Stokes' Theorem

Stokes’ Theorem is an integral theorem that connects the flux integral of the curl of a vector field (on a surface) to the vector curve integral over the boundary curve of the surface. It is a generalisation of Green’s theorem, lifted to higher dimensions.

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Definition §

Stokes' Theorem ^theorem

Let $S$ be an oriented surface that is (potentially piecewise) smooth, oriented with the unit normal vector field $\hat{n}$. Then let $C=\partial S$ be the boundary curve of $S$ such that $S$ induces a positive orientation on $C$. Finally, let $\vec{F}$ be a $C^1$ vector field defined on a domain in ${\mathbb{R}}^3$ that $S$ is a subset of.

Stokes’ Theorem states:

$${\iint }_S(\nabla \times \vec{F})\cdot \text{d}\vec{S}={\int }_C\vec{F}\cdot \text{d}\vec{c}$$

• The left hand side is a flux integral over $S$
• The right hand side is a (vector field) curve integral over $C$

Surface Independency §

Stokes’ theorem is extremely useful due to the fact that it is independent of the choice of surface $S$! That is, for a given boundary curve $C$, we can choose any surface $S_i$, as long as it is bound by $C$ and induces a positive orientation on $C$.

Mathematically:

$${\int }_C\vec{F}\cdot \text{d}\vec{c}={\iint }_{S_1}(\nabla \times \vec{F})\cdot \text{d}\vec{S}={\iint }_{S_2}(\nabla \times \vec{F})\cdot \text{d}\vec{S}$$

This also means any two arbitrary surfaces $S_1$ and $S_2$ that share the same boundary curve and induce the same orientation on it have the same value of ${\iint }_S(\nabla \times \vec{F})\cdot \text{d}\vec{S}$.

Thus, it helps to use the simplest surface possible.

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Zero On Closed Surfaces §

Stokes’ Theorem also implies that any closed surface (surface with no boundary curve) $S$ always has the flux integral of $\nabla \times \vec{F}$ being zero.