Curl measures local rotational tendency, while circulation measures the accumulated tangential effect around a closed curve. Stokes' theorem connects these local and global descriptions.

Let $S$ be an oriented surface with consistently oriented boundary $\partial S$. Then

$$\oint_{\partial S}\mathbf F\cdot d\mathbf r

\iint_S(\nabla\times\mathbf F)\cdot\mathbf n,dS.$$

Local rotation adds up to boundary circulation

Imagine dividing the surface into many tiny patches. Each patch has a small circulation around its boundary. Along edges shared by neighboring patches, the two boundary directions are opposite, so the internal contributions cancel.

After all cancellations, only the circulation around the outer boundary remains. The curl measures the infinitesimal rotational contribution whose surface integral produces that total.

Orientation

The orientation of the boundary and surface normal must agree by the right-hand rule. Reversing the surface normal also reverses the positive direction around the boundary, so both sides change sign together.

The surface is not unique

Different surfaces can share the same boundary curve. Under the hypotheses of the theorem, the flux of curl through any such compatible surface gives the same boundary circulation.

This freedom can make a difficult integral much simpler: choose either the line integral around the boundary or the curl flux through a convenient spanning surface.

Stokes' theorem is another local-to-global principle, turning distributed infinitesimal rotation across a surface into circulation along its edge.