The Green’s theorem converts a line integral over a closed curve in 2D to a double integral of the curl in the interior area.
Circulation form of the Green's theorem
Given a simple closed curve
(simple means it never intersects itself) that encloses a region , is counterclockwise oriented, and is piecewise smooth, and given a vector field defined & differentiable in , then
Example
Let
be a circle of radius 1 centered at counterclockwise, compute
Note if we want to compute this directly, we can parameterize based on the circle
and . Using Green’s theorem, we will instead compute double integral
Then,
We know that the
coordinate of the center of mass is , thus If we don’t have this intuition, we can also evaluate this double integral manually.
Flux Form of Green’s Theorem
The Green’s theorem relates a line integral to double integral. Similarly, the flux form of the Green’s theorem relates the two-dimensional flux
Flux form of the Green's theorem
The flux form can be derived from the circulation form
Let
and , then
Note on the Applicability of Green’s Theorem
It is important to note that Green’s Theorem won’t apply if
Example:
.
for everywhere where is defined, but is not defined at the origin. If we have a curve not enclosing the origin, then the Green’s theorem applies and
. However, if we have a curve enclosing the origin, then we can’t apply the Green’s theorem.
Applications
- conservative vector fields and curl
- The planimeter was a device to determine the area of an arbitrary two-dimensional shape. It does that by computing the line integral
, which by Green’s theorem is
Related
- Stoke’s theorem - can be seen as a 3D generalization of the Green’s theorem