\begin{equation*}\oint_{C} \textbf{F} \cdot d\textbf{r}= 2(\text{Area of pond region D})\end{equation*}

This formulation shows that the total circulation around the boundary depends on the cumulative rotational behavior inside the region. For the pond example, it explains how pollutants rotate and spread, with the internal flow structure determining large-scale transport patterns.