Stokes' theorem

In vector calculus, Stokes’ theorem (also Green’s theorem in circulation form) tells us that the total circulation of a vector field $\vec{F}$ along a closed contour $C$ is equal to the curl of the field integrated over a region enclosed by the contour (a flux).

$${\oint }_{C=\partial S}\vec{F}\cdot \vec{T}dS={\iint }_S\nabla \times \vec{F}\cdot \hat{n}dS$$

$\nabla \times \vec{F}$ is a flux density. The contour must be closed. If it isn’t, we can’t directly use the theorem as stated.

Geometrically we can think about it as stating how much the vector field rotates within the surface is the work done to move an object on the boundary contour.

The right-hand side (excluding the normal vector) is $\vec{J}$, a flux density.

Computations

To determine the direction of ${\hat{n}}_{S}dS$, we can use the right-hand curl rule, where the fingers are in the direction of $\vec{T}dS$ and the thumb is in the direction of ${\hat{n}}_{S}dS$.

Recall:

$${\oint }_C\vec{F}\cdot \vec{T}dS={\oint }_C\vec{F}\cdot {\vec{r}}^{\prime }dt$$

To compute ${\hat{n}}_{s}dS$ (from the flux page), we can take the cross product of the parameterisations:

$${\hat{n}}_{S}dS=S_u\times S_v$$

Or alternatively:

$${\hat{n}}_{S}dS=\frac{\nabla F}{\Vert \nabla F\Vert }dS$$

where $dS$ could be the differential element for a given parameterisation.

See also

• Cauchy-Goursat theorem, the complex analysis equivalent
• Divergence theorem, the dot product (divergence) equivalent
• Ampere’s law, an application of Stokes’ theorem to find the magnetic field