Green's, Stokes', and divergence theorems

For a bounded planar region with a simple piecewise smooth boundary oriented counterclockwise, and a C¹ field (P,Q) on an open neighborhood of the closure, Green's circulation theorem equates the boundary integral P dx+Q dy with the area integral of Qx−Py. A sufficient classroom setting is a region assembled from finitely many ordinary smooth pieces with no pathological boundary. Internal edges appear twice with opposite orientations and cancel, explaining why only the external boundary remains. With holes, include every boundary component and orient inner boundaries clockwise.

The field must be defined on the interior as required by the theorem. The punctured-plane field (−y/r²,x/r²) has zero scalar curl away from the origin but nonzero circulation around the unit circle. Applying Green's theorem to the full disk would be invalid because the field is undefined at its center. Applying it to an annulus is valid, and the clockwise inner-boundary contribution cancels the outer one. Removing a singular point is therefore not harmless merely because a point has zero area; its surrounding topology and field behavior matter.

For an oriented piecewise smooth surface S with piecewise smooth boundary and a C¹ vector field on a neighborhood of S, Stokes' theorem equates boundary circulation with surface flux of curl. A standard sufficient setting uses finitely many regular smooth patches with compatible orientations and well-behaved edges. The boundary direction follows the right-hand rule relative to the normal. Reversing the normal also reverses the boundary direction, so both sides change sign together. Choosing a normal and then independently choosing the opposite boundary convention breaks the identity's bookkeeping.

Stokes' theorem can replace a difficult surface by an easier one with the same oriented boundary, provided the field is regular on the replacement and each surface meets the hypotheses. It does not authorize stretching a surface through an excluded singularity. A closed surface has empty boundary, so its curl flux is zero under these assumptions. This is consistent with divergence of curl being zero for sufficiently smooth fields, although the two statements are at different levels: one integrates a first derivative over a surface, and the other differentiates again pointwise.

The divergence theorem equates total outward flux across the complete boundary of a bounded solid region with the volume integral of divergence. A standard sufficient region has a piecewise smooth boundary and can be decomposed into finitely many regular pieces; the field is C¹ on a neighborhood of its closure. Internal faces cancel as in the planar argument. If a calculation asks for flux through only a curved cap, adding a flat closing disk may make the volume integral easy, but the closing disk's oriented flux must then be subtracted.

A theorem-selection checklist begins with the quantity being integrated. Tangential circulation around a planar loop suggests Green; circulation around a spatial boundary suggests Stokes; outward flux through a closed surface suggests divergence. Next verify the full boundary, orientation, and field domain. Finally choose the easier side to compute. These results do not imply that every field with zero boundary integral is zero inside, because cancellations can occur. They relate totals of specific derivatives and boundary components, not a complete reconstruction of the field from one number.