Stokes' theorem is a fundamental result in differential geometry and vector calculus that relates a surface integral over a surface \( S \) to a line integral over the boundary curve \( \partial S \) of that surface. It provides a powerful way to convert between the two types of integrals and is an essential tool in both mathematics and physics.
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Let be an oriented smooth surface with piecewise smooth boundary, and let be a continuously differentiable vector field on a neighborhood of . With the compatible boundary direction determined by the right-hand rule,For on the horizontal unit disc with upward normal, the left side is times its area, giving . The counterclockwise boundary has tangential field of magnitude one and circumference , giving the same line integral. Reversing the surface normal reverses the boundary direction and both signs. The theorem replaces a circulation calculation with a surface integral of the curl, whichever is easier.