Stokes theorem

2020 Mathematics Subject Classification: Primary: 58A [MSN][ZBL]

The term refers, in the modern literature, to the following theorem.

Theorem Let $M$ be a compact orientable differentiable manifold with boundary (denoted by $\partial M$) and let $k$ be the dimension of $M$. If $\omega$ is a differential $k-1$-form, then $$\int_M d \omega = \int_{\partial M} \omega$$ (see Integration on manifolds for the definition of integral of a form on a differentiable manifold).

The theorem can be considered as a generalization of the Fundamental theorem of calculus. The classical Gauss-Green theorem and the Stokes formula can be recovered as particular cases. The latter is also often called Stokes theorem and it is stated as follows.

Theorem Let $\Sigma\subset \mathbb R^3$ be a compact regular $2$-dimensional surface $\Sigma$ that bounds the $C^1$ curve $\gamma$ and let $v$ be a $C^1$ vector field. Then $$\begin{equation}\label{e:Stokes_2} \int_\Sigma (\nabla \times v) \cdot \nu = \int_\gamma \tau \cdot v\, , \end{equation}$$ where

• $\nu$ is a continuous unit vector field normal to the surface $\Sigma$
• $\tau$ is a continuous unit vector field tangent to the curve $\gamma$, compatible with $\nu$
• $\nabla \times v$ is the curl of the vector field $v$.

The right hand side of $\eqref{e:Stokes_2}$ is called the flow of $v$ through $\Sigma$, whereas the left hand side is called the circulation of $v$ along $\gamma$. The theorem can be easily generalized to surfaces whose boundary consists of finitely many curves: the right hand side of $\eqref{e:Stokes_2}$ is then replaced by the sum of the integrals over the corresponding curves.

References

[Ap]	T.M. Apostol, "Calculus" , I , Blaisdell (1967) MR0214705 Zbl 0148.28201
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin (1965) MR0209411 Zbl 0141.05403