Green, Stokes, Divergence: one theorem

The total on the boundary equals the derivative total inside

Fix a planar vector field $F(x,y) = \big(P(x,y),\,Q(x,y)\big)$ and a region $R$ with a closed boundary $\partial R$. The circulation is a boundary total: walk once around $\partial R$, and at each step add up the component of $F$ that points along your motion. Green's theorem says that total equals the integral over the interior of a single number, the curl, which measures the local spin of the field:

The two sides are computed in unrelated ways. The left is a sum of dot products along a curve; the right is an area integral of a derivative sampled point by point. Drag the control points to reshape the blob, change its size with the scale slider, and change how finely the boundary is sampled with the segments slider. The readout reports both totals and their difference, which falls toward zero as the sampling refines.

Both totals scale together

Scaling the region about its centre by a factor $k$ multiplies every length by $k$ and therefore every area by $k^2$. Both sides of Green's theorem contain exactly one area, so both totals must multiply by $k^2$ in lockstep. This is a cheap but sharp test: if the two sums were not both area integrals of the same kind, they would drift apart under scaling. The chart below plots the boundary total (solid) and the interior total (circles) against the scale; the slider moves the vertical marker and the readout reports the equality at that scale. Keep this $k^2$ factor in mind — it is the same area-scaling that becomes the Jacobian determinant in Part 5.

Outflow through the boundary equals divergence inside

Circulation asked how much the field points along the boundary. Ask instead how much it points through the boundary, and the same structure appears with a different derivative. Walk the boundary, project $F$ onto the outward normal $n$, and the total outflow equals the integral of the divergence over the interior:

This is the two-dimensional case of the divergence theorem, and it is Green's theorem with the components rotated a quarter turn. Switch the field between a source (positive divergence), a sink (negative), and a uniform drift (zero), and read both totals. The equality holds in every case, including when both sides are zero.

And a one-dimensional ancestor

Green, Stokes and the divergence theorem look like three results because they live on loops, surfaces and solids. Written side by side, they are one sentence: the total of a field on a boundary equals the total of its derivative over the interior. Pick a statement; the canvas draws its geometry and the readout fills in the relevant numeric totals. The field buttons change the field used by the planar statements.

In the FTC the boundary is the pair of endpoints $\{a,b\}$ and the interior is the interval between them; the derivative total accumulates the same change the boundary points report directly. Each theorem above adds one dimension of interior and one dimension of boundary, and keeps the same balance.

Why they are one statement is easiest to see by tiling the interior. Cut $R$ into many small cells. Each interior edge is shared by two cells, and the two cells traverse it in opposite directions, so their boundary contributions cancel: what survives the telescoping sum is only the outer boundary. On the boundary side we already summed the whole loop; on the interior side each cell contributes its own tiny boundary integral, and a cell's boundary total divided by its area is exactly the curl (for circulation) or the divergence (for flux). Sum the cells and the cancellation leaves Green's theorem. The same tiling argument, one dimension up, is Stokes and the divergence theorem. This is the FTC of Volume I, Part 9 in higher dimensions — there the interior was a single interval, the only interior point cancelled pairwise, and only the endpoints survived.

The boundary integral is signed

A loop has no intrinsic direction, so a boundary integral must be told which way to go. Reverse the walk and every step changes sign, so the circulation flips. The interior integral of the curl does not flip — the region is the same region. This is not a contradiction; orientation is part of the statement of the theorem, and a surface's normal is the matching choice that fixes it. The one-dimensional echo is $\int_a^b f' = -\int_b^a f'$, with $f(b)-f(a)$ flipping to $f(a)-f(b)$.

The engine behind change of variables and optimisation

Green's theorem is the reason the change-of-variables formula works: the determinant of the Jacobian is exactly the factor that makes the sub-totals on the two sides agree, which is the subject of Part 5: The Jacobian determinant. It is also why Part 6: Multiple integrals & coordinates can swap a double integral for a polar or spherical one: the theorems tell you what a coordinate change does to a total. In Nonlinear Optimization the same structure appears as the conservative-field test — a field is a gradient exactly when its curl is zero everywhere, so path-independence, potential surfaces and the guarantee that a loss landscape has no circulation are all one statement read off the boundary of a loop. The two-link arm's Jacobian, its singular configurations and the reachable set are all built from these boundary/interior identities; Part 5 makes that explicit.

• 3Blue1Brown, Stokes, divergence and Green — the same three costumes, with the tiling picture animated.
• MIT 18.02SC, Multivariable Calculus — the course that runs alongside this volume, theorems and all.
• Khan Academy, Multivariable calculus — extra practice on Green's, Stokes' and the divergence theorem.