For an upward-oriented band of the circular elliptic paraboloid between radii

r_{0}

and

R

, the vector field

B = (- y^{3}, x^{3}, z^{3})

has vertical curl

3 (x^{2} + y^{2})

. Its surface integral is

3 π (R^{4} - r_{0}^{4}) /2

. Stokes theorem gives the same value from the counterclockwise outer circle minus the counterclockwise inner circle. Equivalently the induced inner boundary direction is clockwise. Forgetting the inner boundary or its reversed orientation changes the answer.

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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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Stokes theorem

Stokes flux through an annular paraboloid (I=3π(R4−r04​)/2)

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Stokes flux through an annular paraboloid ( $I = 3 π (R^{4} - r_{0}^{4}) /2$ )

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