# classical Stokes’ theorem

Let $M$ be a compact  , oriented two-dimensional differentiable manifold (surface) with boundary in $\mathbb{R}^{3}$, and $\mathbf{F}$ be a $C^{2}$-smooth vector field defined on an open set in $\mathbb{R}^{3}$ containing $M$. Then

 $\iint_{M}(\nabla\times\mathbf{F})\cdot d\mathbf{A}=\int_{\partial M}\mathbf{F}% \cdot d\mathbf{s}\,.$

Here, the boundary of $M$, $\partial M$ (which is a curve) is given the induced orientation from $M$. The symbol $\nabla\times\mathbf{F}$ denotes the curl of $\mathbf{F}$. The symbol $d\mathbf{s}$ denotes the line element $ds$ with a direction parallel    to the unit tangent vector $\mathbf{t}$ to $\partial M$, while $d\mathbf{A}$ denotes the area element  $dA$ of the surface $M$ with a direction parallel to the unit outward normal $\mathbf{n}$ to $M$. In precise terms:

 $d\mathbf{A}=\mathbf{n}\,dA\,,\quad d\mathbf{s}=\mathbf{t}\,ds\,.$

The classical Stokes’ theorem reduces to Green’s theorem on the plane if the surface $M$ is taken to lie in the xy-plane.

The classical Stokes’ theorem, and the other “Stokes’ type” theorems are special cases of the general Stokes’ theorem involving differential forms  . In fact, in the proof we present below, we appeal to the general Stokes’ theorem.

(To be written.)

## Proof using differential forms

The proof becomes a triviality once we express $(\nabla\times\mathbf{F})\cdot d\mathbf{A}$ and $\mathbf{F}\cdot d\mathbf{s}$ in terms of differential forms.

###### Proof.

Define the differential forms $\eta$ and $\omega$ by

 $\displaystyle\eta_{p}(\mathbf{u},\mathbf{v})$ $\displaystyle=\langle\operatorname{curl}\mathbf{F}(p),\mathbf{u}\times\mathbf{% v}\rangle\,,$ $\displaystyle\omega_{p}(\mathbf{v})$ $\displaystyle=\langle\mathbf{F}(p),\mathbf{v}\rangle\,.$

for points $p\in\mathbb{R}^{3}$, and tangent vectors $\mathbf{u},\mathbf{v}\in\mathbb{R}^{3}$. The symbol $\langle,\rangle$ denotes the dot product  in $\mathbb{R}^{3}$. Clearly, the functions $\eta_{p}$ and $\omega_{p}$ are linear and alternating  in $\mathbf{u}$ and $\mathbf{v}$.

We claim

 $\displaystyle\eta$ $\displaystyle=\nabla\times\mathbf{F}\cdot d\mathbf{A}$ on $M$. (1) $\displaystyle\omega$ $\displaystyle=\mathbf{F}\cdot d\mathbf{s}$ on $\partial M$. (2)

To prove (1), it suffices to check it holds true when we evaluate the left- and right-hand sides on an orthonormal basis $\mathbf{u},\mathbf{v}$ for the tangent space of $M$ corresponding to the orientation of $M$, given by the unit outward normal $\mathbf{n}$. We calculate

 $\displaystyle\nabla\times\mathbf{F}\cdot d\mathbf{A}(\mathbf{u},\mathbf{v})$ $\displaystyle=\langle\operatorname{curl}\mathbf{F},\mathbf{n}\rangle\,dA(% \mathbf{u},\mathbf{v})$ definition of $d\mathbf{A}=\mathbf{n}\,dA$ $\displaystyle=\langle\operatorname{curl}\mathbf{F},\mathbf{n}\rangle$ definition of volume form $dA$ $\displaystyle=\langle\operatorname{curl}\mathbf{F},\mathbf{u}\times\mathbf{v}\rangle$ since $\mathbf{u}\times\mathbf{v}=\mathbf{n}$ $\displaystyle=\eta(\mathbf{u},\mathbf{v})\,.$

For equation (2), similarly, we only have to check that it holds when both sides are evaluated at $\mathbf{v}=\mathbf{t}$, the unit tangent vector of $\partial M$ with the induced orientation of $\partial M$. We calculate again,

 $\displaystyle\mathbf{F}\cdot d\mathbf{s}(\mathbf{t})$ $\displaystyle=\langle\mathbf{F},\mathbf{t}\rangle\,ds(\mathbf{t})$ definition of $d\mathbf{s}=\mathbf{t}\,ds$ $\displaystyle=\langle\mathbf{F},\mathbf{t}\rangle$ definition of volume form $ds$ $\displaystyle=\omega(\mathbf{t})\,.$

Furthermore, $d\omega$ = $\eta$. (This can be checked by a calculation in Cartesian coordinates  , but in fact this equation is one of the coordinate-free definitions of the curl.)

The classical Stokes’ Theorem now follows from the general Stokes’ Theorem,

 $\int_{M}\eta=\int_{M}d\omega=\int_{\partial M}\omega\,.\qed$

## References

• 1 Michael Spivak. Calculus on Manifolds. Perseus Books, 1998.
Title classical Stokes’ theorem ClassicalStokesTheorem 2013-03-22 15:27:52 2013-03-22 15:27:52 stevecheng (10074) stevecheng (10074) 6 stevecheng (10074) Theorem msc 26B20 GeneralStokesTheorem GaussGreenTheorem GreensTheorem